MSc Physics and Astronomy

Theoretical Physics

Master Thesis

Geometric quantum computing withsupersymmetric lattice models

by

Diko Hemminga10680268

August 25, 2018

60 EC

September 2017 - July 2018

Supervisor:prof.dr. Kareljan Schoutens

Examiner:dr. Marcel Vonk

Institute for Theoretical Physics AmsterdamInstitute of PhysicsFaculty of Science

University of Amsterdam

AbstractIn this thesis we investigate quantum control of qubits defined by a super-symmetric lattice model. Periodic chains with length L = 3n produce twodegenerate ground states, which can be interpreted as the computationalstates of a qubit. After the introduction of a staggering parameter in thelattice model we can choose a closed adiabatic path in this parameter spacewhich results in a non-Abelian Berry phase. This unitary can be interpretedas a geometric quantum gate operation.

The triangle (L = 3) and hexagon (L = 6) lattice configurations both definea single qubit and allow us to investigate the construction of one-qubit quan-tum gates. We show that we can define the phase shift gate and rotationgate by a geometric procedure, which are sufficient for the construction of ageneral one-qubit gate.

In the bow tie lattice, defined as two connected triangles (qubits), we in-vestigate the construction of a non-trivial two-qubit quantum gate. Weexplore its non-trivial nature with the entanglement entropy as measure.We find that the procedure based on the non-Abelian Berry phase can pro-duce near-maximal entanglement entropy. While the non-trivial nature ofthe constructed two-qubit quantum gate is shown, the proof of equivalenceto known non-trivial (universal) two-qubit quantum gates is not given inthis work.

Title: Geometric quantum computing with supersymmetric lattice modelsAuthor: Diko Hemminga, diko.hemminga@casema.nl, 10680268Supervisor: prof.dr. Kareljan SchoutensExaminer: dr. Marcel Vonk

Project conducted between September 2017 and July 2018Date final version: August 25, 2018

Institute for Theoretical Physics AmsterdamUniversity of AmsterdamScience Park 904, 1098 XH Amsterdam

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AcknowledgementsI would like to express my gratitude to Kareljan Schoutens, for being willingto supervise my project, supplying me with suggestions and keeping me onthe right track during the process. I would like to thank Marcel Vonk, foragreeing to be my second supervisor and helping with the mathematicalpicture. Many thanks are due to Herma, my mum, for the endless supportand all the encouragements. Naturally also thanks are due to Haijo, mydad, for planting the seeds for my interest in physics when I was young andbeing there along the way. Lastly, I would like to thank Patrick, for beingsomeone to talk to during lunch, and the other inhabitants of the MasterStudents room, for providing a nice place to work on this project.

Nederlandse populair wetenschappelijke samenvattingIn dit project werk ik een nieuw idee uit voor een quantumcomputer. Debasiselementen van een quantumcomputer zijn qubits, die de wetten van dequantummechanica volgen. Dit in tegenstelling tot de bits in een klassiekecomputer, zoals je laptop of smartphone. De quantummechanica voegtnieuwe mogelijkheden toe, zoals superposities en verstrengeling van qubits.Daarvan gebruikmakend zal een quantumcomputer bepaalde rekentaken veelsneller kunnen uitvoeren. Het basiselement, de qubit, is een natuurkundigsysteem. In mijn project gebruik ik een kleine keten van drie roosterpunten(een driehoekje). Er zijn regels waar een deeltje zich op het rooster magbevinden. Deze keuze wordt ook wel een supersymmetrisch roostermodelgenoemd. Het blijkt dat dit model twee mogelijke grondtoestanden voort-brengt, die we kunnen interpreteren als de toestanden van een qubit.

Om berekeningen te kunnen uitvoeren, moeten we de toestand van mijn‘rooster-qubit’ gecontroleerd kunnen veranderen. Daarvoor introduceer ikeen parameter in ons model, waarmee de grondtoestanden kunnen wordenaangepast. Aangezien het veranderen van de parameter kan worden gezienals het volgen van een gesloten pad in de parameterruimte, gebruiken weconcepten uit de meetkunde om het resultaat te beschrijven.

Met goede keuzes van de parameter kunnen we twee operaties (of quantumlogische poorten) op een enkele qubit definieren. Dit is zelfs genoeg om elkemogelijke operatie uit te voeren door de twee operaties te combineren. Devolgende stap is het bekijken van een systeem van twee qubits. We koppelentwee driehoekjes tot een nieuw rooster dat lijkt op een vlinderdas. In ditsysteem onderzoeken we of we de twee qubits kunnen verstrengelen, watniet mogelijk is met operaties op maar een van de qubits tegelijk. Opnieuwgebruiken we een pad in de parameterruimte van het roostermodel. Wevinden inderdaad een verstrengelende operatie op twee qubits, maar kunnendeze moeilijk vergelijken met operaties die bekend zijn uit de literatuur.

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Contents

List of Figures ix

1 Introduction and summary 1

2 Supersymmetric lattice models 52.1 Features of Mk models . . . . . . . . . . . . . . . . . . . . . . 82.2 Staggering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Staggering in the triangle and hexagon lattices . . . . 102.3 Features of different chain lengths . . . . . . . . . . . . . . . . 112.4 Lattice symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Supersymmetry in particle physics . . . . . . . . . . . . . . . 15

3 Qubits and quantum computing 173.1 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Noisy Intermediate Scale Quantum . . . . . . . . . . . . . . . 193.3 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Quantum error correction . . . . . . . . . . . . . . . . 223.3.2 Our approach to fight decoherence . . . . . . . . . . . 23

3.4 Quantum control . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . . 263.6 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.7 Gate universality . . . . . . . . . . . . . . . . . . . . . . . . . 313.8 Many-body strategies for multiqubit gates . . . . . . . . . . . 34

4 Geometric phases and fibre bundles 354.1 Quantum mechanical phase . . . . . . . . . . . . . . . . . . . 354.2 Adiabaticity and the quantum adiabatic theorem . . . . . . . 36

4.2.1 Adiabatic quantum computing . . . . . . . . . . . . . 374.3 Abelian Berry phase . . . . . . . . . . . . . . . . . . . . . . . 374.4 Non-Abelian Berry phase . . . . . . . . . . . . . . . . . . . . 414.5 Mathematical background of the non-Abelian Berry phase . . 43

4.5.1 Gauge structure . . . . . . . . . . . . . . . . . . . . . 434.5.2 Projective spaces . . . . . . . . . . . . . . . . . . . . . 484.5.3 Fibre bundle structure . . . . . . . . . . . . . . . . . . 50

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4.6 Mathematical interpretation of geometric quantum computa-tion on the supersymmetric lattice . . . . . . . . . . . . . . . 54

4.7 Literature review of geometric quantum computation . . . . . 59

4.7.1 Zanardi and Rasetti (1999) . . . . . . . . . . . . . . . 59

4.7.2 Pachos et al. (1999) . . . . . . . . . . . . . . . . . . . 60

4.7.3 Ekert et al. (2000) . . . . . . . . . . . . . . . . . . . . 61

4.7.4 Jones et al. (2000) . . . . . . . . . . . . . . . . . . . . 61

4.7.5 Duan et al. (2001) . . . . . . . . . . . . . . . . . . . . 62

4.7.6 Solinas et al. (2003) . . . . . . . . . . . . . . . . . . . 64

4.7.7 Ruseckas et al. (2005) . . . . . . . . . . . . . . . . . . 65

4.7.8 Sjoqvist et al. (2016) . . . . . . . . . . . . . . . . . . . 66

4.7.9 Leroux et al. (2018) . . . . . . . . . . . . . . . . . . . 68

5 Triangle and hexagon lattices 71

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 The triangle lattice . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.1 Staggering . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.2 Ground states on the triangle lattice . . . . . . . . . . 74

5.2.3 Berry connections associated to the triangle lattice . . 76

5.2.4 Berry curvature and Chern number associated to thetriangle lattice . . . . . . . . . . . . . . . . . . . . . . 79

5.3 The hexagon lattice . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.1 Staggering . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.2 Ground states on the hexagon . . . . . . . . . . . . . . 84

5.3.3 Berry connections associated to the hexagon lattice . . 85

5.3.4 Berry curvature and Chern number associated to thehexagon lattice . . . . . . . . . . . . . . . . . . . . . . 86

5.4 The Berry phase associated with the triangle and hexagonlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4.1 Closed path in parameter space . . . . . . . . . . . . . 88

5.4.2 Non-Abelian Berry phase: mixing and non-mixing kinds 89

5.4.3 Interpretation as a solid angle . . . . . . . . . . . . . . 93

5.5 The one-qubit quantum gates in the triangle and hexagonlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.5.1 Phase shift quantum gate . . . . . . . . . . . . . . . . 95

5.5.2 Rotation quantum gate . . . . . . . . . . . . . . . . . 96

5.5.3 General one-qubit quantum gate . . . . . . . . . . . . 98

5.5.4 Construction of general angles for a general one-qubitgate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Bow tie lattice 103

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2 The bow tie lattice . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.1 Staggering . . . . . . . . . . . . . . . . . . . . . . . . . 105

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6.2.2 Mirror symmetry . . . . . . . . . . . . . . . . . . . . . 1066.2.3 Ground states on the bow tie . . . . . . . . . . . . . . 1086.2.4 Berry connections . . . . . . . . . . . . . . . . . . . . 110

6.3 The Berry phase associated to the bow tie lattice . . . . . . . 1126.4 The two-qubit quantum gate in the bow tie lattice . . . . . . 116

6.4.1 Two uncoupled triangles . . . . . . . . . . . . . . . . . 1176.4.2 Entanglement after application of two-qubit gate . . . 120

7 Discussion, outlook, and conclusion 1337.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.1.1 Parameter choices . . . . . . . . . . . . . . . . . . . . 1337.1.2 Noisy Intermediate Scale Quantum . . . . . . . . . . . 1347.1.3 Mathematical point of view . . . . . . . . . . . . . . . 1357.1.4 Entanglement entropy . . . . . . . . . . . . . . . . . . 1377.1.5 The two-qubit system . . . . . . . . . . . . . . . . . . 1387.1.6 Experimental discussion . . . . . . . . . . . . . . . . . 139

7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Bibliography 147

Appendix 153

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viii

List of Figures

2.1 Schematic view of the (empty) triangle and hexagon lattices,periodic chain of three and six sites, respectively. The edgescorrespond to neighboring sites in the lattice model. . . . . . 6

2.2 Schematic view of the (empty) bowtie and trident lattices.The edges correspond to neighboring sites in the lattice model. 6

2.3 The triangle lattice with the site-dependent staggering para-meter ~λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Reduction of the staggering parameters on the hexagon latticeusing translational symmetry. . . . . . . . . . . . . . . . . . . 14

2.5 Reduction of the staggering parameters on the trident latticeusing rotational symmetry. . . . . . . . . . . . . . . . . . . . 15

2.6 Reduction of the staggering parameters on the bow tie latticeusing mirror symmetry. . . . . . . . . . . . . . . . . . . . . . 15

3.1 Representation of the pure-state space of a single qubit alsoknown as the Bloch sphere. . . . . . . . . . . . . . . . . . . . 18

4.1 The Bloch sphere S2 representing the state space of a singlequbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 The staggering parameter space for ~λ ∈ S2. . . . . . . . . . . 56

4.3 Schematic view of the map between the path in staggeringparameter space and the space of lattice states. . . . . . . . . 59

4.4 Schematic view of a conical path in a spherical parameter space. 62

4.5 Schematic representation of the tripod configuration (left)and the Λ configuration (right) of a atomic level scheme. . . . 68

5.1 Schematic view of the |1〉 state on the triangle lattice, a pe-riodic chain of three sites. Empty sites are represented bywhite circles and occupied sites by black circles. The edgescorrespond to neighbouring sites in the lattice model. . . . . . 72

5.2 The triangle lattice with staggering parameter ~λ included(left) and the space of the staggering parameter S2 (right). . 74

5.3 Schematic view of the hexagon lattice, a periodic chain of sixsites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

ix

5.4 The hexagon lattice with the staggering parameter ~λ showingits reduction by translational symmetry. . . . . . . . . . . . . 82

5.5 Plots of a path on the sphere; triangular loop and a circlearound the equator. . . . . . . . . . . . . . . . . . . . . . . . 88

5.6 Plots of a path on the sphere; a circle through both Northand South Pole and an orange-sliced-shaped loop. . . . . . . . 89

5.7 Triangular path in the staggering parameter space . . . . . . 95

6.1 Schematic view of the bow tie lattice configuration. . . . . . . 1046.2 The bow tie lattice with the definition of the staggering para-

meter on this lattice. . . . . . . . . . . . . . . . . . . . . . . . 1076.3 Construction of the bow tie lattice by two triangle lattices and

therefore the creation of a two-qubit system by connecting twoone-qubit systems. . . . . . . . . . . . . . . . . . . . . . . . . 117

6.4 The entanglement entropy as a function of dµb generated bya geometric two-qubit quantum gate defined by path para-meters θb and φb. . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.5 The entanglement entropy as a function of θb (a,b) or φb (c,d)generated by a geometric two-qubit quantum gate defined bypath parameters θb, φb and µb. . . . . . . . . . . . . . . . . . 128

6.6 The entanglement entropy S as a function of parameters θband φb for three values of the parameter dµb = 1, 10 and 1000. 130

6.7 The entanglement entropy as a function of dµb generated bya geometric two-qubit quantum gate defined by path para-meters θb and φb. . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.8 The entanglement entropy as a function of dµb generated bya geometric two-qubit quantum gate defined by path para-meters θb and φb. . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.1 Examples of longer lattice chains . . . . . . . . . . . . . . . . 1427.2 Two new lattice configurations as ansatz of two qubits (a) or

one qubit (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

A.1 Two perspectives at the trident lattice configuration. . . . . . 154A.2 Tripod atomic level scheme as investigated by Leroux et al.

[29]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.3 Path in (φ1, φ2) phase parameter space as implemented by

Leroux et al. [29]. . . . . . . . . . . . . . . . . . . . . . . . . . 160

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Chapter 1

Introduction and summary

In this thesis we will describe our research into geometric quantum com-putation based on supersymmetric lattice models. The elementary unit ofquantum computation, the qubit, is defined in terms of the states of thelattice model. We use a special feature of the supersymmetric lattice model:For periodic chain with a length a multiple of three lattice sites the modelproduce two zero-energy degenerate ground states [16, 18]. This space de-fines the computational qubit space. We will introduce the supersymmetriclattice model in detail in chapter 2. The focus of this work is the investiga-tion of quantum control of the qubit defined in the supersymmetric latticemodel. A general introduction to quantum computation will be given in 3.Quantum control is an essential component of an implementation of quan-tum computation in a physical system [12]. To this end, we introduce ex-plicit parameter dependence in the supersymmetric lattice model, by meansof a staggering parameter on the lattice sites. This translates to parameterdependence in the lattice ground states and therefore in the qubit definitionon the lattice. We consider a closed path in parameter space, transversedadiabatically. This action results in a geometric phase, known as the Berryphase, on the states in the ground state subspace. As the two ground statesare degenerate, we find a matrix-valued non-Abelian Berry phase. We willsee that the non-Abelian Berry phase can be interpreted in the context ofgauge theory and fibre bundles [9, 43]. We will go into this mathematicalstructure in chapter 4.

The non-Abelian Berry phase on the lattice subspace after a closed adia-batic path naturally defines a unitary operation on the computational qubitspace. In this way the non-Abelian Berry phase enables us to investigatequantum gate operations on the qubits after a geometric path in parameterspace. This is definition of geometric or holonomic quantum computation.Our work can be of interest for the research to limit the influence of decoher-ence on quantum computation. As the size of contemporary experimental

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implementations of quantum computation are not suitable for fault-tolerantquantum computation yet [38], it is useful to investigate approaches whichlimit decoherence. To this end, we combine the supersymmetric latticemodel with geometric quantum computation. The supersymmetric prop-erties produces protection of the ground states (the qubit states here) aslong as supersymmetry is not broken. Geometric schemes are more resis-tant to small errors as the gate operation depends on a geometric quantitysuch as an enclosed surface (solid angle) of the path in parameter space.Geometric quantum computation can seen as an intermediate step towardstopological quantum computation [46, 47].

We will implement the described method for geometric quantum computa-tion with a supersymmetric lattice model in the second part of this thesis.In chapter 5 we start with the explicit construction of the lattice groundstates in the two simplest lattice yielding two degenerate ground states: thetriangle and the hexagon lattices, periodic chains of three and six sites re-spectively1. We introduce two choices for the staggering parameter on thelattice and parametrize it by the standard spherical angles. This allows usto define a path in parameter space, with freedom in the surface enclosed bythe path. The unitary matrices found by the non-Abelian Berry phase canbe interpreted directly as the matrix form of known quantum gates. The twostaggering choices result in the definition of the phase shift quantum gateand the rotation quantum gate. This set is sufficient for one-qubit quantumcontrol as stated by quantum gate universality [36].

In chapter 6 we extend our discussion to a new lattice configuration. Weattach two triangle lattices by an edge to create the bow tie lattice. Thenatural interpretation as two-qubit system allows for the investigation of anon-trivial two-qubit quantum gate. The additional parameter on the con-nection between the triangles acts as a coupling between the qubits definedon each of the triangles. We investigate the non-trivial nature of the uni-tary constructed by the non-Abelian Berry phase in the bow tie lattice. Anon-trivial two-qubit quantum gate can not be decomposed into one-qubitquantum gates and is as such essential for quantum gate universality [36]. Itsconstruction would enable us to perform any n-qubit operation, as the tri-angle lattice has been shown to be able to produce any one-qubit operation.The constructed two-qubit quantum gate unitary matrices are investigatedby measuring the entanglement entropy of the final state after application ofthe quantum gate. As entanglement can not be produced by one-qubit oper-ations, it can give an indication for the wanted non-trivial behaviour of thetwo-qubit operations. We find that geometric paths can be chosen whichproduce a final state with near-maximal entanglement entropy. However,

1The author has previously taken interest in small lattices in Hemminga [24].

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the dependence of the matrix form of the quantum gate on the parametersdetermining the geometric path is highly non-linear. This makes optimiza-tion of the geometric path to achieve the maximal entanglement entropydifficult. Although our analysis shows that the produced two-qubit quan-tum gate is non-trivial, we are not able to transform the found expressionto any known universal two-qubit gate such as CNOT.

In conclusion, we show that for geometric quantum computation definedon the supersymmetric lattice we are able to construct a general one-qubitquantum gate and we are also able to find non-trivial two-qubit quantumgates. However, to be able to achieve quantum gate universality more re-search into the geometric two-qubit quantum gate is needed.

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4

Chapter 2

Supersymmetric latticemodels

This chapter about the supersymmetric lattice models is the first of thethree theoretically oriented chapters. We will introduce the supersymmetriclattice model including staggering on the lattice sites. Then we will considerdifferent supersymmetric Mk models and lattice symmetries.

The supersymmetric lattice model describes spin-less fermions on adjacentlocations, called sites. We will first consider periodic chains, one-dimensionallattices. Each site has two neighbors. In particular, we will be investigatingthe periodic chain with three and six sites, named the triangle lattice andthe hexagon lattice. These configurations are given in figure 2.1. Second, wewill also consider other configurations. We combine two triangles by addingan edge between them to create a new lattice configuration, called the bowtie lattice. In the appendix we consider the trident, where three sites areonly adjacent to a fourth site ‘in the middle’. These configurations are givenin figure 2.2.

As is standard in condensed matter physics, the model is described by itsHamiltonian. The Hamiltonian is defined in terms of operators Q and Q†,which are called supersymmetric charges. The notation Q− = Q and Q+ =Q† will be used in this text. The operators Q± change the number offermions on the lattice by ±1. The Hamiltonian H is defined as the anti-commutator of the supersymmetric charges,

H = Q,Q† = Q−, Q+ = Q+Q− +Q−Q+. (2.1)

The supersymmetric charges are nilpotent, they square to zero: (Q−)2 = 0and (Q+)2 = 0 [54]. The result is that the following relations hold for thecommutators with the Hamiltonian,

[H,Q] = [H,Q†] = 0. (2.2)

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Site 1 Site 2

Site 3

Site 1 Site 2

Site 3

Site 4Site 5

Site 6

Figure 2.1: Schematic view of the (empty) triangle and hexagon lattices,periodic chain of three and six sites, respectively. The edges correspond toneighboring sites in the lattice model.

Site 1

Site 2

Site 3

Site 4

Site 5

Site 6

Site 1

Site 2Site 3

Site 4

Figure 2.2: Schematic view of the (empty) bowtie and trident lattices. Theedges correspond to neighboring sites in the lattice model.

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Another example of nilpotent operators in physics are the Grassmann vari-ables in the path integral formulation of fermions [2].The spectrum of the Hamiltonian H is positive semi-definite. Eigenstates ofthis Hamiltonian are divided into doublet pairs and singlets. The doubletsconsist of two states with a difference in fermion number of one, but withthe same energy (by eq. (2.2)). For example, a state |Ψ〉 forms a doubletwith its supersymmetric partner |χ〉 = Q+ |Ψ〉. As part of the definition weknow that Q− |Ψ〉 = 0 and Q+ |χ〉 = 0 must hold as annihilation properties.The ground states of a supersymmetric lattice model are the singlet states.By definition of a singlet state |Φ〉, it holds that Q+ |Φ〉 = 0 and Q− |Φ〉 = 0.We find that its eigenvalue of the Hamiltonian, the energy, is equal to zero:H |Φ〉 = Q−, Q+ |Φ〉 = 0 = 0 |Φ〉. A supersymmetric lattice model givesrise to singlet ground states with exactly zero energy. This special featureprompts us to use analytical calculations in this work and at the same timeallows this.

In a more general formulation, the supersymmetric N = 2 algebra is formedby the set of operators Q−, Q+, H, F, where F is the fermion number

operator. It is defined by F =∑N

j=1 c†jcj , effectively counting the num-

ber of fermions on the chain. The operators cj , c†j are the standard an-

nihilation and creation operator, respectively. These satisfy the fermionicanti-commutation relations ci, c†j = δij , c†i , c

†j = 0 and ci, cj = 0 [18].

We can write the supersymmetric charge operators Q− and Q+ in termsof the fermionic operators [25]. We use also the expression P<i> which isdefined by

P<i> =∏

j next to i

Pj =∏

j next to i

(1− c†jcj). (2.3)

The operator Pj = 1 − Fj = 1 − c†jcj is called a projector, as we can see

that P = P 2. As Fj is the fermion number operator on site j the projectorPj returns 1-# number of fermions on site j. Therefore the expression forP<i> in eq. (2.3) requires all sites adjacent to site i to be empty to be non-zero. In the case of the supersymmetric lattice model with nearest neighborexclusion, later referred to as M1, the supersymmetric charge operators aredefined as

Q =∑i

c†iP<i> = Q+, Q† =∑i

ciP<i> = Q−. (2.4)

With these definitions we can rewrite the (M1) Hamiltonian completely interms of fermionic operators, the result is as given in Fendley et al. [18] is

H = Q†, Q (2.5)

=∑i

∑j next to i

P<i>c†icjP<j> +

∑i

P<i>. (2.6)

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On the periodic chain we can write out the sum over j, with index i to beunderstood with i mod N ,

H =N∑i=1

[Pi−1

(c†ici+1 + c†i+1ci

)Pi+2 + Pi−1Pi+1

], (2.7)

as presented in Fendley et al. [18].

2.1 Features of Mk models

Mk models describe interacting spin-less fermions by an explicit N = 2supersymmetric lattice model. These models are often considered on a one-dimensional lattice, such as the periodic chain, but can also be extendedto higher dimensions. The index k refers to the used exclusion rule, whichallows a group of at most k fermions on neighboring sites [19]. A modelwith nearest neighbor exclusion is therefore classified as M1, as it does notallow two adjacent sites to be occupied. Equivalent terms used in literatureare fermions with hard-core exclusion or ‘fat’ fermions. We can summarizethe condition as PjPj+1 = 0,∀j [16].

We need to extend the definition of the supersymmetric charges to the Mk

model. This will be done in terms of the constrained fermionic creationand annihilation operators d†[a,b],j and d[a,b],j . The operator d†[a,b],j , wherea, b = 1, . . . , k and b ≤ a, creates a fermion at lattice site j in such a way thata string of a particles is formed, with the newly created particle at positionb in the string. For the M1 model we find the relation d†[1,1],j = Pj−1c

†jPj+1.

The interpretation is that d†j creates a fermion on position j if the two

adjacent sites j − 1 and j + 1 are empty. The supersymmetric charge Q+

for a periodic chain of length L can now be written as

Q+ =

L∑j=1

∑a,b

d†[a,b],j , (2.8)

with Q− defined equivalently.It is possible to generalize the definition of the supersymmetric charges Q±

more [17]. For the Mk model we define a Hilbert space Hk. Now we can

define subspaces H(f)k defined by the eigenvalue f of the fermion number

operator F . In other words we partition the Hilbert space according tothe number of fermions in the states. The supersymmetric charges Q±

define maps from the subspace with fermion number f to the subspaces withfermion number f + 1, respectively f − 1, while preserving supersymmetry

Q− : H(f)k → H(f−1)

k , (2.9)

Q+ : H(f)k → H(f+1)

k . (2.10)

8

In other words the supersymmetric chargesQ± define maps from the fermionicsector of the Hilbert space Hk to the bosonic sector and vice versa [54].The ground states allow for a description by cohomology theory, whichmakes it a tool to compute the number of ground states. The ground statesare the states |s〉 that satisfy Q+ |s〉 = 0, but also cannot be written ina form |s〉 = Q+ |s′〉 for any state |s′〉. Therefore the ground states formthe cohomology of operator Q+ and the number of ground states is thedimension of this space [16, 18].

2.2 Staggering

In the definition of the supersymmetric charges introduced above each latticesite is treated equally. By introducing staggering we can be more general.We construct an inhomogeneous lattice model where the probability ampli-tudes are explicitly site dependent. If we consider the simplest example ofthe M1 model on the periodic chain, we introduce staggering as follows. Weadd a site-dependent parameter λj to our definition of the supersymmetriccharges. Note that λj = 1, ∀j reproduces the expression stated before [16].The nilpotency of the supersymmetric charges is conserved after introduc-tion,

Q− =∑N

j=1 λjPj−1cjPj+1, (2.11)

Q+ =∑N

j=1 λ∗jPj−1c

†jPj+1. (2.12)

For the general case of the Mk model we can extend the definition of stag-gering parameter λj to λ[a,b],j . Directly from the definitions in eq. (2.8) andeq. (2.12), as also presented in Fokkema [19], we can write

Q+ =L∑j=1

∑a,b

λ[a,b],jd†[a,b],j . (2.13)

If we consider the M2 model the set of λ[a,b],j can be simplified to twoparameters, λj and µj . We define

λ[1,1,],j = λj , λ[2,1,],j = λjµj , λ[2,2,],j = λjµj−1. (2.14)

So we can interpret this as introducing the staggering parameter µj specif-ically for cases with neighboring particles, which are allowed by the M2

model. Note that the index j − 1 is natively defined mod L in a chain oflength L, but can mean any adjacent site. We can interpret the structureas λj to be defined on site j and µj on edges between sites.Several staggering choices are described in literature, such as in [16, 19].Dependent on the lattice configuration the staggering parameter is oftenchosen with a certain configuration too. For example, in Fokkema [19] the

9

M2 model is considered with λ[1,1],j =√

2λj and λ[2,1],j = λ[2,2],j = λj .Additionally the λj are staggered with period two: λ2n+1 = λ1, λ2n = λ2,where we could set one to 1 and the other to λ, to effectively consider theratio of λi. Two limits are discussed: λ→ 0 is the extreme staggering limitand for λ = 1 the model is critical. An example of staggering with a period ofthree sites in a M1 supersymmetric model is λ3i = λ3i+2 = y, λ3i+1 = 1,∀i,which is described in Fendley and Hagendorf [16].

2.2.1 Staggering in the triangle and hexagon lattices

In this thesis we will first consider two chains of length three and six, coinedthe triangle and the hexagon. The reason will be elaborated on below. Thissuggests to take the staggering parameter with period three too. Keeping inmind our wish to perform an adiabatic path through parameter space, wechoose the vector describing the staggering parameters as the unit vectornormal to the surface of the sphere at the point (θ, φ). The spherical coordi-nates (r, θ, φ) satisfy the ISO1 convention commonly used in physics; radialdistance r, polar angle θ (theta), and azimuthal angle φ (phi). Explicitly,our choice for the staggering parameter is the three-dimensional vector

~λ =

sin θ cosφsin θ sinφ

cos θ

(2.15)

so we choose λ3n+1 = sin θ cosφ, λ3n+2 = sin θ sinφ and λ3n = cos θ. The

length∣∣∣~λ∣∣∣ is unity. We will give the point θ = 0, the North Pole on the

sphere, a special function in our procedure. As ~λ∣∣∣θ=0

= (0, 0, 1), indepen-

dently of φ, this is an ideal point to start and end our adiabatic path throughparameter space. Another example is constant staggering, without site de-pendence, which can be achieved using θ = cos−1( 1√

3) and φ = π

4 . In that

case we find ~λ = 1√3(1, 1, 1). The triangle lattice with the added staggering

parameter ~λ = (λ1, λ2, λ3) is shown in figure 2.3.

Effectively the ‘staggering parameters’ are θ and φ. We could introduce atime dependence t in θ and φ, such that the adiabatic path in parametrizedby t. We choose to keep θ and φ as independent variables, writing allderivatives and integrals in terms of these parameters. We will show thatthe staggering parameter ~λ will lead to a transformation matrix known asa rotation matrix, using the non-Abelian Berry phase, in chapter 5. It isuseful to also think about a different, independent, choice for the staggering

1International Organization for Standardization

10

λ1 λ2

λ3

Figure 2.3: The triangle lattice with the site-dependent staggering para-meter ~λ.

parameter. Let us define

~λΩ =

0− sin(θ/2) exp(iφ)

cos(θ/2)

. (2.16)

We will show that this choice leads to a transformation matrix introducinga phase shift in section 5.5.

2.3 Features of different chain lengths

In literature it is described that different lattice configurations and super-symmetric lattice models allow for different numbers of supersymmetricground states [52, 17]. The main tool is the Witten index. Its generaldefinition is

W = Tr[(−1)F exp−βH], (2.17)

where F is the number of fermions, β is the inverse temperature and Hdenotes the Hamiltonian. Note that it can be thought of as a modificationof the standard partition function, with the difference in the included factor(−1)F [2].In a supersymmetric theory the expression for the Witten index is indepen-dent of β. The reason for this simplification is the following, which is aresult of the level structure of a supersymmetric theory. The allowed statesare arranged in doublets and singlets. The states in a doublet both have thesame energy but a fermion number differing by one, therefore canceling eachother in the calculation of the Witten index. The singlets are states withzero energy so also resulting in terms independent of β. So the expressioncan be written as

WS =∑

(−1)F , (2.18)

where the sum goes over the possible states on the lattice and F denotesthe number of fermions in each of those states. The Witten index defines a

11

lower bound to the number of zero-energy ground states and is effectivelyequal to the number of bosonic ground states minus the number of fermionicground states. In other words the absolute value |W | equals the number ofzero-energy ground states [17], as in the cases considered here the groundstates lie in a single sector. In our work we find the number of ground statesusing the Witten index; we write out all states allowed by the supersym-metric model and sum the number of states for each allowed fermion numberwith the suitable sign. The value of the Witten index is independent of allsupersymmetry preserving deformations [54].

If we consider a M1 chain, a one-dimensional periodic lattice with hard-corefermions, with a length of a multiple of three, the number of ground statesis equal to two. By definition both have zero energy. This result is shownin Fendley and Hagendorf [16], Fendley et al. [18].

A argument for this feature is given by energy minimization. Consider theHamiltonian in eq. (2.7): The first term allows fermions to hop to neighbor-ing sites (while maintaining nearest-neighbor exclusion). The second termfavors adding more fermions, as long as they can be more than two sitesapart. This can be interpreted as an additional repulsive potential [18]. Sothe energetically optimal lattice configuration consists of fermions three sitesapart. In other words the number of fermions in the ground states is N fora periodic lattice chain of length 3N [16]. By taking every third site in thelattice chain as a sublattice, one can show that any periodic lattice chain oflength 3N produces two supersymmetric ground states [18].

Number of ground states for the triangle and hexagon lattices

We can also calculate the number of ground states by using the Witten in-dex explicitly. In the triangle (figure 2.1) there exist one state with zerofermions, the empty state, which corresponds to the empty lattice withoutfermions. Moreover, there exist three states with one fermion, which alreadycorresponds to the maximum occupation. The M1 condition, nearest neigh-bor repulsion, forbids us to add another particle to the lattice. The Wittenindex (eq. (2.18)) is equal to W = 1 − 3 = −2. The supersymmetric M1

model on the triangle has two fermionic ground states. By looking at thestructure of doublets and singlets, we find that the ground states are stateswith one fermion, one doublet is formed between the empty state and oneof the one-fermion states.

We can also do the counting in the hexagon (figure 2.1): There is one statewith zero fermions on the chain, the empty chain. Six states with onefermion, as the fermion can be one each of the vertices. There are ninestates with two fermions; six states with the fermions two sites apart andthree states with the fermion three sites apart. There are also two states

12

with three fermions. The M1 condition results in half filling (three fermionson six sites) as the maximal occupation. We can now find the Witten indexW = 1− 6 + 9− 2 = +2 by eq. (2.18), so the supersymmetric M1 model onthe hexagon has two bosonic ground states. The structure of doublets givesthat the ground states are two-fermion states. The ground states on thetriangle lattice and the hexagon lattice are investigated in detail in Chapter5.

Number of ground states for the trident and bow tie lattices

In the trident lattice configuration (figure 2.2) we can also easily do thecounting. We imposed the M2 model here. The possible states are anempty state (zero fermions), four states with one fermion, six states withtwo fermions and one state with three fermions. Three fermions is the max-imum density for M2 on the trident lattice. The Witten index results inW = 1− 4 + 6− 1 = +2. So the supersymmetric M2 model on the tridenthas two bosonic ground states. The structure of doublets gives that theground states are two-fermion states. The trident lattice is investigated inthe appendix.

The bow tie lattice (figure 2.2) is constructed by connecting two triangles byone of their vertices. We will impose a combined M1 +M2 supersymmetricmodel: In each of the triangles we allow at most one fermion (M1), but thefermions may both be on the sites connected by the added edge between thetriangles (M2). We can see in two ways that the ground states are stateswith two fermions. The ground states of each of the triangles are one-fermionstates and adding the extra edge (with the M2 condition) does not changethis. Therefore combined there are four ground states on the bow tie, withtwo fermions on the lattice. The number four is simply calculated as thereare two possibilities on each of the two triangles. The second method is toagain make use of the Witten index: There is one state with zero fermions(the empty state), six states with one fermion and nine states with twofermions. Therefore the Witten index is equal to W = 1 − 6 + 9 = 4. Twofermions on the bow tie is the maximum occupation. The ground states onthe bow tie lattice are investigated in detail in Chapter 6.

2.4 Lattice symmetry

In our discussions and calculations for the different lattice configurations wewill use lattice symmetries explicitly. We assume that the operator corre-sponding to each lattice symmetry commutes with the Hamiltonian. Peri-odic lattice chains can be equipped with translational symmetry. In the casewe will consider this is a symmetry for translation over three sites, as wechose our staggering parameter with a period of three sites. The hexagon,

13

λ1 λ2

λ3

λ4λ5

λ6

λ1 λ2

λ3

λ1λ2

λ3

Figure 2.4: Reduction of the staggering parameters on the hexagon latticeusing translational symmetry.

the chain with six sites, is effectively staggered with staggering parameter

~λsix = (sin θ cosφ, sin θ sinφ, cos θ, sin θ cosφ, sin θ sinφ, cos θ). (2.19)

The symmetry corresponds to a certain lattice operator, so the states onthe lattice can be classified by their eigenvalue of this operator. On multi-ple occasions this will help to reduce the total Hilbert space to the Hilbertsubspace which contains the ground states. It is then natural to take as thebasis for our problem the set of eigenstates of this operator. For the transla-tion operator over three sites, denoted T3, we need that on a chain of length3n T3 satisfies Tn3 = 1. In the triangle, n = 1, the translation operator istrivial. If we look specifically at the hexagon, we can easily see that thetranslation operator T3 has eigenvalues Λ = ±1. The operator works on thelattice states explicitly by T3 |i〉 = |i+ 3〉, where i mod 6 (we use site index6 instead of 0). The reduction of the staggering parameter on the hexagonlattice is shown in figure 2.4.

In the trident configuration, where three sites are all connected to a fourthone but not each other, we can use rotational symmetry. Starting from theset of staggering parameters λi, i = 1, . . . , 4;µi, i = 1, . . . , 3, the rotationalsymmetry reduces this to λ, λ4, µ. The reason is that we have to identifythe three sites and their links to the fourth one under rotational symmetry.The reduction is also shown in figure 2.5.

In the bowtie lattice, constructed by connecting two triangles, we will intro-duce a mirroring operator. You can imagine the mirror placed in the middleof the connection between the triangles. While at first sight the problemis described by two sets of the spherical coordinates (θi, φi), i = 1, 2, themirror operator identifies the sets with each other. Additional to stagger-ing parameter ~λ on the sites of both triangles, we define a parameter µon the connection between triangles. This is conform the definition of a M2

14

µ1

µ2µ3

λ1

λ2λ3

λ4

µ

µµ

λ

λλ

λ4

Figure 2.5: Reduction of the staggering parameters on the trident latticeusing rotational symmetry.

λ1

λ2

λ3

λ4

λ5

λ6

µ3

λ1

λ2

λ3

λ1

λ2

λ3

µ3

Figure 2.6: Reduction of the staggering parameters on the bow tie latticeusing mirror symmetry.

model between the triangles as given in eq. (2.13) with the specific M2 choiceeq. (2.14).

2.5 Supersymmetry in particle physics

The theory of supersymmetry is most often associated with particle physics,so we can not neglect mentioning this. The theory of supersymmetriccharges in a supersymmetric algebra is especially applicable. Supersym-metry (SUSY) describes a relation between fermions and bosons, the twoclasses of elementary particles with half-integer and integer spin, respec-tively. The theory states that each particle has an associated particle inthe other class. This superpartner, with a spin differing by a half-integer,would be new and undiscovered particles. In a perfectly unbroken super-symmetry, pairs of superpartners would share the same mass and have thesame internal quantum numbers except for their spin. As the particles butno superpartners are discovered, if supersymmetry exists, it will be a spon-taneously broken symmetry, which would allows superpartners to differ inmass. At this moment there is no evidence whether supersymmetry is cor-rect, but it is an attractive solution to some of the major concerns in particlephysics. More information can be found in Martin [33].

15

16

Chapter 3

Qubits and quantumcomputing

This chapter about quantum computation is the second of three theoreticallyoriented chapters. In this chapter we will introduce concepts of quantumcomputation, such as the qubit, entanglement and quantum gates. We willalso consider limiting decoherence and achieving quantum gate universality.

3.1 Qubits

The quantum bit, or qubit, is the fundamental unit in a quantum computer.A classical bit can take the values zero and one, or off and on. Importantdifferences between a classical and a quantum mechanical system (or bit)are the concepts of superposition and entanglement. A quantum bit canalso take values ‘in between’, that is, its state can be a superposition of zeroand one. In the qubit the zero and one are written as quantum mechanical‘ket’ states: |0〉 and |1〉. A superposition is defined as follows

|Ψ〉 = c0 |0〉+ c1 |1〉 , (3.1)

where the state is normalized by |c0|2 + |c1|2 = 1, c0, c1 ∈ C. We can vi-sually imagine these states as a vector from the origin to the surface of theunit sphere. The sphere in this context is called the Bloch sphere, shownin figure 3.1, where |0〉 lies on the North Pole and |1〉 on the South Pole.In other words the Bloch sphere is a geometrical representation of the purestate space. The Bloch sphere is equivalent to the complex projective line,CP1. This definition will be explored further in chapter 4.

While the concept of superposition can be present in the state of a singlequbit, to find the meaning of entanglement we will look at a two-qubit sys-tem. The general state of two qubits is a four-dimensional complex vector in

17

|1〉

|0〉

Figure 3.1: Representation of the pure-state space of a single qubit alsoknown as the Bloch sphere.

the two-qubit basis |00〉 , |01〉 , |10〉 , |11〉. The standard notational simpli-fication used here is |ij〉 = |i〉 ⊗ |j〉. These states are generically entangled,such that they can not be written as a product between two one-qubit states.Let us look at two examples

|P 〉 =1√2

(|00〉+ |01〉) = |0〉 ⊗ 1√2

(|0〉+ |1〉), (3.2)

|E〉 =1√2

(|00〉+ |11〉), (3.3)

where |P 〉 is a product state, as it is written as a product of one-qubit states,and |E〉 is an entangled state. The state |E〉 is also known as one of thetwo-qubit Bell states [36]. Entanglement will be discussed further in section3.6.

A qubit can in principle be realized in any quantum mechanical two-levelsystem, although we have to be able to define initialization, control andread out of the system [12]. This will be discussed in more detail in section3.4. We will discuss now a couple of examples of physically realized qubits.Three simple physical systems which realize a qubit are a single electronspin, the ground state and an excited state of an atom and the polarizationof a photon. The spin of an electron, ~S = 1/2, has two possible values forits projection with respect to a chosen z-axis: Sz = ±1/2. We can choosethese values as qubit states, for example |0〉 = |S = 1/2, Sz = −1/2〉 = |↓〉and |1〉 = |S = 1/2, Sz = 1/2〉 = |↑〉. The photon can be measured to havetwo possible polarizations, clockwise and counterclockwise, which can alsobe used as qubit states. The best hardware available today are qubit imple-mentations based on trapped-ions [4] and superconducting circuits [6]. Thiswork is loosely related to the concept of the topological qubit [48]. We willwork on geometrical (or holonomic) quantum computation, which can beseen as a intermediate step towards topological quantum computing. Ourwish and expectation is that this concept is more tolerant to errors thanquantum computation based on other concepts.

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Definition of a qubit with a supersymmetric lattice model

After discussing general definitions and implementations of the qubit, let usdefine our qubit on the supersymmetric lattice. The main point of our workis to investigate the supersymmetric lattice model in a configuration whichproduces two zero-energy ground states. This is directly related to our wishto define a qubit. Each of the degenerate ground states will correspond toa qubit state, |0〉 or |1〉. In our work we let the lattice model depend on astaggering parameter, so we find also a dependence on parameters θ, φ inthe explicit expression of the ground states. We choose to define the qubitstates in terms of the coefficient vectors of the ground states on the NorthPole, θ = 0. We will define quantum gates on these qubit states using apath through parameter space. It is natural to let these paths start and endon the North Pole, where we have made a clear interpretation of the groundstates as qubit states.

3.2 Noisy Intermediate Scale Quantum

In the recent proceedings Preskill [38] introduces the terminology of NoisyIntermediate Scale Quantum (NISQ) technology. He describes NISQ tech-nology as a pivotal new era in quantum technology. ‘Intermediate scale’refers to quantum computers of a size of fifty to a few hundred qubits whichwill be available in the next few years. Fifty qubits is a significant mile-stone, as they can outperform the most powerful (classical) supercomputers.‘Noisy’ emphasizes imperfect control over the qubits, the noise will place se-rious limitations on quantum technology in the near future. Preskill doesnot expect NISQ to change the world by itself; it should be regarded as astep towards more powerful quantum technology. The NISQ era describesquantum computers with noisy gates unprotected by quantum error correc-tion. The error gate per gate will put a bound on the maximum circuitsize and therefore the computational power of NISQ technology. Moreover,we need to prepare and measure qubits accurately. From an engineeringpoint of view the manufacturing constraints of number of connections andreliability should be kept in mind.

In the near future NISQ quantum computers may be able to outperformclassical computers. Its power is based on quantum complexity, with theunderlying concept of quantum entanglement. Important is also quantumerror correction, which will determine the scalability of quantum computa-tion.

Preskill gives three reasons why quantum computers may have capabilitiessurpassing a classical computer. Quantum algorithms have been discov-ered for problems that are believed to be hard for classical computers. Thewell-known examples are Shor’s factoring algorithm [42] and Grover’s search

19

algorithm [22]. The second argument comes from complexity theory: ‘easyquantum states’ have superclassical properties. The possibly most persua-sive argument is that there is no known classical algorithm to simulate aquantum computer. For more details on quantum supremacy, see Harrowand Montanaro [23].

For a physicist the natural problem to look at is simulating a many-particlequantum system. In such a quantum system entanglement has profoundconsequences. Especially dynamics of highly-entangled many-particle sys-tems is a promising arena where quantum computers may have a significantadvantage over classical computers. The size of the quantum mechanicalHilbert space grows exponentially with the number of particles. This fun-damental fact makes the calculations of large quantum mechanical systemsdifficult. Let us compare the simulation of a quantum mechanical system inmore precise terms. Consider the memory requirements of a system of Nspin-1/2 particles. Each state can be in two states; |↑〉 or |↓〉, and each clas-sical bit can also be in two states: 0 and 1. So we can describe the system ofN ‘classical’ spins by N bits, bi ∈ 0, 1, i = 1, . . . , N . However, in the quan-tum case we allow superpositions (linear combinations) of arrays of N spins,which means that we need 2N complex numbers, αj ∈ C, j = 1, . . . , 2N , todescribe the state of the system. Note the exponential scaling with the sys-tem sizeN . This is the simplest argument to research quantum computation,to simulate large quantum mechanical systems. As the quantum computeris itself quantum mechanical, it benefits from the same exponential scaling[36]. While an universal quantum computer has not been developed (yet),for the research into quantum many-body systems there has been progress inquantum simulation. The main idea is to use a quantum mechanical systemover which you have a lot of control to simulate a Hamiltonian of a systemyou want to investigate. These controllable systems can be realized in labsetup, with for example laser-induced periodic potentials.

Quantum computing is hard as a result of a fundamental feature of thequantum world: we cannot observe a quantum system without disturbingthe system. That means that we need to keep the system nearly perfectlyisolated from the environment, if we want to use the system to store andprocess quantum information. At the same time, the qubits must stronglyinteract with each other to perform computation and we need to control thesystem from the outside. Eventually we also want to read out the resultof the computation. It is very challenging to build a quantum system thatsatisfies all the criteria. The expectation is that eventually the quantum sys-tems can be protected using the principle of quantum error correction andthen quantum computers can be scaled up. Entanglement lies at the basis ofquantum error correction as it is possible to encode quantum information ina highly entangled state. Unfortunately, there is a significant overhead cost

20

for quantum error correction; it requires many additional physical qubits.For more detail see Steane [49]. Clever design of noise-resilient algorithmsmay extend the computational power of NISQ devices.

Let us stress the fundamental difference of measuring a classical bit and aquantum bit. The classical bit takes values 0 and 1. If we measure thevalue of a bit equal to 0 we will find 0. A qubit can also be in superpositionstates, in general |B〉 = c0 |0〉+c1 |1〉, where ci ∈ C and |c0|2+|c1|2 = 1. Thismeans that if we measure |B〉, that we can find |0〉 and |1〉 with a certainprobability. These probabilities are precisely |c0|2 and |c1|2, so the secondrequirement makes sure the probabilities sum to one (Born’s rule). However,measurement destroys the superposition state, measuring the same qubitagain will always give the same result as measuring the first time. Thisis a result of the quantum mechanical nature of the qubit. At the otherhand, this is also what is meant in popular science by saying that the qubitis in the two states 0 and 1 at the same time. From the point of view oferror correction this nature is very inconvenient. In a classical system it ispossible to measure a bit and put its value in another bit, effectively copyingthe value. The single bit error known as ‘bit-flip’ can then be corrected bycomparing to the reference bit(s). In a quantum system, as it is impossibleto measure the qubit without destroying its superposition value, it is notpossible to clone the qubit. This is the premise of the research field ofquantum error correction [36].

3.3 Decoherence

The concept of decoherence is very important to consider for quantum com-putation. Decoherence, or quantum noise, can be seen as loss of informationfrom the system (the quantum computer) to the environment. The ideal pic-ture of a closed quantum system is not true in reality. The coherence, orphase relation between different states, can be lost in a decay process. Thecharacteristic time is called the decoherence time. During the process thesystem becomes coupled to the environment and entanglement is created,in some unknown way, between the system and the environment. In otherwords, quantum information is shared or transferred to the environment.

Decoherence can be modeled as non-unitary dynamics of the system alone.The combined system with environment still evolves unitarily. The systemundergoes a irreversible (non-unitary) transformation acting on its Hilbertspace. The irreversibility underlines the fact that information is lost. Ex-amples include dissipative and dephasing contributions [30].

Decoherence is also studied in the context of quantum measurement andthe transition from quantum to classical behavior. The existence of deco-

21

herence poses a challenge for physical realizations of quantum computation.It is necessary to manage its influence as the quantum computer relies onpreserving the interference and entanglement between quantum states toperform computation. A solution is presented in the next subsection calledquantum error correction. Another approach to solving the decoherenceproblem is the theory of decoherence-free subspaces and subsystems. For areview see Lidar and Whaley [30].

One of the challenges of an experimental realization of the quantum com-puter is therefore its coupling to the environment. A quantum computer hasto be well isolated to retain its quantum properties, but at the same timeits qubits have to be accessible. The qubits may need to be manipulatedto perform computation or read out the results. A realistic implementationmust achieve a delicate balance in its coupling to the environment [36].

3.3.1 Quantum error correction

Quantum error correction is a method to perform quantum information pro-cessing in a reliable manner in the presence of noise. The goal of quantumerror correction is to increase the fidelity with which quantum information isstored or communicated. Quantum-error-correcting codes work by encodingquantum states in a special way, to make them noise resilient. Fault-tolerantquantum computation concerns the protection of quantum information as itdynamically undergoes computation. The theory of fault-tolerant quantumcomputation produces the remarkable ‘threshold theorem’: provided thatthe noise in individual quantum gates is below a certain constant thresholdit is possible to efficiently perform an arbitrarily large quantum computation[36].The key idea of (classical) protection against noise is to encode the mes-sage by adding redundant information to the message. So if the encodedmessage is affected by noise, the redundancy allows to decode and thus re-cover the message. However, there are some important differences betweenthe classical and quantum information related to error correction. Firstly,while creating redundancy we have to take the no-cloning theorem in mind.Secondly, the errors are continuous, instead of a classical (discrete) bit fliperror. And thirdly, measurement destroys the quantum information, we can-not recover the quantum state in this way. In Nielsen and Chuang, Chapter10 is shown that these differences do not prove fatal for quantum error cor-rection. The general approach to creating redundancy uses physical ancillaqubits used for measurement in addition to the physical code qubits storingthe quantum information.

22

3.3.2 Our approach to fight decoherence

Let us discuss the motivation of the approach to quantum computation pre-sented here, in the context of protection against quantum errors. We lookinto geometric quantum computation on the supersymmetric lattice. Bothmain ingredients are important here. We use the fact that the supersym-metry protects the computational states as motivation for using this latticemodel. If the supersymmetry is unbroken, the system is robust with respectto dissipation and decoherence. The supersymmetry of a quantum systemis called unbroken if the ground state energy is zero. In other words, thesupercharges Q,Q† annihilate all ground states. So in this work we assumeunbroken supersymmetry. Moreover, the dimension of the degenerate spaceof ground states is given by the Witten index and therefore has the propertyof topological invariance. More information can be found in Tomka et al.[50].

By definition topological invariant quantities can not be changed by smallperturbations. This makes quantum computation based on topology morerobust to errors, although the details depend on the experimental realiza-tion. This feature has incited research into a form of fault-tolerant quantumcomputation known as topological quantum computation. For more infor-mation, see Nayak et al. [35].

Our work presents a theoretical ansatz for geometric quantum computation.Geometric quantum computation inherits protection against errors from itsgeometric principles. It can be seen as an intermediate step towards topolog-ical quantum computation. In the next chapter we will introduce the non-Abelian Berry phase, which will allows us to construct geometric quantumgates. The non-Abelian Berry phase does not depend on the specifics of thepath, only on its geometry such as the area spanned by a loop. This makesgeometric phases robust to noise in the classical control parameters [48]. Itis also flexible in the way a computation is implemented, as there many dif-ferent paths possible to find a certain value of the solid angle spanned [31].Experimental implementations of geometric quantum computation will beshown in section 4.7. This literature also discusses the protection againsterrors that is provided by the geometric approach.

3.4 Quantum control

The work in this thesis will focus on the possibilities of quantum controlof the qubits defined on a supersymmetric lattice. The basis of quantumcontrol consists of a set of operations we are able to perform on the qubit.We will talk about quantum gates and their matrix representations in thenext section. The main goal of this research is to find which quantum gatescan be performed on the supersymmetric lattice based qubits. As will be

23

introduced in chapters 5 and 6, we take a geometrical approach to definethe quantum gate operations.

To be able to speak about quantum computation on a supersymmetric lat-tice, we have to satisfy a set of criteria, also known as DiVincenzo’s criteria.In his paper DiVincenzo [12] describes five requirements for the physical im-plementation of quantum computation and adds two additional requirementsfor quantum communication. This set of requirements can be summarizedas follows

1. A scalable physical system with well-characterized qubits2. The ability to initialize the state of the qubits to a simple fiducial

state, such as |000 . . .〉3. Long relevant decoherence times, much longer than the gate operation

time4. A ‘universal’ set of quantum gates5. A qubit-specific measurement capability6. The ability to interconvert stationary and flying qubits7. The ability faithfully to transmit flying qubits between specified loca-

tions

DiVincenzo’s seven requirements are aimed towards experimental implemen-tations of quantum computation and quantum communication [12]. Ourwork is primarily aimed at presenting a theoretical ansatz which can beused for geometric quantum computation. The requirements stated can notbe satisfied from a theoretical discussion alone, only a experimental imple-mentation will give definite answers of the possibilities. From our point ofview we will discuss the relevant requirements. We will not touch uponquantum communication, described by requirements 6 and 7.

The first requirement

The supersymmetric model is used to provide qubits; the two degenerateground states define the computational space. We know from the supersym-metric theory that other states have positive energy, such that we assumethat in combination with the adiabatic approach the probability of populat-ing these states is small. The set of lattice states are also a finite set; it isconvenient to consider a Hilbert space of finite size, for example to minimizedecoherence [36]. We have to note that this system does not corresponddirectly to a physical system, which would mean an experimental imple-mentation. A first step towards the scalability of the supersymmetric modelbased qubits is investigated in the bow tie lattice. Two qubits, defined oneach of the triangles, are coupled to form a two-qubit system. In principlethis approach could be extended to coupling more triangles (qubits) in thismanner.

24

The second and third requirements

The specifics of the initialization of the qubit state are also dependent onthe physical implementation. The construction of a simple initial state isvital for quantum computation. To extract all information from the resultof a calculation we need to be able to repeat the process, with the same ini-tial state each time. In the approach presented in this work the staggeringparameter enables us to initialize the lattice quantum states in the compu-tational subspace. However, this will not in general be a simple state, suchas |0〉, it could be some superposition of |0〉 and |1〉. Therefore initializationon the supersymmetric lattice to a simple state will require a extra protocol.Note that a commonly used cooling approach [12, 36] may not work as thecomputational subspace coincides with the ground state manifold. We willcontinue the discussion in chapter 7. The requirement of long decoherencetimes will also naturally depend on the experimental implementation. Ourapproach is thought to provide some protection to decoherence by combina-tion of the supersymmetric lattice model and the adiabatic approach. Wehave considered this in section 3.3.2.

The fourth requirement

The fourth requirement we will consider in detail. The main focus of ourwork is defining a universal set of quantum gates for the supersymmetriclattice based qubits. We want to construct the necessary unitary opera-tions by a geometric protocol. Our approach will be to construct unitaryoperations by the non-Abelian Berry phase. More information on universalquantum control can be found in sections 3.5 and 3.7. We consider differentchoices for the staggering parameter in the supersymmetric lattice modelon the triangle to find the rotation and phase shift quantum gates, as pre-sented in chapter 5. Moreover, we investigate the possibilities of creatingentanglement with a two-qubit unitary operation in chapter 6. We note thatin the system considered for two-qubit operations also the one-qubit opera-tions are still possible, if we address the staggering parameter in each of thetriangles separately. In theory we have the capability to address any set oftwo adjacent qubits by means of turning on the parameter on the connectionbetween the triangles. However, we have not investigated beyond the bowtie configuration.

The fifth requirement

The final (computation) requirement concerns a qubit-specific measurementcapability, which is (again) dependent on the experimental implementation.You could imagine that this means that we translate the computationalqubit states back to lattice states and measure the probability of finding afermion on each of the sites.

25

(3.4)

Measurement is naturally the final element of quantum computation anddenoted by the meter symbol shown in 3.4. This pictorial notation willalso be used in the next section (section 3.5) to show the construction ofquantum circuits by quantum gates. It corresponds to a projective measure-ment in the computational basis, whereas more general measurements canbe represented by unitary transforms followed by projective measurements.A projective measurement corresponds to what is known as a collapse of thewave function [36]. A measurement with (near) perfect quantum efficiencycan also be used for state preparation in a quantum computation imple-mentation [12]. Projective measurements are often difficult to implement.It is required that the coupling between the quantum and classical systemis large and also switchable. Unwanted (projective) measurements can beconsidered as a decoherence processes. Measurement can be viewed as aninterface between the quantum and the classical world, it is generically con-sidered to be an irreversible operation: Measurement is destroying quantuminformation and replacing it by classical information. Only in special con-structed cases, such quantum teleportation and quantum error-correction,this does not need to hold; in these instances the measurement result doesnot reveal information about the identity of the quantum state being mea-sured. Measurement can be thought of as a process of coupling one or morequbits to a classical system, such that after some time the state of the qubitsis indicated by the state of the classical system [36].

An interesting and important observation is that the constraints on thephysical realization of the quantum computer are opposing in general. Aquantum computer has to be well-isolated in order to retain its quantumproperties, that is reduce decoherence, but at the same time the system hasto be accessible so that it can be manipulated to perform a computationor to read out the results. A realistic implementation must find a balancebetween these constraints [36].

3.5 Quantum gates

Quantum gates, or generally quantum operations, are the extension of classi-cal computer gates to the quantum computer. In classical computers, basedon bits valued zero or one, the operations are called logic gates. Apart fromthe trivial identity operator, leaving its input unchanged, the only non-trivial one-bit operator is the negation gate, NOT, taking zero to one and oneto zero. We will see that in the language of the quantum gates, the set of

26

one-qubit operators is extensively increased. The standard two-bit opera-tors are AND and OR. Combining these two-bit operators with the negationgate NOT, creates the two-bit operators NAND and NOR. Rounding out the setof two-bit gates are XOR and XNOR. It is shown that circuits of only NOR gates(Pierce, 1880) or alternatively only NAND gates (Sheffer, 1913) can reproducethe functions of all the two-bit gates. Consequently, these gates are calleduniversal logic gates. We are interested in this notion of universal gates inthe case of quantum gates: we will introduce the equivalent definition insection 3.7 and aim for defining operations on the lattice creating this set.

One-qubit gates

The one-qubit gates perform a operation on one qubit at the time. Thestate of a qubit can be represented by a point on the Bloch sphere, so theset of possible one-qubit operations can be viewed as rotations to anotherpoint on the Bloch sphere. Therefore quantum states and quantum gatesare often defined in terms of a vector basis. A single qubit is represented asa vector

|Ψ〉 = c0 |0〉+ c1 |1〉 ⇔ Ψ =

(c0

c1

)(3.5)

and a one-qubit gate by a 2× 2 unitary matrix. In general a gate acting onk qubits is represented by a 2k × 2k unitary matrix in the vector basis of kqubits |Ψk〉. For example if k = 2 the two-qubit gate is defined in the basisof

|Ψk=2〉 = c00 |00〉+ c01 |01〉+ c10 |10〉+ c11 |11〉 ⇔ Ψk=2 =

c00

c01

c10

c11

(3.6)

The set of possible operations is extensively increased with respect to theclassical bit. The operation related to the classical NOT gate is now calledthe Pauli X gate, as it can be represented as the Pauli matrix σx eq. (3.7)applied to the qubit vector [36].The classical negation operation is embedded as |0〉 = (1, 0) is changed to|1〉 = (0, 1) and vice versa. However, the quantum gate can be applied toany quantum state |Ψ〉 = c0 |0〉+c1 |1〉, with the action X |Ψ〉 = c0 |1〉+c1 |0〉.The Pauli z-matrix defines the Pauli Z gate, which add a minus sign to |1〉but leaves |0〉 unchanged. This gate is also called the phase-flip gate andhas no classical analog. Its matrix form equals the Pauli z-matrix eq. (3.7).The third Pauli gate is Y, defined by the Pauli matrix σy 3.7, which by theproperties of the Pauli matrices can also be written as Y = iYZ. The Pauligates NOT = X, Y and Z are shown in quantum circuit notation in 3.8.

σx =

(0 11 0

)σy =

(0 −ii 0

)σz =

(1 00 −1

)(3.7)

27

X Y Z (3.8)

An important gate in quantum computation algorithms is the Hadamardgate H. It is particularly known for its function in creating entangled states.Entanglement is the most important quantum mechanical concept for quan-tum computing. The Hadamard gate can be represented in the standard|0〉 , |1〉 one-qubit basis as

H =1√2

(1 11 −1

). (3.9)

H(3.10)

We can see that the Hadamard gate is related to the Pauli gates by H =1√2(X+Z). The application of this gate creates a superposition state, special

to quantum computing compared to classical computing. The superpositionis a linear combination of the classical zero and one states.

H |0〉 = 1√2(|0〉+ |1〉), H |1〉 = 1√

2(|0〉 − |1〉) (3.11)

In the Bloch sphere picture the vector in moved from the poles to the equa-tor by the Hadamard gate [36]. The simple quantum circuit notation isshown in 3.10.

A more general one-qubit gate is the phase shift gate, which constitutes afamily of gates parameterized by α. Its matrix definition is equal to

Phaseα =

(1 00 eiα

)(3.12)

where α is called the phase shift.

Phase(α) = Rα (3.13)

Common examples are taking α = π/4 and α = π/2, which are known as T

and S. The phase shift α = π reproduces the Pauli Z gate, which is thereforealso known as the phase flip gate. The quantum circuit notation is shownin 3.13.

The rotation quantum gate also constitutes a family of one-qubit quantumgates, parametrized by β. Its matrix representation in the one-qubit basisis equal to the rotation matrix in two-dimensions over an angle β

Rotationβ =

(cosβ − sinβsinβ cosβ

). (3.14)

28

Rot(β)

(3.15)For example the angle choice β = π

2 leads to the iY, the Pauli y-matrixmultiplied by the imaginary unit i. It quantum circuit representation isgiven in 3.15.

Two-qubit gates

Two-qubit gates perform an action on two qubits. This is often representedin the two-qubit basis by a 4×4 unitary matrix. In quantum circuit notationthe gate is extended to two qubits, as shown in 3.16.

|0〉U2|0〉

(3.16)A simple example is the SWAP two-qubit gate, which swaps the values of twoqubits. Its matrix representation in the 00, 01, 10, 11 basis is equal to

SWAP =

1 0 0 00 0 1 00 1 0 00 0 0 1

(3.17)

An important family of two-qubit gates are the controlled gates. The basicfunction is applying a operation on the second qubit if the first qubit equals|1〉 and leaving the second unchanged if the first equals |0〉. This opera-tion can for example the Pauli gates X(= NOT), Y, Z. The general form of acontrolled gate CU where U is the ‘controlled’ operator is given by

CU =

1 0 0 00 1 0 00 00 0

U

(3.18)

The CNOT quantum gate, where U = X, is generally used to create entan-glement between qubits as discussed in the next section 3.6. The controlledgate CU is given in quantum circuit notation in 3.19 and the special notationfor the well-known CNOT quantum gate is shown in 3.20.

•U

(3.19)

29

• ≡ •X

(3.20)

3.6 Entanglement

The concept of entanglement is key in quantum information and quantumcomputation. It is one of the particularities of quantum mechanics comparedto classical theories. Well-known examples include quantum teleportationand quantum error correction. Preskill [38] gives the example of a very spe-cial and fictional book. In a normal classical book you can get informationabout the contents of the book by reading the pages one by one. After read-ing all the pages, you know the complete contents. In this special, entangled,book if you read a page you understand nothing about the contents, andafter reading all pages you still don’t. Only if you could look at all pages atonce, you can extract the information hidden in the book.

The basic property of an entangled state is the fact that it cannot be writtenas a product state. Consider the following example, as introduced in section3.1: Let the first qubit be in the state |0〉 and the second qubit in the state

1√2(|0〉+ |1〉) then the two qubits together are in the product state

|Ψ〉 = |ψ1〉 ⊗ |ψ2〉 = |0〉 ⊗ 1√2

(|0〉+ |1〉) =1√2

(|00〉+ |01〉). (3.21)

Now consider the following two-qubit state

|Ψ〉 =1√2

(|00〉+ |11〉) (3.22)

which is a well-known example of an entangled state. It is one of the Bellstates, which are maximally entangled. It can not be written as productstate of two one-qubit states. This Bell state can be generated from theinitial two-qubit state |00〉 by applying the Hadamard gate (3.10) on thefirst qubit and the CNOT (3.20) gate on both qubits, as given in the quantumcircuit in 3.23. The three other, maximally entangled, Bell states can begenerated by this quantum circuit from the initial states |10〉 , |01〉 , |11〉.

H •

(3.23)

30

Entanglement entropy

A measure for the entanglement of the quantum state of a multi-qubit systemis given by the Von Neumann entanglement entropy. It is valued betweenln(d) for a maximally entangled state in the Hilbert space of d qubits andzero for a product state. The definition of the entanglement entropy is givenby

S(ρ) = −Tr(ρ ln ρ) (3.24)

where Tr denotes the trace and ln needs to be evaluated explicitly as matrixlogarithm. The object ρ is called a density operator, which is a differentrepresentation of a quantum mechanical state. We define 0 ln 0 ≡ 0. Purestates, where the state |ψ〉 is known exactly, are given by |ψ〉 〈ψ|. Mixedstates, which are ensembles of different pure states, can be incorporated inthe same language. A density matrix ρ satisfies the trace condition Tr ρ = 1,it is a positive operator and Tr ρ2 ≤ 1 with equality only for a pure state.The reduced density operator is a descriptive tool for subsystems and in thatmanner used to measure the entanglement between subsystems. Considera two-part system of which the state is described by the density operatorρAB. Then the reduced density operator for the subsystem A is defined by

ρA = TrB(ρAB) (3.25)

where TrB denotes the partial trace over system B. This operation is alsocalled ‘tracing out’ subsystem B. If we consider a two-qubit system its stateψ is (generally) written in the basis |00〉 , |01〉 , |10〉 , |11〉. The matrix formof the density operator can be represented as

〈00| 〈01| 〈10| 〈11||00〉 a b c d|01〉 e f g h|10〉 i j k l|11〉 m n o p

(3.26)

Now we can explicitly find the matrix form of the reduced density operatorof subsystem A. The result is

ρA = TrB(ρAB) =

(a+ f c+ hi+ n k + p

). (3.27)

We will use this form in the calculation of the entanglement entropy gener-ated by the non-Abelian Berry phase with the bow tie lattice.

3.7 Gate universality

Quantum computation is defined by qubits which carry the quantum in-formation and operations on the qubits to perform manipulation on the

31

quantum information. For a general case one might expect that many-qubitgates, which act on several qubits at the same time, are indispensable. How-ever, in this section we will introduce the concept of gate universality. Insimple terms this theorem states that any action on a n-qubit system canbe composed from the gate actions from a universal set. This universal setconsists of only one- and two-qubit gates. In a bit more detail, a set ofquantum gates is said to be universal for quantum computation if any uni-tary operation may be approximated to arbitrary accuracy by a quantumcircuit involving only those gates. An example of such a universal set is thefollowing: the Hadamard gate (H, eq. (3.9)), the π/8 gate (T , eq. (3.12) forφ = π/4) and the CNOT gate (eq. (3.18)) [36].In Nielsen and Chuang [36] three universality constructions are described,which build upon each other to show that the set H,T, CNOT is universal.

First construction

In the first construction Nielsen and Chuang show that an arbitrary unitaryoperator can be expressed exactly in terms of a product of unitary operatorsthat act non-trivially on a two-dimensional computational subspace. Theproven corollary states that an arbitrary unitary matrix on a n-qubit systemcan be written as a product of at most 2n−1(2n−1−1) two-level unitary ma-trices. Naturally, there are specific unitary matrices for which more efficientdecompositions exist. In short, the number of two-level unitary operationsscales as O(22n) = O(4n). This construction is based on Reck et al. [39].

Second construction

The second construction shows that an arbitrary unitary operator on n-qubits can be expressed exactly using single qubit and CNOT gates. For this,with the first construction in mind, Nielsen and Chuang show that singlequbit gates and CNOT gates can be used to implement an arbitrary two-levelunitary operation on the n-qubit space. This result is based on Barencoet al. [5] and DiVincenzo [11]. The number of single qubit and CNOT gatesneeded scales with O(n2). We find that an arbitrary unitary operation onn qubits can be implemented using O(n24n) single qubit and CNOT gates;while this is the wanted result this construction does not provide efficientquantum circuits.

Third construction

The third construction proves that single qubit operation can be approxi-mated to arbitrary accuracy using the Hadamard and π/8 gate. This com-pletes the universality proof of the set H,T, CNOT. The universality proofof this set can also be found in the work of Boykin et al. [8]. The set of uni-tary transformations is continuous, so by a discrete set an arbitrary unitary

32

operation can only be approximated. In a full analysis a measure of erroris a key quantity. The Hadamard gate H and the π/8 gate can be used toapproximate any single qubit unitary operation. Nielsen and Chuang [36]show this by constructing the gate Rn(θ), which corresponds to a rotationover θ about an axis along n. While θ and n are defined in terms of trigono-metric functions of π/8, it can be shown that repeated iteration can be usedto approximate to arbitrary accuracy any rotation Rn(α).

Different options for universality

A second set of universal gates consists of the Hadamard gate, phase gate,the CNOT gate, and the Toffoli gate. The Toffoli gate is also known as CCNOT(controlled-controlled-NOT) gate and acts on three qubits.The universality construction can be related to the ‘definition’ of a generalone-qubit gate. There is another method to find a ‘universal’ one-qubitgate. A general one-qubit unitary U can be constructed by multiplication ofrotation and phase shift gates [36]. Note that the rotation and phase shiftquantum gates can be thought of as continuous gates: For any value of theangles, the definitions in eq. (3.12) and eq. (3.14) provide unitary operations.This allows us to define any one-qubit unitary in terms of these one-qubitgates. To be precise, to construct a unitary U ∈ U(n) SU(n), with andeterminant different from unity, we need to include a global phase factor.The construction leads to the following

Ugeneral = exp(iα)I2 Phase(β)Rotation(δ)Phase(ε), (3.28)

consisting of the global phase factor exp(iα), the phase shift gates Phase(β),Phase(ε)(eq. (3.12)) and the rotation gate Rotation(δ) (eq. (3.14)). This constructionwill be used in section 5.5 to show the construction of this general one-qubitunitary gate for geometric quantum computation on the supersymmetriclattice.

In the construction presented here the universal set is a discrete set. Thereis no free variable, such as an angle, in the definition of the gates. These dis-crete gates can be constructed by taking a certain value in the continuousgates such as the rotation gate and phase shift gate. However, in quan-tum computation implementations it may be easier or more convenient toconsider protocols defining a set of continuous gates. For comparison themeasures of efficiency and control must be considered. For example in thecase of the single qubit gate: A discrete set needs a large quantum circuitto approximate a general unitary operation, while the construction by therotation and phase shift gates, needs only three quantum gates, but needsalso very precise control of the angle parameters.

33

In this work, in the context of holonomic quantum computation, we see thatconstructing continuous single qubit gates is the more natural option. Wewill show that the Hadamard gate, phase gate and π/8 gate can be con-structed by tuning to the correct angles. So we can also connect to theapproach presented here to achieve (single qubit) gate universality.

Following the construction presented in Nielsen and Chuang [36], we haveseen that any unitary operation on n qubits can be approximated by a smallset of universal unitary quantum gates. This is not possible efficiently, asthere does not always exist a quantum circuit of a size which scales poly-nomially in n for any n-qubit unitary operation. A simple argument givenin Nielsen and Chuang [36] is that generating an arbitrary state of n qubitsalso requires exponentially many operations. The universal construction isthought to be close to optimal, to approximate an arbitrary operation. Thismeans that fast quantum algorithms will clearly need a different approachthan is taken in the universality construction. The Solovay-Kitaev theoremstates that an arbitrary single-qubit gate can be approximated to an accu-racy ε using a number O(logc(1/ε)) of gates from the given universal set,where c ≈ 2 is a constant. Therefore to approximate a circuit containing mCNOT and single qubit gates with an accuracy requires O(m logc(m/ε) gatesfrom the discrete universal set. More information can also be found in Knill[27].

Let us also make notice of the proof that almost any two qubit quantumlogic gate is universal, which was independently performed by Deutsch et al.[10] and Lloyd [32]. Deutsch et al. [10] also state that there can not be aone-qubit universal gate, because a one-qubit gate or any number of one-qubit gates cannot place two initially un-entangled qubits into an entangledstate. Likewise no classical gate can be universal, as no superpositions canbe created.

3.8 Many-body strategies for multiqubit gates

For a different view to quantum gate universality we look at Groenlandand Schoutens [21]. This paper considers an approach towards buildinga n-qubit gate without using one- and two-qubit gates as building blocks.The protocol is based on resonant driving of eigenstates of a many-bodyHamiltonian. This is applied to the Krawtchouk spin chain. They show theconstruction of the iSWAPN quantum gate which can be efficiently mappedto a NOT gate or iSWAP2 gate with N − 2 controls. The gate universalitytheorem concludes that any n-qubit quantum gate can be constructed froma universal set of one- and two-qubit gates. The strategy in this paper [21]is geared towards constructing n-qubit gates more efficiently.

34

Chapter 4

Geometric phases and fibrebundles

In this section we will elaborate on some of the mathematics behind quan-tum mechanics. Quantum mechanics takes place in a n-dimensional Hilbertspace, which can be denoted as L2(Cn), the complex inner product space ofsquare-integrable functions. First, we will take a look at quantum mechan-ical phases; a simple phase factor, the dynamical phase and the geometricphase. Second, we introduce the concepts of adiabaticity and focus on ageometric phase which has our special interest, the Berry phase. Third,we will consider some mathematical structure underlying the geometric as-pects. We will end this chapter with a short historical overview of geometricquantum computation.

4.1 Quantum mechanical phase

In quantum mechanics a phase is usually defined as a complex scalar withunity norm, so it can be written as a complex exponential eiθ. If we considermultiplying a quantum mechanical state |Ψ〉 with a phase eiθ, so |Ψ′〉 =eiθ |Ψ〉, we can easily see that this does not alter any expectation values.We have that ∣∣Ψ′⟩ = eiθ |Ψ〉 ⇒

⟨Ψ′∣∣O ∣∣Ψ′⟩ = 〈Ψ|O |Ψ〉 (4.1)

for any operator O. So any physical observables are unchanged. In otherwords, the phase can be chosen arbitrarily for a certain quantum mechanicalstate, we say that we define a wavefunction up to a phase. Therefore thephase is neglected in most cases. Berry [7] pointed out that the phase canhave observable consequences if the system undergoes an adiabatic changearound a closed loop. The key lies in interference of the final state with theinitial state.

35

4.2 Adiabaticity and the quantum adiabatic theo-rem

Let us consider a Hamiltonian H continuously dependent on a set of para-meters ~R, H(~R). Suppose that the parameters ~R change as a function oftime: ~R = ~R(t) along a path C. We assume that the change in parametersis adiabatic. An adiabatic process is recognized by a gradual change of ex-ternal conditions, here parametrized by ~R. The adiabatic theorem statesthat a physical system will remain in the same instantaneous eigenstate if agiven change in external parameters (perturbation) acts slow enough and ifthere is a energetic gap between the eigenvalue (energy) of this instantaneouseigenstate and the rest of the spectrum. In more detail, by the adiabatictheorem, if the Hamiltonian changes from Hi to Hf , the nth eigenstate of Hi

changes to the nth eigenstate of Hf [20]. The energetic gap condition trans-lates to the assumption of a spectrum that is discrete and non-degenerate.We can quantify how slow the change in parameters must be, we see thatH ~/∆E must hold, where ∆E is the smallest energy gap in the spectrumof the system. Note that adiabatic evolution is different from perturbationtheory. During adiabatic evolution the changes in parameters can be verylarge, but they occur very slowly. In perturbation theory only small changesin parameter are considered. The interesting case when a system does notreturn to its original state when transported around a closed loop is callednon-holomorphic.

Roughly speaking, the notion of adiabaticity lies on the border of statics anddynamics, taking into account dynamical effects but in the limit of infinitelyslow changes. A typical situation is a system which can be divided intotwo subsystems with completely different timescales; a slow and a fast sub-system. The adiabatic theorem is more general than Hamiltonians with adiscrete and nondegenerate spectrum. This is stated for Hamiltonian whichfulfill the so-called gap condition: a part of the spectrum is separated by anenergy gap from the other parts. The intrinsic time scale is therefore deter-mined by the energy gaps in the spectrum. The time dependence usuallyenters into the physical Hamiltonian via time dependence of some externalparameters. We suppose that these external parameters parametrize somemanifold M . This is where geometry comes in. Adiabatic evolution definesparallel transport of the vector ψ in the spectral bundle along a curve in thebase manifold M . This parallel transport is realized via the adiabatic timeevolution operator UAD(s) [9]. This mathematical interpretation is exploredfurther in section 4.5.

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4.2.1 Adiabatic quantum computing

We want to emphasize the following concept to clear up a possible confusionthat might arise. In this text we consider circuit-based quantum compu-tation and not adiabatic quantum computation. In circuit-based quantumcomputation the circuit consists of quantum gates, which can be representedby unitary matrices. We use the adiabatic theorem only as requirementfor the definition of the Berry phase, to construct such unitary matrices.Adiabatic quantum computing is based on the principle that a system canadiabatically evolve to a solution of a satisfiability problem. Such a problemis finding a variable assignment (a solution) that minimizes some ‘energy’function. The Hamiltonian H is slowly varying and by the adiabatic theo-rem the ground state follows the evolution. The ground state of H(T ), theevolved Hamiltonian, should encode the solution satisfying the problem [15].For noiseless qubits, a theoretical argument can show that adiabatic quan-tum computing is as powerful as circuit-based quantum computing [1]. How-ever, the form presently used is the quantum annealer, which is a noisyversion of adiabatic quantum computing. The equivalence to circuit-basedquantum computing only holds with a high overhead cost in additional phys-ical qubits such that the adiabatic method is executed correctly. This is dif-ferent from the quantum annealers used today such as the D-Wave machine[38].

4.3 Abelian Berry phase

Consider a Hamiltonian H(~R), where the parameters ~R change adiabaticallyas a function of time: ~R = ~R(t) along a path C. We assume that we can findinstantaneous eigenvalues (eigenenergies En) and orthonormal eigenvectors(eigenstates |n〉) such that

H(~R(t)) |n(~R(t))〉 = En(~R(t)) |n(~R(t))〉 . (4.2)

This equation determines the eigenstates |n(t)〉 up to a phase. The timeevolution of the system is described by the Schrodinger equation

i~d

dt|ψ(t)〉 = H(t) |ψ(t)〉 . (4.3)

where we can write the wavefunction of the general state |ψ〉 as a linearcombination of the eigenstates

|ψ(t)〉 =∑n

cn(t) |n(t)〉 . (4.4)

The eigenstates |n(t)〉 = |n(~R(t))〉 satisfies the Schrodinger equation

i~d

dt|n(~R(t))〉 = H(~R(t)) |n(~R(t))〉 . (4.5)

37

We assume in this section that the instantaneous eigenstates are non-degenerate.In the next section we will consider the degenerate case. Let us prepare thewavefunction of the system, |ψ(t)〉, in an initial state |n(~R(t))〉. It is cru-cial that we prepare the system initially in an eigenstate, as by the adiabatictheorem the system will then stay in an instantaneous eigenstate of H(~R(t))along the path. However, the phase is still a degree of freedom, in fact theonly one. During the adiabatic evolution, the phase θ(t) does not need tobe zero, as we will show. So we write

|ψ(t)〉 = e−iθ(t) |n(~R(t))〉 (4.6)

for the wavefunction of the state during the adiabatic evolution along pathC.

Dynamical phase

We know that the phase θ(t) must contain a dynamical factor related to theenergy of the eigenstate. The dynamical phase factor is the result of theintegration of the Schrodinger equation. Consider for a moment

i~d

dt|ψ(t)〉 = H |ψ(t)〉 = En(t) |ψ(t)〉 (4.7)

then we would find the solution

|ψ(t)〉 = exp

(− i~

∫ t

0En(t′)dt′

)|ψ(0)〉 . (4.8)

The phase factor is called the dynamical phase factor

φdyn =1

~

∫ t

0En(t′)dt′. (4.9)

We will see this same factor appear in the next derivation, with the addi-tion of a second phase. So we find that each eigenstate acquires a phasewhich depends on the corresponding eigenenergy. This provides us with a‘traditional’ method for adding a phase to the wavefunction. If we turn on acertain Hamiltonian for a time t then the wavefunctions acquire a (dynami-cal) phase depending on the eigenenergies of this Hamiltonian and this timet.

Geometric phase

The catch is that there exist another phase factor besides the dynamicalphase factor. This geometric phase factor will be the focus in this work.Consider the time evolution of the system by the Schrodinger equation

H(~R(t)) |ψ(t)〉 = i~d

dt|ψ(t)〉 . (4.10)

38

If we assume that eq. (4.6) holds, working this equation out result in

En(~R(t)) |n(~R(t))〉 = ~(

d

dtθ(t)

)|n(~R(t))〉+ i~

d

dt|n(~R(t))〉 . (4.11)

Next, we take the inner product with the bra of the instantaneous eigenstate,〈n(~R(t))|, and use that the eigenstates are normalized, 〈n(~R(t))|n(~R(t))〉 =1.

En(~R(t)) = ~(

d

dtθ(t)

)+ i~ 〈n(~R(t))| d

dt|n(~R(t))〉 (4.12)

We can rewrite this equation and solve for the phase θ(t)

d

dtθ(t) = En(~R(t))/~− i 〈n(~R(t))| d

dt|n(~R(t))〉 (4.13)

θ(t) =1

~

∫ t

0En(~R(t′))dt′ − i

∫ t

0〈n(~R(t′))| d

dt′|n(~R(t′))〉dt′.(4.14)

The first part of phase θ(t) is the dynamical phase, introduced in eq. (4.9).The second part is called the geometric phase, also known as the Berryphase. We define

γn = i

∫ t

0〈n(~R(t′))| d

dt′|n(~R(t′))〉dt′. (4.15)

The evolved state after time t can then be written as

|ψ(t)〉 = e−iθ(t) |n(~R(t))〉 = e−iφdyne−iγn |n(~R(t))〉 . (4.16)

We can write the Berry phase in a form only dependent on the parameters~R and independent of time t,

γn = i

∫ t

0〈n(~R(t′))| d

dt′|n(~R(t′))〉 dt′ (4.17)

= i

∫ t

0〈n(~R(t′))| ∇~R ·

d~R

dt|n(~R(t′))〉 dt′ (4.18)

= i

∫ ~R(t)

~R(0)〈n(~R)| ∇~R |n(~R)〉 · d~R, (4.19)

where ∇~R denotes the gradient in ~R-space. For a closed loop C in ~R-space,~R(0) = ~R(t), the integral may at first sight seem to vanish. However, theintegrand may not necessarily be a total derivative, so we can find a non-trivial geometrical (or Berry) phase.

This rewritten form of the geometrical phase exemplifies one of its features.While the dynamical phase φdyn is explicitly dependent on the elapsed time,and in particular the final time t, the geometrical phase is not. The geomet-rical Berry phase depends only on the path through parameter ~R-space, noton the time or velocity of the path. This fact shows the geometrical natureof the Berry phase.

39

Abelian Berry connection

We can write the Berry phase after a closed loop C (also) as

γn = i

∮C〈n|∇~Rn〉 · d~R (4.20)

and in this Abelian case we can rewrite this expression using Stokes’ theorem

= i

∫∫ [∇~R × 〈n|∇~Rn〉

]· d~a, (4.21)

where ~a denotes an infinitesimal area element of the area enclosed by thepath C. This equality is a special case of Stokes’ theorem in its general form,which states ∫

∂Ωω =

∫Ωdω, (4.22)

where ω denotes a differential form and d the exterior derivative [34].

The general Stokes’ theorem in eq. (4.22) shows the interpretation of theexpression in the Berry phase as a connection. We call

~Ann = 〈n|∇~Rn〉 (4.23)

the Berry connection or Berry (vector) potential. In this case of the non-degenerate eigenstates, the Berry connection is Abelian; ~Ann is a U(1)-valued one-form. Different notations may include a factor i in the definition,as we will see in 4.5. Now we can write

γn = i

∮C~Ann · d~R (4.24)

= i

∫∫ [∇~R × ~Ann

]· d~a (4.25)

While in general multiplication of a quantum state by a U(1) phase is notobservable, the Berry phase is a relative phase between initial and final state.After an adiabatic closed loop the Hamiltonian returns to its initial form,and the initial eigenstate has followed adiabatically. The final state onlydiffers by a phase, which could be measured in an interference experiment.

The Berry phase is a real quantity, adding a phase to the eigenstates |n(~R)〉and not a decay. We will show that Ann = 〈n|∇~Rn〉 is purely imaginary

by showing Re( 〈n(~R)|∇~Rn(~R)〉) = 0. The normalization condition of the

eigenstates 〈n(~R)|n(~R)〉 = 1 for all values of ~R is used. Then the following

40

holds

0 = ∇~R 〈n(~R)|n(~R)〉= 〈∇~Rn|n〉+ 〈n|∇~Rn〉= 〈n|∇~Rn〉

∗ + 〈n|∇~Rn〉= 2 Re( 〈n(~R)|∇~Rn(~R)〉)→ Re( 〈n(~R)|∇~Rn(~R)〉) = 0.

The geometric phase introduced here was first recognized by Michael Berryas a non-trivial quantum mechanical phase in his 1984 paper [7]. In 1956Indian physicist Pancharatnam had already seen the geometric phase bystudying the interference of polarized light beams through crystals [3]. ThePancharatnam phase is now defined by the relative phase between two lightbeams in different polarization states.

If we apply the definitions derived above to states defined by a supersym-metric lattice model, we find that the zero-energy ground states create atrivial dynamical phase. After a path through parameter space, only thegeometric phase contributes.

We have introduced adiabatic evolution in section 4.2 following the adiabatictheorem, which is the corner stone of the Berry phase. We will use thisto define geometric quantum computation. In literature also descriptionsof non-adiabatic schemes exist, where the evolution does not satisfy theadiabatic theorem. More information can be found in section 4.7. A differentconcept, which may be seen as opposite to adiabatic evolution, is quantumquenching. This process is based on a very fast change of parameters andinvestigating the evolution of the system afterwards. Naturally, the systemgets into a superposition of different eigenstates, whereas during adiabaticevolution the eigenstate ‘follows’ the evolution of the Hamiltonian. After aquantum quench the system loses its information about its initial state.

4.4 Non-Abelian Berry phase

In the previous section (section 4.3) we have introduced the Abelian Berryphase, which is acquired after a closed adiabatic path in a system describedby discrete and non-degenerate eigenstates. Let us now consider the case ofdegenerate eigenstates, as this is the case we are interested in for the twodegenerate ground states of the supersymmetric lattice model. While oneof the trademarks of the adiabatic theorem states that the eigenstates donot mix during the path through parameter space, this is not true in (eachof) the degenerate subspaces. During adiabatic evolution the eigenstatesfollow the perturbation of the Hamiltonian, but they are able to ‘rotate’ in

41

the degenerate subspace. As by the adiabatic theorem the states stay in thesame subspace.

The non-Abelian construction of the Berry phase is also known as theWilczek-Zee phase. Wilczek and Zee [53] have shown in their article ‘Appear-ance of Gauge Structure in Simple Dynamical Systems’ that the non-Abelianphase arises in adiabatic evolution [53]. Consider n degenerate levels whichare mapped back onto themselves after adiabatic evolution. For n = 1,a single level, the situation is described by the Abelian Berry phase, themapping is a simple phase multiplication. Assume that a given space ofdegenerate levels does not cross other levels.Consider the time-dependent Schrodinger equation

i∂ψ

∂t= H(~λ(t))ψ. (4.26)

Following the derivation of Wilczek and Zee [53], we choose an arbitrarysmooth set of basis states ψa(t) for the space of degenerate levels, such that

H(~λ(t))ψa(t) = 0 (4.27)

We write the solutions of the Schrodinger equation in eq. (4.26) as

ηa(t) = Uab(t)ψb(t) (4.28)

with the initial condition ηa(0) = ψa(0). Here we assumed adiabaticity. Nowwe want to determine U(t). The solutions to the Schrodinger equation ηa(t)must remain normalized in time, so writing the inner product in bra-ketnotation, we find

〈ηa|ηa〉 = 1 and 〈ηa′ |ηa〉 = 0 (4.29)

d

dt〈ηb|ηa〉 = 0, b = a, a′ (4.30)

〈 d

dtηb|ηa〉+ 〈ηb|

d

dtηa〉 = 0 (4.31)

So using that⟨ηb∣∣ d

dtηa⟩

= 0 and inserting the definition of ηa, we find

0 = 〈ηb|d

dtηa〉 = 〈ηb|

d

dt(Uac)ψc〉+ 〈ηb|Uac

d

dtψc〉 . (4.32)

This equation can be rewritten to

(U−1 d

dtU)ba =

⟨ψb

∣∣∣∣ d

dtψa

⟩≡ Aab, (4.33)

which defines the anti-Hermitian matrix Aab, the non-Abelian Berry connec-tion or Wilczek-Zee connection. This equation in eq. (4.33) can be solved[53] in terms of a path-ordered integral

U(t) = P exp

[∫ t

0A(t′)dt′

]. (4.34)

42

The path-ordering operator is included for the correct evaluation of thematrix exponential of the non-Abelian Berry connection matrix, which isnecessary if connection matrices on different parts of the closed loop do notcommute.

The derivation shown here has some historical value for the definition of thenon-Abelian Wilczek-Zee phase, as being the first to propose the extensionof the (Abelian) Berry phase to degenerate (thus non-Abelian) eigenspaces.We have presented the derivation as it is given in Wilczek and Zee [53]. Analternative derivation based on a different approach is given in the appendix,which is due to Rezakhani et al. [40]. During our calculations in the nexttwo chapters 5 and 6 we will use an approach where we first calculate thenon-Abelian Berry connection matrices A, perform the integration to findthe non-Abelian Berry phase γ and then matrix exponentiate to find thecorresponding unitary U . The necessary definitions will be re-introduced inchapter 5.

4.5 Mathematical background of the non-AbelianBerry phase

The mathematical structure of the Berry phase is in the first place situated ingauge theories. Both the Abelian and non-Abelian case can be interpretedin this manner. In the second place we can look into the Berry phase inthe context of a fibre bundle structure. In the next section we will thinkabout the mathematical structure underlying the procedure described in thisthesis: geometric quantum computation on the supersymmetric lattice.

4.5.1 Gauge structure

Gauge structure in physics depends on the fundamental fact that the physicsof Nature should not depend on how the theory describes it. A centralconcept is the gauge choice. We can choose a gauge in our theoretical de-scription, but the physical (measurable) quantities should not change fordifferent gauge choices [34].

Let us immediately consider the Abelian Berry phase as an example. TheAbelian Berry phase introduced in section 4.3 corresponds to a U(1) gaugestructure. The gauge choice in a system is like choosing the zero of yourcoordinate system. In this case it means that we can choose the phase of thewavefunction arbitrarily. However, as the Berry phase shows, if the Hamil-tonian traverses over a path through parameter space, the phase at the endof this closed path cannot be chosen arbitrarily. An important notion isthat a gauge choice does not influence physical observables. For example

43

the origin of a coordinate system does not change the size of the force be-tween objects and the phase of a wavefunction in quantum mechanics doesnot change expectation values.

Another well-known example is the theory of electromagnetism. We considerflat space with c = 1. The magnetic field ~B and the electric field ~E can beexpressed in terms of the vector potential Aµ, µ = 0, . . . , 3. Introducing the

four-vector notation Aµ = (φ, ~A), the expressions are

~B = ∇ ~A, (4.35)

~E = ∂∂t~A−∇φ. (4.36)

The physics of electromagnetism is described by the Maxwell equations andone can show that these equations are invariant under the gauge transfor-mation

Aµ → A′µ = Aµ + ∂µχ, (4.37)

where χ denotes a scalar function. This is an example of an Abelian gaugetheory [34]. To make the discussion complete let us also introduce the elec-tromagnetic field tensor Fµν , which is defined by

Fµν = ∂µAν − ∂νAµ (4.38)

with the explicit relations Ei = F0i and Bi = −12εijkF

jk, where εijk is theLevi-Civita symbol. The electromagnetic field tensor Fµν is invariant underthe gauge transformation in eq. (4.37).

These two examples are part of general gauge theory. The two main ele-ments in the theory are the connection A and the curvature F . These areclosely related to the vector potential and field tensor introduced before.The components of the connection A and the curvature F can be written intensor notation as Aµ and Fµν , respectively. In other words the connectionA can be written as

A = Aµ dxµ (4.39)

= Aθ dθ +Aφ dφ (4.40)

where the second expression assumes xµ = θ, φ, which is the case we will con-sider in this work. The relation between the connection A and the curvatureF is given by the exterior derivative

F = dA (4.41)

= ∂∂xµAν dxµ ∧ dxν (4.42)

= Fµν dxµ ∧ dxν (4.43)

44

The wedge product ∧ is defined as the totally antisymmetric tensor product.This form shows directly that F is invariant under transformations. Inmore detail, if we take the gauge transformation in eq. (4.37), we find thatA′ν = Aν + ∂νχ results in

∂

∂xµA′ν =

∂

∂xµ(Aν + ∂νχ) = ∂µAν +

∂

∂xµ∂

∂xνχ. (4.44)

However, the second term ∂∂xµ

∂∂xν χdxµ ∧ dxν must vanish as ∂

∂xµ∂∂xν χ is

symmetric, but dxµ ∧ dxν is anti-symmetric with respect to interchangingµ↔ ν.

Aharonov-Bohm effect

The electric and magnetic fields ~E and ~B are defined in terms of Fµν and areat the center of electromagnetism. In combination with quantum mechanicsthere are situations where Fµν is not sufficient to describe the phenomena,

and the vector potential and scalar potential A = (A0, ~A) are needed. Oneof common examples is the Aharonov-Bohm effect. We consider a beamof electrons split in two and passing a solenoid of infinite length on bothsides. A shield prevents electrons from penetrating into the solenoid. Sothe electron do not feel a magnetic field during their flight. The result isan interference pattern on the screen. We have to conclude that the gaugefield Aµ is important here. To be more precise, the wavefunction depends on

the vector potential ~A even though ~B = 0. We can distinguish two paths,i = I, II, passing the solenoid on different sides. If the vector potential isnon-zero, the wavefunction is given by the gauge-transformed form

ψAi (~r) ≡ exp

(ie

∫ ~r

P

~A(~r′) · d~r′)ψi(~r) (4.45)

where P is a reference point far from the apparatus. If we consider a super-position of wavefunctions ψAI +ψAII , which satisfy ψAI (P ) = ψAII(P ), then wefind the following result. The amplitude at a point Q on the screen is givenby

ψAI (Q) + ψAII(Q) = exp

(ie

∫γI

~A(~r′) · d~r′)ψI(Q) + exp

(ie

∫γII

~A(~r′) · d~r′)ψII(Q)

= exp

(ie

∫γII

~A(~r′) · d~r′)[

exp

(ie

∫γ

~A(~r′) · d~r′)ψI(Q) + ψII(Q)

](4.46)

where γ = γI − γII . We see that the interference wavefunction dependsexplicitly on the vector potential ~A [34].

45

Gauge potential

The anti-Hermitian matrixA as defined (for a particular basis ψa) in eq. (4.33)plays the role of gauge potential. The specific form Aab in eq. (4.33) dependsupon the choice of basis ψa. However, A depends only on the geometry ofthe space of degenerate levels. For a different choice of basis, we find thatA transforms as a proper gauge potential. ψ′(t) = Ω(t)ψ(t) implies thatA′(t) = d

dt(Ω)Ω−1 + ΩAΩ−1. The path-ordered integral (eq. (4.34)) trans-forms trivially under this new choice of basis [53].

Consider the space with the parameters ~λ = (λ1, . . . , λµ, . . . ) as coordinates.Then we can define the gauge potential Aµ on this space by the explicitexpression Aµ =

⟨ψ∣∣ ∂∂λµψ

⟩[53]. Using this new definition we can rewrite

eq. (4.34) as follows

U(t) = P exp

[∫ t

0Aµ(~λ(t))dλµ

](4.47)

and for a closed path one obtains a Wilson loop

U = P exp

[∮Aµdλµ

]. (4.48)

We find that the ordered integral only depends on the path, not on theparametrization of the path [53].

The theory of (gauge) connections is also related to the theory of Riemanniangeometry. In the treatment of curved spaces the connection specifies howtensors are transported along a curve. An important notion is the definitionof parallel transport, which tells us how to compare vectors at differentpoints in the manifold. The derivative of a vector field V = V µeµ naivelygiven by ∂

∂xν Vµ is replaced by the covariant derivative of V defined by

∇νV µ =∂

∂xνV µ + V λΓµνλ. (4.49)

If the manifold is endowed with a metric, there is a preferred choice of Γµνλ,called the Levi-Civita connection. We can also say that the connection co-efficients Γµνλ specify how the basis vectors change from point to point onthe manifold. Consider a curve in a manifold M , then the parallel transportof a vector X along the curve is given by the condition ∇VX = 0 where Vis the tangent vector to the curve. If the tangent vector V is itself paralleltransported along the curve, if ∇V V = 0, then the curve is called a geodesic.It can be seen as the ‘straightest possible curve’. This is an important objectin a theory of Riemannian geometry such as general relativity [34].

46

Non-Abelian gauge theory

So far we have talked about Abelian gauge theories, such as the transfor-mations in electromagnetism and transformations based on the U(1) group.Yang and Mills (1954) introduced non-Abelian gauge transformations, whichhave since taken a central role in elementary particle physics. For non-Abelian gauge theories the gauge potential or Yang-Mills gauge field Aµtakes values in the Lie algebra of a compact semi-simple Lie group G. Stan-dard examples for G are SO(N) or SU(N), the special orthogonal group orthe special unitary group of dimension N . In other words the gauge poten-tial Aµ can be expanded in terms of the anti-Hermitian generators Tα.These generators satisfy the commutation relations

[Tα, Tβ] = fγαβTγ (4.50)

where the numbers fγαβ are called the structure constants of Lie group G.The terminology of generators is related to the fact that an element U of Gnear the unit element can be written as

U = exp(−θαTα) (4.51)

Looking back at the notation in components, the curvature in the non-Abelian case is given by

Fµν = ∂µAν − ∂νAµ + [Aµ, Aν ], (4.52)

where [Aµ, Aν ] denotes the commutator. For Abelian connections this com-mutator is zero.

Differential forms

We have introduced the notations of a connection A and curvature F re-lated by the exterior derivative d. Let us consider the general case. Animportant class of tensor fields on a differential manifold are the differentialforms. A antisymmetric tensor of the type (0, k) is called a differential formof order k (or simply a k-form). The two basic operations are the wedgeproduct and exterior derivative. Differential forms occur implicitly in allbranches of physics as natural objects appearing as integrands of line, sur-face, volume and n-dimensional integrals. The known, convenient notationdV = dx1 . . . dxn actually denotes an n-form dV = dx1 ∧ · · · ∧ dxn, withwedge product ∧ [9]. This definition leads for example to the general formof Stokes’ theorem ∫

Mdω =

∫∂M

ω (4.53)

47

First Chern number

An example for a topological invariant we will consider is the first Chernnumber c1. This quantity can be calculated from the expression of the Berrycurvature, which is a gauge invariant quantity defined in terms of the Berryconnections. In general, the Chern invariant is rooted in the mathematicaltheory of fiber bundles, which will be discussed in section 4.5.3. If we knowthe explicit form of the Berry curvature (from the explicit form of the Berryconnections), which is the case in this work, we can find the first Chernnumber by taking the trace of the Berry curvature and integrating this overthe whole parameter space [9]. In mathematical notation

c1 =i

2π

∫P

TrF (4.54)

4.5.2 Projective spaces

The theory of projective spaces is applicable to quantum mechanics, whichmakes it interesting to consider in this work. We will also interpret the de-generate subspace of ground states in the context of Grassmann manifolds.

The complex projective space CPn defines a subspace of Cn+1. The n-tuplez = (z0, z1, . . . , zn) ∈ Cn+1 determines a complex line through the origin (ifwe take z 6= 0). Let us define an equivalence relation ∼ by the following:z ∼ w if ∃a ∈ C, a 6= 0 such that w = az. Then the complex projectivespace is defined by CPn ≡ (Cn+1 − 0)/ ∼. On the complex projectivespace [z0, z1, . . . , zn] are called the homogeneous coordinates, where z =(z0, z1, . . . , zn) is identified with λz (for λ 6= 0). The equivalence classesof the relation ∼ are sometimes called (projective) rays. For example thecomplex projective plane CP 2 is the two-dimensional complex projectivespace, which is described by three complex coordinates.A different construction of the complex projective space is given by thefollowing. Consider the sphere

S2n+1 = ~z ∈ Cn+1||~z| = 1

and the equivalence relation ∼ for two points ~z1 and ~z2 from S2n+1

~z1 ∼ ~z2 if and only if ∃α ∈ R, ~z2 = eiα~z1

The set of equivalence classes S2n+1/ ∼ is the complex projective spaceCPn [9]. We have a natural action of a Lie group U(1) on S2n+1, that isU(1)× S2n+1 → S2n+1 defined by (eiα, ~z) → eiα~z for any ~z ∈ S2n+1. Thecomplex projective space CPn coincides with the space of orbits of the aboveU(1) action

CPn ∼= S2n+1/U(1) ∼=U(n+ 1)

U(n)× U(1). (4.55)

48

Projective Hilbert space

In the context of quantum mechanics the projective Hilbert space is an ex-ample of a complex projective space. Given a complex Hilbert space H, theprojective Hilbert space P (H) is the set of equivalence classes of vectors inH. The general definition of a Hilbert space H is a real or complex innerproduct space which is a complete metric space with respect to the norminduced by the inner product.

If the complex Hilbert space H = Hn is finite-dimensional, one acquires theequality P (Hn) = CPn−1. For example, the Hilbert space describing thestate of one qubit, H2, can be identified with the complex projective lineCP 1. This object is more commonly known as the Bloch sphere. We haveintroduced the Bloch sphere in section 3.1 (figure 3.1) which is a visual-ization of the space of a two-level quantum system, such as a qubit or anatomic structure.

The physical significance of this projective structure is the fact that wave-functions ψ and λψ correspond to the same physical states, for any value ofλ 6= 0. It is convention to choose a normalized wavefunction, such that ψhas unit norm 〈ψ|ψ〉 = 1, this corresponds to choosing ψ in an equivalenceclass (or ray). However this norm constraint does not completely determineψ within the ray. A global phase λ = exp(iφ), with unity norm, can alsobe added. The reason is as follows. In the standard approach to quantummechanics, pure quantum states are represented by vectors in a complexHilbert space H. Each vector ψ ∈ H describes a state by a collection ofexpectation values.

〈ψ|A |ψ〉〈ψ|ψ〉

.

For this reason, two vectors describe the same physical state if and only ifthey are linearly dependent. If we normalize the state vector, there is stilla freedom to choose an overall phase factor eiα. Hence, such a phase factorhas no physical meaning. For this reason we may equivalently representquantum states as one-dimensional projectors in H

Pψ = |ψ〉 〈ψ|

However, the relative phase controls a key effect of quantum mechanics;quantum interference. A superposition of two states leads to a interferenceformula which enables one to measure the relative, but not the overall, phase[9].

49

Grassmann manifolds

Complex Grassmann manifolds are generalizations of complex projectivespaces. An element of a complex projective space CPn is a one-dimensionalsubspace of Cn+1. The Grassmann manifold, denoted by Gk,n(C) is the setof complex k-dimensional subspaces of Cn. Therefore CPn = G1,n+1(C)holds. The definition in terms of equivalence classes can be seen as follows[34]. Let Mk,n(C) be the set of k × n matrices of rank k (k ≤ n). Fortwo matrices A,B ∈ Mk,n(C) let us define an equivalence relation. A ∼ Bif there exists a g ∈ GL(k,C) such that B = gA. So we can identify theGrassmann manifold Gk,n(C) by the coset space Mk,n(C)/GL(k,C).[34]

Sjoqvist et al. [45] connects geometric quantum computation with complexGrassmann manifolds. The tripod atomic structure defined by two compu-tational states |0〉 and |1〉, an ancilla state |a〉 can be identified as the Grass-mann manifold G2,3(C). The computational manifold is two-dimensionalinside the three-dimensional ground state manifold. The tripod structureis also equivalent to the ground states on the supersymmetric triangle lat-tice: In the three-dimensional space of one-fermion states we construct twodegenerate ground states.

4.5.3 Fibre bundle structure

The mathematical concepts of the Berry phase can be embedded in thetheory of fibre bundles. A fibre bundle can be thought of as a topologicalspace which looks locally like a direct product of two topological spaces.Let us start with the mathematical definition as stated in Nakahara [34]. Adifferentiable fibre bundle (E, π,M,F,G) consists of the following elements:

1. A differentiable manifold E called the total space.2. A differentiable manifold M called the base space.3. A differentiable manifold F called the fibre (or typical fibre)4. A surjection π : E → M called the projection. The inverse imageπ−1(p) = Fp ∼= F is called the fibre at p.

5. A Lie group G called the structure group, which acts on F on the left.6. A set of open covering Ui of M with a diffeomorphism φi : Ui×F →π−1(Ui) such that π φi(p, f) = p. The map φi is called the localtrivialization since φ−1

i maps π−1(Ui) onto the direct product Ui×F .7. For φi(p, f) = φi,p(f), the map φi,p : F → Fp is a diffeomorphism. OnUi ∩Uj 6= ∅, we require that tij(p) ≡ φ−1

i,p φj,p : F → F be an elementof G. Then φi and φj are related by a smooth map tij : Ui ∩ Uj → Gas φj(p, f) = φi(p, tij(p)f). The maps tij are called the transitionfunctions.

The shorthand notation Eπ−→M is common for a fibre bundle (E, π,M,F,G).

If all the transition functions can be taken to be identity maps, the fibre

50

bundle is a trivial bundle. A trivial bundle is equal to the direct productM × F .Consider a fibre bundle E

π−→M , then a section (or cross section) S : M → Eis a smooth map which satisfies π s = IM . So s(p) = s|p is an element ofFp = π−1(p). The set of sections on M is denoted by Γ(M,F ).

A fibre bundle Eπ−→M whose fibre is a vector space is called a vector bun-

dle. So F equals Rk or Ck. The transition functions belong to GL(k,R)or GL(k,C) respectively, since they map a vector space onto another vectorspace of the same dimension isomorphically. A vector bundle whose fibre isone-dimensional, F = R or C, is called a line bundle. The structure groupis then Abelian [34]. As an example, non-relativistic quantum mechanicsin three-dimensional space is associated to the (trivial) complex line bundleL = R3× C. The wavefunction ψ(x) is equal to a section of L.

A principal bundle has a fibre F which is identical to the structure groupG. A principal bundle P

π−→ M is also denoted by P (M,G). Given a prin-cipal bundle, we may construct an associated fibre bundle. Let G act ona manifold F on the left. We define an action of g ∈ G on P × F by(u, f)→ (ug, g−1f) where u ∈ P and f ∈ f . Then the associated fibre bun-dle is an equivalence class P×F/G in which two points (u, f) and (ug, g−1f)are identified.

It is possible to define a connection on a principal bundle. This is realizedconcretely by the introduction of a connection one-form. This object has alocal form known as a gauge potential. A connection on a principal bundlenaturally defines a covariant derivative in the associated vector bundle. Aloop γ : [0, 1] → M at p = γ(0) = γ(1) generically defines a transformationτγ : π−1(p)→ π−1(p) on the fibre. For a point u ∈ P with π(u) = p considerthe set of loops Cp(M) at p

Cp(M) ≡ γ : [0, 1]→M |γ(0) = γ(1) = p. (4.56)

The set of elements

Φu ≡ g ∈ G|τγ(u) = ug, γ ∈ Cp(M) (4.57)

is a subgroup of the structure group G and is called the holonomy groupat u [34]. We can now write the following. Let Ai = Aiµ dxµ be a gaugepotential over Ui and γ a loop in Ui. Let τγ(u) = ugγ , u ∈ P, gγ ∈ G. Thenwe can write

gγ = P exp

(−∮γAiµ dxµ

)(4.58)

51

The proper phase space of a quantum system is the space of rays in H:P(H) = H/ ∼, which is called a projective Hilbert space, as introducedin the previous subsection. We can define a canonical projection Π : H →P(H), such that [ψ] = Π(ψ) corresponds to a ray in H passing through ψ.The above construction defines a vector bundle over P(H) with a typicalfibre F = C, and a structure group G = GL(1,C). The fibres are one-dimensional, so this is called a complex line bundle.If we consider a (n + 1)-dimensional Hilbert space H ∼= Cn+1, the corre-sponding quantum phase space P(H) is a complex projective space CPn =S2n+1/U(1), where S2n+1 is a unit sphere in Cn+1 [9].

Connections on fibre bundles

In this section the (Abelian) Berry phase is placed in the context of connec-tions on fibre bundles. We consider a Hamiltonian dependent on parameters~R, where ~R changes adiabatically as a function of time, ~R = ~R(t). After aclosed loop in parameter ~R-space, the system acquires the Berry phase inaddition to the dynamical phase. Simon [43] first recognized the geometricmeaning of the Berry phase. It originates from the holonomy in the para-meter space. Consider a manifold M describing the parameter space and let~R = (R1, . . . , Rk) be the local coordinate. A quantum state |n; ~R〉 denotesthe normalized nth eigenstate of the Hamiltonian H(~R) at each point ~Rof M . We can distinguish a state |n; ~R〉 from a state exp(iφ) |n; ~R〉 so aphysical state is expressed by an equivalence class

[ |~R〉] = g |~R〉 |g ∈ U(1). (4.59)

In other words, at each point ~R of M we have a U(1) degree of freedom:we have principal U(1) bundle over the parameter space M , P (M,U(1)).The projection is given by π(g |~R〉) = ~R. Fixing the phase of |~R〉 at eachpoint ~R ∈ M amounts to choosing a section. In this language of bundlesthe Berry connection is defined by

A = Aµ dRµ ≡ 〈~R| d |~R〉 = 〈~R| (d |~R〉) = −(d 〈~R|) |~R〉 (4.60)

where d = (∂/∂Rµ) dRµ denotes the exterior derivative in ~R-space. TheBerry connection is anti-Hermitian. A component of the Berry connectioncan be written as Aµ(~R) = 〈~R| ∂

∂Rµ |~R〉. The field strength F of A is calledthe Berry curvature and is given by

F = dA = (d 〈~R|) ∧ (d |~R〉) =

(∂ 〈~R|∂Rµ

)(∂ |~R〉∂Rµ

)dRµ ∧ dRν . (4.61)

The Berry phase can be written as

η(t) = i

∫ t

0ds

dRµ

ds〈~R(s)| ∂

∂Rµ|~R(s)〉 = i

∫ ~R(t)

~R(0)〈~R| d |~R〉 (4.62)

52

and Nakahara [34] shows that it is a holonomy associated with the connec-tion on P (M,U(1)).

Therefore the geometric phase finds a natural description in terms of certainfibre bundles. The connection is a geometric object which allows one tocompare different fibres of the bundle and to transport elements from onefibre to another. It plays a crucial role in the theory of geometric phases. Aconnection provides us with a rule of parallel transporting the fibre F alongthe path γ [9].In more general terms the geometric phase can be interpreted as follows.Simon [43] first observed that the Berry phase has an elegant mathematicalinterpretation as the holonomy of a certain connection in the appropriatefibre bundle. The base space is a manifold of external parameters M andthe fibre at x ∈ M is a complex line Hn(x), that is the nth eigenspace ofHamiltonian H(x). Let us first assume the nth eigenvalue is nondegenerate.For normalized vectors in Hn, we can consider a fibre at x defined by

Fx = eiα |n(x)〉 |α ∈ R ∼= U(1). (4.63)

This construction results in the principal U(1)-bundle with the total space

P =⋃x∈M

Fx.

A vector |h〉 is called horizontal if it is orthogonal (in the sense of the scalarproduct in H) to the corresponding fibre Fx. This provides a natural wayof defining a connection on the fibre bundle. This connection is often calleda Berry-Simon connection. The Berry phase factor for a closed curve corre-sponds to an element from the holonomy group of the Berry-Simon connec-tion [9].

Spin-12 in a magnetic field

A common example is a spin-12 particle in a magnetic field. The parameter ~R

corresponds to the applied magnetic field. Consider the Hamiltonian givenby H(~R) = ~R · ~σ. This two-level system has eigenvalues ±|~R|. For the de-generate case of ~R = 0, the singularity behaves like a magnetic monopole in~R-space. This analogy can be made clear by introducing polar coordinates,~R = (R sin θ cosφ,R sin θ sinφ,R cos θ). We find that we cannot define aglobal connection in all ~R-space. If we fix the amplitude R of the magneticfield, the parameter space is restricted to S2. At each point ~R of S2, thestate has U(1) phase degree of freedom. This corresponds to the U(1) bun-dle P (S2, U(1)), which also describes the magnetic monopole. The fibre at~R consists of the equivalence class [ |R〉] defined in eq. (4.59) correspondingto the nth eigenstate |n; ~R〉. The projection π maps a state to the corre-sponding parameter, so π : eiα |~R〉 → ~R ∈ S2. The bundle is not trivial as

53

it cannot be described by a single connection.

The non-Abelian Berry phase or Wilczek-Zee factor may also be reformu-lated as a holonomy element in an appropriate fibre bundle. Each pointin parameter space, x ∈ M , gives rise to an N -dimensional spectral spaceHn(x). Hence we define the following spectral bundle over M

E(n) =⋃x∈MHn(x), (4.64)

with a typical fibre F = CN . The nth spectral bundle E(n) is an associatedvector bundle to a U(N)-principal bundle P (n). This principal bundle isconstructed by first fixing N unit vectors φi ∈ Hn(x) and defining a fibre as

F (n)x =

∑b

Uabφb|U ∈ U(N) ∼= U(N). (4.65)

The U(N)-principal bundle over M is then defined by

P (n) =⋃x∈M

F (n)x . (4.66)

Holonomic quantum computation

In the framework of holonomic quantum computation the requirements forimplementing the universal quantum computer can be expressed in termsof the availability of closed paths in M . Universality is the (experimental)capability of driving the control parameters along a minimal set of loops,which generate the basic quantum gates. Then it should be possible to ap-proximate any unitary operation. The holonomy group is a subset of theunitary group U(n). For equality between the groups the connection Ais called irreducible. Therefore we can reformulate the universality condi-tion as: holonomic quantum computation is universal if the correspondingconnection is irreducible. The Ambrose-Singer theorem implies that whenF spans the whole Lie algebra u(n) then the connection A is irreducible.By taking the n-dimensional complex projective space as the control para-meter space M , in principle, F will span the whole algebra u(n), hence thecorresponding adiabatic connection is irreducible and holonomic quantumcomputation is realized [9].

4.6 Mathematical interpretation of geometric quan-tum computation on the supersymmetric lat-tice

In this chapter we have introduced the mathematical concepts relevant forgeometric quantum computation. In this section we want to introduce and

54

|1〉

|0〉

Figure 4.1: The Bloch sphere S2 representing the state space of a singlequbit.

explain the mathematical structure which will be (implicitly) dealt with inthis work.

The state space of a single qubit is in the first place equal to C2, as |ψ〉 =α |0〉+ β |1〉, with α, β ∈ C correspond to all possible states. In other wordsthe Hilbert space H is the two complex dimensional space C2.

However, as we have stated two quantum states are equivalent if they onlydiffer by a complex 0 6= c ∈ C. By considering the projection of |ψ〉 toa representative [|ψ〉], we naturally construct the projective Hilbert spaceP(H). This projection defines a vector bundle over P(H) with a typicalfibre F = C, and a structure group G = GL(1,C). The fibres are one-dimensional, so this is a complex line bundle by definition.

We can also consider the space of normalized representatives of the quan-tum states instead. So the space is reduced to CP 1 ∼= S2, as the complexprojective line is equivalent to the two-sphere. This definition of the statespace of a single qubit is also known as the Bloch sphere representation,which is shown in figure 4.1.

The complex projective line corresponds to the terminology of the quantumphase space P(H). Note that the normalization condition does not constrainthe freedom of adding a (global) phase factor eiΦ ∈ U(1). Therefore wecan interpret the quantum phase space (or projective Hilbert space) as aprincipal U(1) bundle over the Bloch sphere S2; P (S2, U(1)). Note alsothat U(1) ∼= S1, so P (S2, S1) is an equivalent notation.

By the equivalence of CPn = S2n+1/S1, for the complex projective lineCP 1 we find that CP 1 = S3/S1. However, we have also stated that CP 1 ∼=S2 (remember the Bloch sphere). Combining these two results results inS3/S1 ∼= S2 which defines a Hopf fibration S3 → S2.

The definitions can be extended to the two-qubit state space with the stan-dard basis |00〉 , |01〉 , |10〉 , |11〉. The Hilbert space corresponds to the fourdimensional space C4. We can also consider the projective Hilbert space

55

λ2

λ3

λ1

Figure 4.2: The staggering parameter space for ~λ ∈ S2.

P (H4) = CP 3, which can be written in terms of a complex projective space.The complex projective space CP 3 is equivalent to S7/S1. It can be shownthat if the the two-qubit is not entangled, if the separability condition issatisfied, this structure produces the Hopf fibration S7 → S4 [9].

Another important space we consider in this work is the space of stag-gering parameter. In the definition of the supersymmetric model on thetriangle and hexagon we consider staggering modulo three sites, so the stag-gering parameter will (in general) be defined by three complex components~λ = (λ1, λ2, λ3) ∈ C3. We impose a constraint of unit norm on the stagger-ing vector ~λ. Next, we make further constraining choices for the staggeringparameter. In the first case we take the staggering parameter to be real(and still normalized), so ~λ ∈ S2. It is natural to parametrize by the (real)spherical angles θ, φ. We imagine the staggering parameter to be a vector inthe origin pointing to the point (θ, φ) on the 2-sphere S2. In the second casewe take the staggering parameter to have a total of three real dimensionsand again impose the normalization constraint. Again the staggering para-meter is parametrized by the two (real) spherical angles θ, φ. The staggeringparameter space is shown in figure 4.2.

For the case of the bow tie lattice, we introduce an extra staggering para-meter, µ. We choose to take its value real and positive; µ ∈ R>0. Notethat the situation of two uncoupled triangles corresponds to the case µ = 1.For the bow tie lattice the staggering parameter space is effectively givenby S2 × [0,∞), which is the Cartesian product of the two-sphere and theinfinite half-line.

Lattice spaces

Let us consider the relevant lattice subspaces. We are particularly interestedin the two-dimensional manifold of the degenerate ground states, for itsinterpretation as computational subspace. However, it is also importantrealize the differences between the lattice ground state manifold and thecomputational manifold. The spaces considered here are vector spaces; they

56

are spaces of quantum mechanical states. Let us first consider the trianglelattice. The ground state manifold is a two-dimensional subspace of thespace of one-fermion states which is spanned by |1〉L , |2〉L , |3〉L for arbitraryvalues of the staggering parameter. The computational manifold for a singlequbit is spanned by |0〉 , |1〉. In our analysis we have defined a direct relationbetween the ground states and the computational basis states in a singlepoint in the staggering parameter space, the North Pole (θ = 0), wherethe ground state manifold is spanned by |1〉L and |2〉L. By definition thelattice ground states are dependent on the staggering parameters and willchange during a path in staggering parameter space. There is an interestingcorrespondence between the three-dimensional space of one-fermion statesand the qubit space spanned by |0〉 , |1〉 , |a〉, where |a〉 denotes an ancilla.This notation is for example used in geometric quantum computation in thetripod level scheme [44, 29]. Both spaces now consist of a two-dimensionalspace of interest embedded in a three-dimensional space: ground states inthe space of one-fermion states and computational states in the space withthe ancilla added.

We consider the two-dimensional vector spaces spanned by the degenerateground states for all values of the staggering parameter. These spaces aresubspaces of the three-dimensional vector space spanned by |1〉L , |2〉L , |3〉L.The structure of the space of all such subspaces is equivalent to a Grass-mannian manifold, G3,2(C) to be precise. In context of geometric quantumcomputation this identification is made by Lloyd [31]. Wilczek and Zee [53]identify the subspace of two degenerate levels in a three-dimensional spaceof energy levels by SU(3)/(SU(2) ⊗ U(1)) = CP 2. The equality can forexample be found from the correspondence in eq. (4.55), as stated in Chrus-cinski and Jamiolkowski [9]. For the case of n = 2 in eq. (4.55) we find thespecific equality stated in Wilczek and Zee [53].

So far we have considered the space of lattice states of the triangle, let uscontinue to the other lattices we use in this work. For the hexagon latticethe supersymmetric ground states are situated in the space of two-fermionstates. Therefore the ground state manifold is a two-dimensional (vector)subspace of a nine-dimensional vector space. Again for a path in staggeringparameter space this subspace moves through the space of all two-fermionstates. Equivalently to the triangle lattice, we interpret this space of allsubspaces as a Grassmannian manifold G9,2(C).

The analysis can be repeated for the case of the bow tie lattice. Conformexpectations, the bow tie lattice can accommodate four degenerate groundstates. The ground states are part of the nine-dimensional space of two-fermion states on the bow tie. Consequently, we interpret this space as theGrassmannian manifold G9,4(C).

57

Quantum gates

The matrix representations of the one-qubit quantum gates are generally el-ements of the two-dimensional unitary group U(2). By consideration of thespecial unitary group SU(2), with unit determinant, we can also conservenormalization of the quantum states. Our construction of the possible uni-tary gates is based on the non-Abelian Berry phase. So the unitary matricesare found by matrix exponentiation of anti-Hermitian matrices found fromthe connections. For the first choice of the staggering parameter, defined bya unit vector on the sphere, we actually find the group of two-dimensionalrotations SO(2), the special orthogonal group. This corresponds to the realanalogue of SU(2) and note that SO(2) ∼= U(1) ∼= S1. For this choice thestates are rotated in the two-dimensional space. To be able to achieve singlequbit gate universality, we consider also a second staggering choice. Thisstaggering choice produces a diagonal complex matrix, which does not com-mute with a generic element of SO(2). For this choice the states acquired arelative phase, one could say the space is split in two one-dimensional sub-spaces. By the theorem introduced in section 3.7 in eq. (3.28), we find thatthese two matrices are sufficient to find any element from SU(2). Theseresults are shown in chapter 5.

The matrix representations of two-qubit quantum gates are elements of thefour-dimensional special unitary group SU(4). Again we base the construc-tion on a scheme with the non-Abelian Berry phase. We consider splittingthe four-dimensional space into a three-dimensional subspace and a one-dimensional subspace, according to the eigenbases of a mirror operator onthe lattice. We find that the one-dimensional subspace is invariant underthe adiabatic closed path in staggering parameter space. We also find thegenerating group of the three-dimensional subspace is spanned three (out ofeight) generators of SU(3). This set together does not span a subgroup ofSU(3). These results are shown in chapter 6.

Map between spaces

We can also consider maps between the relevant spaces introduced here.Naturally, we can think about the projection maps defined in fibre bundles.An interesting topic is the map between the path space of the staggeringparameter space and the space of ground states for the corresponding valueof the staggering. This map is schematically shown in figure 4.3. As weconsider adiabatically transversing a closed loop in the staggering parameterspace, this map will define a closed loop in state space. At each point ofthe path in state space the space of ground states is two-dimensional. Wecould consider it to be equivalent to the complex project line CP 1 ∼= S2

(the Bloch sphere). Note that each path in the staggering parameter space

58

λ2

λ3

λ1

S2CP 1

Figure 4.3: Schematic view of the map between the path in staggering para-meter space and the space of lattice states.

maps to a different set of CP 1, as the ‘basis’ is directly dependent on thestaggering parameter. The non-trivial action of the adiabatic closed path instaggering parameter space, i. e. the non-Abelian Berry phase, is related tothe transformation which is applied to the CP 1 spaces over the closed loopin state space.

4.7 Literature review of geometric quantum com-putation

In this section we want to highlight some literature describing experimen-tal approaches to geometric (or holonomic) quantum computation. We willpresent the main points from these papers chronologically. We want to gaininsight in the research into geometric quantum computation and find inspi-ration for an implementation of geometric quantum computation with thesupersymmetric lattice models.

4.7.1 Zanardi and Rasetti (1999)

Zanardi and Rasetti [55] consider holonomic quantum computation, which isbased on the non-Abelian geometric phase. The space of control parametersM defines a family of Hamiltonians Hλλ∈M. They consider a (smooth)path γ : [0, T ] → M defining an adiabatic loop in M. For a closed loopγ(T ) = γ(0) holds and it is assumed that the Hamiltonian family has thesame degeneracy structure along the loop. This defines a smooth family ofHamiltonians H(t) = Hγ(t) along with a unitary Uγ .

For general Hamiltonians degeneracy is a singular case, due to the symmetryconstraints it involves, and non-degeneracy is generic. Therefore the degen-eracy requirement for the existence of non-Abelian holonomies is stringent

59

from a geometric point of view. On the other hand, physical systems oftenprovide the required symmetries for having degenerate eigenspaces.

Zanardi and Rasetti [55] show that two generic loops generate a universalset of gates over the quantum code. Their proof is based on the holonomygroup of the connection A. Define the loop space by

Lλ0 = γ : [0, 1]→M|γ(0) = γ(1) = λ0 (4.67)

This implies physically that, as long adiabaticity holds, the evolution doesnot depend on the rate at which γ is traveled but just on its geometry. Itis shown that the set Hol(A) = ΓA(Lλ0) is a subgroup of U(n), known asthe holonomy group of the connection A. Then it is shown that the con-nections associated with non-Abelian geometric phases are irreducible, suchthat Hol(A) = U(n). An argument equivalent to Lloyd [32] then shows thata set of two generic unitaries Ui = ΓA(γi) results in G = Hol(A) = U(N):the closure of the set of transformations G obtainable by composing the Ui(along with their inverses) equals the set of unitary matrices of dimensionN .

4.7.2 Pachos et al. (1999)

The conceptual framework described by Pachos et al. [37] is referred to asholonomic quantum computation and builds upon the work of Zanardi andRasetti [55]. The unitary transformations for computation are realized bythe holonomy of the connection. The connection describes the geometry ofthe bundle of eigenspaces associated with parameter space. Quantum in-formation is encoded in a degenerate eigenspace of the Hamiltonian. TheHamiltonian is part of a family F parameterized by λ ∈M. The λ’s repre-sent the control parameters, which can be driven to manipulate the codingstates. Pachos et al. [37] choose CPn as control parameter manifold. Con-sider a loop C in the control manifold and assume the evolution along theloop is adiabatic. One finds that after the control process the initial stateacquires a dynamical phase and a holonomy ΓA(C) ∈ U(n). Its appearanceaccounts for the nontriviality of the bundle of eigenspaces over M. Theset of holonomies of all possible loops C for a connection A is called theholonomy group Hol(A) = ΓA(C)C ⊂ U(n). If Hol(A) = U(n) holds theconnection A is called irreducible and all unitary operations can be gener-ated.

For generating a quantum gate g ∈ U(n) one has to determine a loop Cgin M such that ΓA(C) = g. To tackle this problem, Pachos et al. [37]consider specific families of loops which lie in two-dimensional submanifolds.By smart choice of the positions of the two-dimensional planes one can setconnections to zero and avoid the path ordering problem. They show that

60

four families can be identified, producing the Pauli matrices and the identity,which form a basis for u(2). They extend the general scheme to a two-qubitsystem and show all possible U(4) rotations can be obtained. Both singlequbit and two-qubit rotations are constructed.

4.7.3 Ekert et al. (2000)

Ekert et al. [14] describe the implementation of a conditional (geometric)phase shift as two-qubit operation for universal quantum computing. Theirmethod is based on geometric phases rather than the more conventional useof dynamical phases. They note that as the conditional phase only dependson the geometry, this provides resilience to certain errors and may providefault-tolerant quantum computation. The geometric quantity in this spe-cific case corresponds to the area spanned by a loop. Geometric phases arerobust to noise in the classical control parameters in the Hamiltonian, butEkert et al. [14] do not speak about how geometric phases behave in thepresence of decoherence or depolarization.

The adiabatic case is important because experimentally the Hamiltoniancan be controlled much easier than the actual state of the system. However,adiabaticity also means that the processes take a long time compared todynamical timescales. This method based on the adiabatic case is muchslower than dynamic methods of generating phases.

For the construction of a conditional phase gate using geometric phasesonly, it is necessary to eliminate the dynamic phase. In the Bloch spherepicture representing a spin state one approach is spin-echo. By applying thecyclic evolution twice, while including a swap transformation (|↑〉 ↔ |↓〉),the dynamical phases cancel but the geometric phases add up.

Ekert et al. [14] use the geometric phase to implement a two-qubit controlled-phase gate. They use both a static and a rotating magnetic field to tunethe transition resonance frequency. The Berry phase acquired by a spindepends directly on this quantity, such that the phase acquired by the firstspin depends on the state of the second spin.

4.7.4 Jones et al. (2000)

Jones et al. [26] use nuclear magnetic resonance (NMR) techniques for quan-tum information processing. They construct a controlled phase shift gate byan NMR experiment using the conditional Berry phase. So this approach isbased on geometrical phases to achieve conditional quantum dynamics: Thestate of one spin determines the Berry phase acquired by the other spin intheir experiments. Jones et al. [26] choose a conical evolution defining thesubtended solid angle. The schematic view of such a conical path (on the

61

Figure 4.4: Schematic view of a conical path in a spherical parameter space.

sphere) is given in figure 4.4. The NMR experiment consists of an ensem-ble of spin half particles influenced by a magnetic field and radio-frequencyradiation field. The path is defined by adiabatically applying a frequencysweep and phase sweep of the radiation field. An amplitude sweep can alsobe used. This procedure is used to define a conditional Berry phase ap-plied to one spin, where the size of the phase shift depends on the otherspin. The differential Berry phase shift is independent of how the process iscarried out. The duration must be slow enough to be adiabatic, but rapidcompared with the decoherence times.

The (inhomogeneous) dynamical phase is dealt with by the spin echo ap-proach. This approach performs the radio-frequency phase sweep twice inopposite directions, with the addition of π pulses. In this way the geometricphases add up and the dynamical phases cancel out. To measure the sizes ofthe Berry phase shifts it is convenient to apply the procedure to a spin in acoherent superposition of states. As the two states of the spin acquire equalbut opposite phases, and the pulse sequence generates twice the phase shift,the total phase different is equal to four times the Berry phase. The desiredphase shift is found by choosing the appropriate values of the frequency de-tuning from the transition frequency and the maximum radio-frequency fieldstrength. Jones et al. [26] state that this approach to geometric quantumcomputation does not have a particular advantage over more conventionalmethods of NMR quantum computation. However, they stress its possibleadvantage for fault tolerant quantum information processing.

4.7.5 Duan et al. (2001)

Duan et al. [13] describe an implementation of all-geometric quantum com-putation based on laser manipulation of a set of trapped ions. They realizeboth Abelian and non-Abelian holonomies. For universal quantum comput-ing, Duan et al. [13] consider constructing looped paths in parameter spaceto achieve a desired geometric evolution, as then a composition of theseparameter loops suffices to obtain an arbitrary unitary evolution. Theystate that (to their knowledge) their proposal is the first realistic one that

62

achieves all the elements of holonomic quantum computation and is feasiblewith current technology. Duan et al. [13] choose the following universal setof gate operations

U(j)1 = exp

(iφ1 |1〉j 〈1|

), (4.68)

U(j)2 = exp

(iφ2σ

yj

), (4.69)

U(jk)3 = exp

(iφ3 |11〉jk 〈11|

), (4.70)

where |0〉j and |1〉j constitute the computational basis for qubit j, σyj de-notes the Pauli-y operator on qubit j and φk, k = 1, 2, 3 are arbitrary phases.The physical system consists of ions confined in a linear Paul trap. Eachion has three ground states |0〉 , |1〉 and |a〉, and one excited state |e〉. Thestate |a〉 will be used as ancillary level. The ground states could be differenthyperfine levels or different Zeeman sublevels. The ground state levels arecoupled to the excited state |e〉 separately by a resonant classical laser witha different polarization or frequency. The structure here matches the tripodlevel scheme which is shown in figure 4.5 and will also be discussed in section4.7.8. The Rabi frequencies serve as control parameters. The parametersare initially chosen such that the computational space spanned by |0〉j and|1〉j is initially an eigenspace of the gate Hamiltonian with a zero eigenvalue.The three Rabi frequencies make an adiabatic cyclic evolution. The adia-batic theorem ensures that the computational space remains the eigenspaceof the gate Hamiltonian with zero eigenvalue, so there is no dynamical phasecontribution. The relative amplitude θ and relative phase φ are the effectivecontrol parameters, and their absolute magnitude is irrelevant as long as itsatisfies the adiabatic condition. A dark state is defined as an eigenstatewith a zero-energy eigenvalue. The parameters θ, φ make a cyclic evolutionwith the starting and ending point θ = 0.

The gate operations U1 (eq. (4.68)) and U2 (eq. (4.69)) are found as fol-lows. The quantum gate U1 is defined in terms of the Berry phase φ1 =∮

sin θ dθ dφ. This phase quantity has the direct geometric interpretation, itis exactly the enclosed solid angle

∮dΩ swept by the vector always pointing

to the (θ, φ) direction. Therefore the gate operation is determined only bya global property, an enclosed area on the sphere, and does not depend onthe details of the evolution along a path in parameter space.The quantum gate U2 works in the space of the two degenerate dark states.Again a cyclic evolution achieves the gate operation U2 with the phaseφ2 =

∮dΩ, which is equal to the swept solid angle by the vector (θ, φ).

For the demonstration of the non-Abelian holonomies, it is found that wedo not need to exploit any interaction between the ions. A combination ofU1 and U2 allows us to implement any single-bit operation.

63

For the construction of the nontrivial two-bit gate U(jk)3 (eq. (4.70)) using

geometric means, the Coulomb interactions between the ions need to beexploited. Duan et al. [13] provide a scheme based on two-color laser ma-nipulation. The transitions |1〉 → |e〉 and |a〉 → |e〉 are driven by a redand a blue detuned laser, with different detuning δ. The relative intensityand phase of the laser beams is written in terms of control parameters θ, φundergoing a cyclic adiabatic evolution from starting θ = 0. The computa-tional bases |00〉 , |01〉 and |10〉 are decoupled from the Hamiltonian, whilethe |11〉 component adiabatically follows and acquires a Berry phase afterthe loop. So we get a conditional phase-shift gate with a purely geometricphase equal to the swept solid angle by the vector (θ, φ).

4.7.6 Solinas et al. (2003)

Solinas et al. [47, 46] consider geometrical quantum information processingbased on semiconductor nanostructures. The quantum processors are verydelicate objects negatively influenced by both coupling to the environment(decoherence) and imperfections in quantum-state control. Several theo-retical solution have been suggested, mainly devoted to stabilize quantuminformation against computational errors induced by coupling with the envi-ronment. These strategies require extra physical resources in terms of eitherqubits or additional manipulations. A different strategy for the stabiliza-tion of quantum information is provided by the topological approach. Inthis case gate operations depend on global features of the control processand are therefore largely insensitive to local inaccuracies. The topologicalschemes are so far an abstract concept. Holonomic quantum computationprovides a significant intermediate step in this direction. Quantum gatesare constructed by driving control parameters along suitable loops using thenontrivial transformations of the initially prepared state. Such transforma-tions are also known as holonomies, which generalize to the non-Abeliancase, and can be computed explicitly by the Wilczek-Zee gauge connection.

Solinas et al. [47, 46] propose the implementation scheme for the realizationof a universal set of non-Abelian holonomic quantum gates. The quantumhardware is given by an array of semiconductor quantum dots and the com-putational degrees of freedom are given by excitonic transitions (interbandoptical excitations). The transitions are driven by ultrafast laser pulses. Anexciton is a Coulomb-correlated electron-hole pair. It is produced by pro-moting an electron from the valence band to the conduction band. This canbe done in a energy-selective fashion by a properly tailored laser excitation.Depending on the polarization of the light three different transitions with thesame energy can be induced. So the same tripod structure as for trapped-ioninterval levels is found here. We find two dark states with zero eigenvalue.These states in a distinguished point in parameter space will encode the

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qubit. The quantum manipulations are realized by the holonomies associ-ated to the Wilczek-Zee connection. The third state will play the role of anancilla qubit. Single qubit universality is achieved by two noncommutingholonomic single-qubit gates.

The two-qubit gate is implemented by exciton-exciton dipole coupling in thesemiconductor quantum dots. In two Coulomb-coupled quantum dots, thepresence of an exciton in one of them produces a shift in the energy levelof the other one. It is possible to produce two excitons by a two-photonprocess. Using different light polarizations all degenerate transitions can beexcited. The result is a three-dimensional dark-state manifold. Solinas et al.[47, 46] state that universal control in this dark space can be achieved in fullyholonomic fashion, as found from the associated u(3)-valued connection. Forexample a controlled-phase shift gate can be realized. Simulations suggestthat one should be able to apply several single-qubit holonomic gates or afew two-qubit gates within the decoherence time.

4.7.7 Ruseckas et al. (2005)

Ruseckas et al. [41] consider an experimental scheme to study the motionand the interaction of neutral quantum gases in non-Abelian gauge fields.This extends the discussion of artificial gauge fields, from Abelian to non-Abelian. The standard example of Abelian gauge fields are electromagneticfields (U(1)). In this optical scheme the adiabatic motion of atoms is inducedby laser fields, which leads to the creation of gauge potentials. A necessarycondition for the proposed scheme is that the atom-laser interaction has de-generate dark eigenstates with a nonvanishing nonadiabatic coupling. Thesimplest system with a nonvanishing adiabatic coupling between degeneratedark states is the so-called tripod scheme (see figure 4.5). In this schemenon-Abelian topological (or Berry) phases have been predicted in the inter-action of an atom with appropriately pulsed laser fields. This level schemecorresponds to that of alkali atoms where the levels are Zeeman componentsof hyperfine levels. Ruseckas et al. show that an effective magnetic fieldcan be generated that has a monopole component. For two degenerate darkstates, the gauge potentials are 2× 2 matrices. A truly non-Abelian situa-tion emerges if the matrices Ax, Ay, Az,Φ do not commute, corresponding

to the 2× 2 matrices of the components of the vector potential ~A and thescalar potential Φ. For this feature it is necessary that the off-diagonal ele-ment is nonzero.

Ruseckas et al. [41] parametrize the Rabi frequencies with angle and phasevariables.

Ω1 = Ω sin θ cosφ expiS1 ,Ω2 = Ω sin θ sinφ expiS2 ,Ω3 = Ω cos θ expiS3 ,(4.71)

65

which allows to write down the adiabatic dark states explicitly. The scalarand vector potentials can also be found. They show that artificial mag-netic fields can be generated and for a different parametrization of the Rabifrequencies also a magnetic monopole.

4.7.8 Sjoqvist et al. (2016)

Sjoqvist et al. [45] references Zhu and Wang [57] who pointed out that nona-diabatic Abelian geometric phases are sufficient for universal all-geometricquantum computation, despite the fact that such phases are U(1) and there-fore commuting. Noncommutativity can be achieved by considering geo-metric phase shift gates in different bases.

Sjoqvist et al. [45] describes the elimination of dynamical phase effects onthe gate operation. It may involve specific paths in state space or tuningapplied fields in certain ways. There is a distinction between dynamicalphases that do not affect the gate operation and phases that are necessaryto compensate for. A general linear combination of the computational basisstates does not necessarily traverse a closed path on the Bloch sphere, inthe time interval in which the basis states perform a cycle. The relateddynamical and geometric phases are global phases and irrelevant to the gateoperation.

The Zhu-Wang (ZW) scheme is described, which uses Abelian nonadia-batic geometric phases to achieve universal geometric quantum computa-tion. The geometric gate can be tuned to the geometric phase shift gateand the Hadamard gate, which are known to be universal for a single qubit.Nonadiabatic geometric phases can be used to achieve fast universal geo-metric quantum computation. In its simplest form it can be constructed ina three-level Λ configuration, which is also shown in figure 4.5. Here a two-dimensional ground state manifold (the qubit state space) is coupled to anauxiliary excited state. Sjoqvist et al. [45] show how the ZW scheme and thethree-level Λ system are related. The gates based on non-Abelian geometricphases can also be interpreted as ZW gates. The gate is holonomic, thetime evolution is purely geometric and depends only on the cyclic evolutionpath in the space of two-dimensional subspaces of the full three-dimensionalHilbert space of the system: the Grassmannian manifold G3,2(C).

For the adiabatic evolution of a degenerate subspace, the resulting dynami-cal phases are the same for all states belonging to the subspace. Thereforethey can be factored out making the gate operation purely geometric. Anexample are the two parameter-dependent dark states of a tripod configu-ration. The non-Abelian geometric gate depends on the loop in parameterspace for this adiabatic scheme. In the non-adiabatic method energy degen-eracies play no role. In this case the computational system lies in a subspace

66

of the Hilbert space, where the Hamiltonian acts trivially during the gateexecution. The Hamiltonian does not need to return to its initial form orevolve slowly. The requirement is that the initial and final subspaces coin-cide. The unitary gate is determined by the loop performed by the subspace.The scheme based on the structure of the three-level Λ system is the mostbasic form of non-adiabatic non-Abelian geometric quantum computation.

The adiabatic and non-adiabatic approaches to the implementation of non-Abelian geometric gates compare as follows. In the adiabatic case the run-time is used to factor out the dynamical phase and construct the purely geo-metric action of the gate. This is achieved by taking the run-time to infinity,the adiabatic limit. In the non-adiabatic case, the run-time must ensure theevolution is cyclic. The run-time is finite and can be short. The adiabaticgate is exact only in the (mathematical) limit of the run-time tending toinfinity. In experiments (finite time) there will be non-adiabatic corrections,which can be arbitrarily small but remain non-zero. The non-adiabatic geo-metric gate is exact for finite time, if the Hamiltonian satisfies the geometrycondition. The physical control parameters induce the geometric phase inthe adiabatic case by changing slowly around a loop. In the non-adiabaticcase they play a passive role. The loops of the adiabatic and non-adiabaticcase are traced out in different spaces. In the adiabatic case the loop lies inthe space of slow control parameters, while in the non-adiabatic case in aGrassmann manifold. However, also the adiabatic geometric phase dependson a path in a Grassmann manifold; the energy eigensubspace traverses aloop in the corresponding Grassmannian.

The tripod and Λ configurations, shown in figure 4.5, define a structurewhere the detailed nature of the underlying physical system is not important.It can for example be a trapped ion addressed by laser fields, a transmonqubit or a nitrogen-vacancy (NV) center in diamond driven by microwavefields. In all cases the approximate Hamiltonian can be written as

Htripod = ∆0 |0〉 〈0|+ ∆1 |1〉 〈1|+ ∆a |a〉 〈a|+Υ (ω0 |e〉 〈0|+ ω1 |e〉 〈1|+ ωa |e〉 〈a|+ h.c.) ,

(4.72)

which is based on the rotating wave approximation (RWA) in the interactionpicture.

In the adiabatic tripod scheme a degenerate pair of energy eigenstates isconstructed of the form c0 |0〉+ c1 |1〉+ ca |a〉. These states are called darkstates as they do not involve the excited state |e〉. This excited state ispotentially unstable and thus can produce light.

It can be found [44, 51] that there is a pair of degenerate dark energy eigen-states |D0(ω)〉 and |D1(ω)〉, for all ω = (ω0, ω1, ωa). There are also two

67

Ω2

Ω1 Ω3

|2〉|1〉 |3〉

|0〉

Ω2Ω1

|2〉|1〉

|0〉

Figure 4.5: Schematic representation of the tripod configuration (left) andthe Λ configuration (right) of a atomic level scheme.

non-degenerate bright states. Now we assume that ω = ω(s) varies arounda loop Cp in parameter space and that Υ = Υ(s) is nonzero on s ∈ [0, 1]. Inthe adiabatic regime, the gate run-time is so large that transitions betweenthe dark subspace and the two bright states become negligible. The condi-tion is that the time is large compared to the inverse of the minimum valueover the path of the bright state energy gap. When the adiabatic conditionis satisfied, the loop in the space of slowly changing parameters ω approxi-mates the nontrivial action of the time evolution operator. This correspondsto a gate acting on the computational space by choosing ω(0) = (0, 0, 1).The space of all dark subspaces is the Grassmannian manifold G(3; 2). It isthe space of the two-dimensional subspaces spanned by |D0(ω)〉 , |D1(ω)〉of the three-dimensional vector space spanned by |0〉 , |1〉 , |a〉. The loop inthe space of slow parameters ω induces a loop in the Grassmannian manifoldG3,2(C), with the initial point mapped to the computational space spannedby |0〉 , |1〉. The states on the supersymmetric triangle lattice can be de-scribed in a similar manner, as shown in section 4.5. For the non-adiabaticcase we need to satisfy the restrictions on the parameters, which have simplesolution for Υ, ω0, ω1 independent of s. It can be shown [44] that a universalpurely geometric single-qubit gate can be realized by choosing the run-timecorrectly. We see that the control parameters play a passive role and caneven be taken constant during the execution of the gate.

The existence of a wide range of conceptually different schemes impliesthat geometric quantum computation can be implemented in many differ-ent physical systems. It also provides possibilities for addressing differenttypes of errors. For example, adiabatic approaches can be used if parametricnoise is present, and non-adiabatic schemes can shorten the execution timeto reduce decoherence.

4.7.9 Leroux et al. (2018)

Leroux et al. [29] study a noninteracting cold fermionic gas of strontium-87

68

atoms coupled to laser fields through a four-level resonant tripod scheme.As stated by Wilczek and Zee [53] the symmetry of the degenerate subspaceof states leads to a non-Abelian gauge field structure. A four-level resonanttripod scheme is used of the 1S0, Fg = 9/2 → 3P1, Fe = 9/2 intercombi-nation line at λ = 689 nm. We have the levels with mg = 5/2, 7/2, 9/2coupled to a common excited state me = 7/2. The effective Hilbert spacedefined by the four coupled bare levels contains two degenerate dark states.The dark states do not couple to the excited state and are therefore pro-tected from spontaneous emission decay. Leroux et al. [29] investigate thegeometric non-Abelian unitary operator U acting on the dark-state mani-fold when the relative phases of the tripod beams are adiabatically sweptalong some closed loop in parameter space. The operator U is given by theloop integral along the loop of the 2× 2 Mead-Berry 1-form ω. This systemrealizes a universal geometric single-qubit gate by implementing adiabaticgeometric transformations acting on two degenerate dark-states. The resultis a general unitary operator of the form(

α β−β∗ α∗

)(4.73)

with |α|2 + |β|2 = 1. The non-Abelian (SU(2)) character is shown by theexplicit dependence on path ordering, which is needed during the explicitcalculation of the unitary operator U . In the appendix we perform thecalculation of the ansatz presented in Leroux et al. [29] explicitly, startingfrom the dark states to the unitary operator U .

69

70

Chapter 5

Triangle and hexagon lattices

5.1 Introduction

In this chapter we consider two examples of a one-dimensional periodic lat-tice (or chain) equipped with a staggered M1 supersymmetric lattice model.Recall that M1 corresponds to nearest neighbor exclusion. We will makeuse of the definitions introduced in chapters 2 and 4, of the supersymmetriclattice model, staggering, and non-Abelian Berry phase.

In particular, we consider two chains with a length equal to a multiple ofthree, with a staggering with a period of three. This length is of interest asthis result implies two ground states on the lattice [16]. First we will analyzethe triangle, a (periodic) chain with three sites, and second the hexagon, achain with six sites. In the next chapter we will consider connecting twotriangles to form another geometry, coined the bow tie. We will show ex-plicit calculations of the ground states on the triangle and the hexagon usingexplicit choices for the staggering. We calculate the Berry phase induced bya closed path in the space of the staggering parameter.1 To conclude thischapter, we interpret the action of the Berry phase in terms of a one-qubitquantum gate.

As both the triangle and hexagon lattice provide two degenerate groundstates, the calculated Berry phase is per definition non-Abelian. However,we will find different cases for the two staggering parameter choices ~λ and~λΩ. For ~λΩ we will work with ground states where only one is dependent onthe angular parameters θ, φ. The result is a diagonal Berry phase matrix.So one could argue that the found action is Abelian. As we are not sureif this Abelian nature persists for any choice of vector representations ofthe ground states we will use the terminology of ‘non-mixing’ to denotethis case. The other staggering parameter choice ~λ results in a off-diagonal

1The Mathematica files can be obtained by an email to the author.

71

Site 1 Site 2

Site 3

Figure 5.1: Schematic view of the |1〉 state on the triangle lattice, a peri-odic chain of three sites. Empty sites are represented by white circles andoccupied sites by black circles. The edges correspond to neighbouring sitesin the lattice model.

Berry phase matrix, so we will use the term ‘mixing’ for that case. Thisterminology is inspired by Solinas et al. [48].

5.2 The triangle lattice

The triangle is the simplest chain of length 3N , that is N = 1, which fea-tures two ground states with zero energy. A simple schematic representationis given in figure 5.1. We work with the supersymmetric Hamiltonian withnearest neighbor exclusion (M1), as introduced in chapter 2.

States on the triangle lattice are written in the basis |1〉 , |2〉 , |3〉, where

1, 2, 3 correspond to the sites in the triangle and |i〉 = c†i |0〉 in terms ofthe fermionic creation operator. In this context does |0〉 denote the emptylattice, the triangle without fermions. However, in later sections, we chooseto use |0〉 to denote the ‘zero’ qubit state in the computational basis, and touse |1〉 to denote the ‘one’ qubit state. To avoid confusion, let us work withthe states on the triangle in terms of their coefficients in the lattice basis|1〉 , |2〉 , |3〉. A general state of the triangle lattice can be denoted by anexpansion in this basis

|ψ〉 = k1 |1〉+ k2 |2〉+ k3 |3〉 (5.1)

which can also be written as

|ψ〉 =

k1

k2

k3

·|1〉|2〉|3〉

. (5.2)

So for our discussion it is sufficient to calculate the coefficients k1, k2, k3 todefine the ground states. This corresponds directly to the elements of a

72

Hilbert space. We will often refer to the vector of coefficients k1, k2, k3 as‘coefficient vectors’.

Let us introduce the following definition for the computational basis statesof the qubit on the triangle,

|0〉comp ≡ |1〉lattice , (5.3)

|1〉comp ≡ |2〉lattice ,

where |i〉comp , i = 0, 1 denote qubit states and |j〉lattice , j = 1, 2, 3 latticestates. The notation of the qubit states in this way is standard, but alsoarbitrary; we could also use |↑〉 , |↓〉. This definition will be made clear insubsection 5.2.2.

5.2.1 Staggering

We choose the staggering parameter on the sites according to the compo-nents of the vector in the origin to the surface of the unit sphere, written inspherical coordinates (θ, φ)

~λ = (sin θ cosφ, sin θ sinφ, cos θ), (5.4)

as introduced in eq. (2.15) in section 2.2.1. We use the ISO2 convention forthe spherical coordinate system, (r, θ, φ), as commonly is used in physics.We set the radial distance r to unity, to consider a unit sphere. Parameterθ describes the polar angle and parameter φ the azimuthal angle. Note that|~λ|2 = 1 and ~λ ∈ R3. If we construct the Hamiltonian of the one-particlesector explicitly from the definition of the staggered supersymmetric charges,denoting the components of ~λ as λi, we find λ2

1 λ1λ2 λ1λ3

λ1λ2 λ22 λ2λ3

λ1λ3 λ2λ3 λ23

. (5.5)

We will use some terminology resembling Earth to describe certain pointson the unit sphere of the parameter space. The point θ = 0 is often calledthe North Pole and the point θ = π the South Pole. The circle withθ = π/2, 0 ≤ φ ≤ 2π might be called the equator. In figure 5.2 we haveshown the implementation of the staggering parameter on the lattice andthe staggering parameter space defined by the choice of ~λ in eq. (2.15).

We consider another choice for the staggering parameter. This choice isinspired by geometric quantum computation literature, such as Solinas et al.

2International Organization for Standardization

73

λ1 λ2

λ3

λ2

λ3

λ1

Figure 5.2: The triangle lattice with staggering parameter ~λ included (left)and the space of the staggering parameter S2 (right).

[46]. Here we choose the staggering parameter complex, but with only twonon-zero components. As introduced in eq. (2.16) in section 2.2.1, let us take

~λΩ = (0,− sin(θ/2) exp(iφ), cos(θ/2)), (5.6)

so with λΩ,1 zero for all values of the (real) parameters θ, φ.

5.2.2 Ground states on the triangle lattice

In eq. (2.12) in section 2.2 we defined the supersymmetric charges Q,Q† withthe inclusion of the staggering parameter ~λ. The ground states are definedby Q |ψgs〉 = 0 and Q† |ψgs〉 = 0. However, if we consider applying Q† to theempty lattice, denoted by |0〉, we find

Q† |0〉 = λ∗1 |1〉+ λ∗2 |2〉+ λ∗3 |3〉 . (5.7)

This one-fermion state is not a ground state; it forms a doublet with theempty lattice. We can see the equivalence of the ground states being or-thogonal to the one-fermion doublet state and the coefficient vectors beingorthogonal to the staggering vector ~λ. Note that the Q† operation resultsin zero on any one-fermion state by definition, as fermions are not allowednext to each other, so a two-fermion state is not possible by the projectionoperators.

We want to choose the coefficient vectors also mutually orthogonal, whichis necessary for the calculation of the Berry phase. The following choice ofcoefficient vectors

f1 = (cos θ cosφ, cos θ sinφ,− sin θ), (5.8)

f2 = (− sinφ, cosφ, 0),

satisfies the criteria ~λ⊥f1, ~λ⊥f2 and f1⊥f2.

74

In this form if we consider the vectors f1, f2 on the North Pole of the unitsphere, corresponding to θ = 0, we find that

f1|θ=0 = (cosφ, sinφ, 0), (5.9)

f2|θ=0 = (sinφ,− cosφ, 0),

which are clearly dependent on angle φ. However, we want to be able todefine the point θ = 0 (the North Pole) unambiguously, that is, independentof φ. So we rotate the coefficient vectors to a form which is independent ofφ on the North Pole.

g1 = cosφf1 − sinφf2, (5.10)

g2 = sinφf1 + cosφf2

The result is

g1 = (cos θ cos2 φ+ sin2 φ, (cos θ − 1) cosφ sinφ,− sin θ cosφ),(5.11)

g2 = ((cos θ − 1) cosφ sinφ, cos θ sin2 φ+ cos2 φ,− sin θ sinφ).

We find now that the vectors g1, g2 are (still) dependent on φ on the SouthPole where θ = π. This dependence may be removed by a different rotationof f1, f2

g1 = cosφf1 + sinφf2, (5.12)

g2 = − sinφf1 + cosφf2

However, we choose to use g1 and g2, eq. (5.11), as the coefficient vectors ofthe one-fermion ground states on the triangle. On the North Pole (θ = 0)the coefficient vectors g1, g2 are (1, 0, 0) and (0, 1, 0). This corresponds byconstruction to the definition of the computational basis states of the qubit.As introduced in eq. (5.3), we can write (1, 0, 0) ≡ |0〉comp and (0, 1, 0) ≡|1〉comp.The need of different sets of coefficient vectors for the North and South Poleis the result of the topology of the sphere S2. There are two coordinatepatches needed to describe the sphere.

A second possible choice for the staggering parameter ~λΩ is given in eq. (5.6).It is easy to find expressions for the coefficient vectors of the ground stateson the triangle for this second choice. Keep in mind that the supersymmetriccharge Q† is defined in terms of the complex conjugate of the componentsof ~λ in eq. (2.12) in section 2.2. So we can see that the following choices aresufficient as they are orthogonal to ~λ∗ and each other.

fΩ,1 = (1, 0, 0), (5.13)

fΩ,2 = (0, cos(θ/2), sin(θ/2) exp(iφ))

75

Coefficient vector fΩ,1 is independent of the parameter angles θ and φ. AlsofΩ,2 is independent of φ on the North Pole, as fΩ,2|θ=0 = (0, 1, 0). We seethat on the North Pole we find that same states: (1, 0, 0) = |0〉comp and(0, 1, 0) = |1〉comp.

5.2.3 Berry connections associated to the triangle lattice

The ground states are described by their coefficient vectors g1 and g2 ineq. (5.11), both three-dimensional vectors dependent on angles θ and φ. Wehave defined the computational qubit states in terms of the (coefficient vec-tors of) the ground states. We are looking to define a one-qubit quantumgate on this computational basis by employing the non-Abelian Berry phase.The coefficient vectors of the ground states, explicitly dependent on para-meters θ and φ, follow a closed path in (θ, φ) parameter space transversedadiabatically. After the closed loop the vectors acquire a non-Abelian phase:they rotate in their two-dimensional subspace upon multiplication by a two-dimensional square matrix. We want to find the explicit expression for thisrotation matrix, which can be directly related to a rotation in the qubitspace.

The non-Abelian Berry connection defined by Wilczek and Zee [53] is givenin eq. (4.33) in section 4.4. We will reintroduce this definition here. Recallthat we defined the staggering parameter ~λ in eq. (5.4). We will considera closed path in this parameter space. We describe the action of the pathby changes in the parameters θ and φ instead of by changes in some timevariable. We are looking calculate the explicit expression for the Berryconnection

(Ap)ij = 〈ψi|∂

∂p|ψj〉 =

⟨ψi

∣∣∣∣ ∂∂pψj⟩. (5.14)

Here p takes the values of parameters θ, φ and i, j = 1, 2 denoting the twoorthogonal lattice states. From the non-Abelian Berry connection A wefind the non-Abelian Berry phase by integration over the path in parameterspace. So if parameter p changes over a certain section, we find

γp = i

∫Ap dp. (5.15)

Then we can find the corresponding unitary matrix by matrix exponentia-tion, while keeping path ordering in mind, the expression is of the form

U = exp(iγ) = P exp

(−∫A

), (5.16)

where A and γ are defined by their components Ap and γp.

76

The first step is to calculate the derivatives of the coefficient vectors. This

is sufficient for the calculation of∣∣∣ ∂∂pψ⟩ as follows from∣∣∣∣ ∂∂pψ

⟩=

∂

∂p|ψ〉 =

∂

∂p

3∑i=1

ki |i〉 =3∑i=1

∂

∂pki |i〉 , (5.17)

where we used the expansion of the state in the basis of one-fermion latticestates. The derivative of the coefficient vectors, ∂

∂p~k, are explicitly given by

∂

∂θg1 =

(− sin θ cos2 φ,− sin θ cosφ sinφ,− cos θ cosφ

), (5.18)

∂

∂θg2 =

(− sin θ sinφ cosφ,− sin θ sin2 φ,− cos θ sinφ

), (5.19)

∂

∂φg1 =

(2(1− cos θ) cosφ sinφ, (cos θ − 1)(cos2 φ− sin2 φ), sin θ sinφ

),

(5.20)

∂

∂φg2 =

((cos θ − 1)(cos2 φ− sin2 φ), 2(cos θ − 1) cosφ sinφ,− sin θ cosφ

).

(5.21)

Next we work out the inner products 〈ψi| ∂∂p |ψj〉, which define the fourentries in the matrix corresponding to the non-Abelian Berry phase. Let usfirst consider the inner products with derivatives to parameter θ, we find

〈ψ1|∂

∂θ|ψ1〉 = (cos θ cos2 φ+ sin2 φ, (cos θ − 1) cosφ sinφ,− sin θ cosφ)

·(− sin θ cos2 φ,− sin θ cosφ sinφ,− cos θ cosφ

)= 0, (5.22)

〈ψ1|∂

∂θ|ψ2〉 = 0, (5.23)

〈ψ2|∂

∂θ|ψ1〉 = 0, (5.24)

〈ψ2|∂

∂θ|ψ2〉 = 0, (5.25)

so we conclude that for the triangle Aθ = 0, the zero matrix. Consider theinner products with derivatives to parameter φ, we find

〈ψ1|∂

∂φ|ψ1〉 = 0, (5.26)

〈ψ1|∂

∂φ|ψ2〉 = 1− cos θ, (5.27)

〈ψ2|∂

∂φ|ψ1〉 = −1 + cos θ = −(1− cos θ), (5.28)

〈ψ2|∂

∂φ|ψ2〉 = 0. (5.29)

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so we conclude that for the triangle

Aφ =

(0 1− cos θ

−1 + cos θ 0

), (5.30)

so the connection matrix for parameter φ is an antisymmetric (or skew-symmetric) matrix, as A>φ = −Aφ. To be more precise it equals Aφ =i(1− cos θ)σy, where σy denotes the Pauli y-matrix. The matrix iσy is alsoknown as the generator of the Lie algebra so(2), of rotations in the plane.The off-diagonal form exemplifies the non-Abelian nature of the connection.A simple analysis shows that the expression 1− cos θ takes values from 0 to2 for values of θ from 0 to π.

Starting from the coefficient vectors in eq. (5.13) of the ground states for thestaggering introduced in eq. (5.6), we can also find the connection matrix.Consider now taking the derivatives to θ, φ of fΩ,1, fΩ,2

∂

∂θfΩ,1 = (0, 0, 0), (5.31)

∂

∂φfΩ,1 = (0, 0, 0), (5.32)

∂

∂θfΩ,2 =

(0,−1

2sin(θ/2),

1

2exp(iφ) cos(θ/2)

), (5.33)

∂

∂φfΩ,2 = (0, 0, i exp(iφ) sin(θ/2)). (5.34)

Next we work out the inner products conform the definition in eq. (5.14),where the complex conjugate is included in the bra 〈ψΩ| state. We find thefollowing for parameter θ

〈ψΩ,1|∂

∂θ|ψΩ,1〉 = 0, (5.35)

〈ψΩ,1|∂

∂θ|ψΩ,2〉 = 0, (5.36)

〈ψΩ,2|∂

∂θ|ψΩ,1〉 = 0, (5.37)

〈ψΩ,2|∂

∂θ|ψΩ,2〉 = 0, (5.38)

so we conclude that the connection matrix AΩ,θ = 0, the zero matrix. For

78

parameter φ we find

〈ψΩ,1|∂

∂φ|ψΩ,1〉 = 0, (5.39)

〈ψΩ,1|∂

∂φ|ψΩ,2〉 = 0, (5.40)

〈ψΩ,2|∂

∂φ|ψΩ,1〉 = 0, (5.41)

〈ψΩ,2|∂

∂φ|ψΩ,2〉 = i sin2(θ/2). (5.42)

We conclude for this different choice of the staggering parameter in thetriangle the connection matrix Aφ is equal to

Aφ =

(0 00 i sin2(θ/2)

)(5.43)

which is a diagonal matrix with a single non-zero entry. The diagonal formdoes not particularly show its non-Abelian nature. One could even say that(for this choice of basis) the Berry connection is Abelian. The expressionsin2(θ/2) takes values from 0 to 1 for values of θ from 0 to π.

5.2.4 Berry curvature and Chern number associated to thetriangle lattice

We have calculated the Berry connections for the triangle lattice explicitly,our final step is to calculate the Berry curvature Fµν . This gauge invariantquantity was introduced in eq. (4.52) in section 4.5.

For the first choice of the staggering parameter ~λ leading to coefficient vec-tors g1, g2 we can find the nonzero component of the Berry curvature Fθφ,the result is

Fθφ =

(0 sin θ

− sin θ 0

)For the second choice of staggering parameter ~λΩ leading to the coefficientvectors fΩ,1, fΩ,2, the result for the Berry curvature FΩ,θφ is equal to

FΩ,θφ =

(0 00 i

2 sin θ

)Now that we have found the explicit form of the Berry curvature, we willcalculate the first Chern number. Taking the trace of the Berry curvaturematrices results in Tr(Fθφ) = 0 and Tr(FΩ,θφ) = i

2 sin θ. So we can see that

for the first staggering choice ~λ the first Chern number is zero, it is trivial.

79

For the second choice we calculate

c1 =i

2π

∫P

TrF =i

2π

∫ π

θ=0

∫ 2π

φ=0dφ dθTrFΩ,θφ

= −1

2

∫ π

θ=0sin θ dθ = −1. (5.44)

The first Chern number for the second choice of staggering parameter ~λΩ isnon-trivial.

5.3 The hexagon lattice

The hexagon lattice or six-chain is the natural next step after the trianglelattice. We will work through the same steps as in the triangle to findexplicit expression for the Berry connections in the hexagon. This closedchain with a length of six sites has again a length of a multiple of three, sowe will find two zero-energy ground states again. We number the sites onthe hexagon by 1, . . . , 6 as shown in figure 5.3. We will write the states onthe hexagon in the following general ket notation

|i, j, . . . , p〉 := c†ic†j . . . c

†p |0〉 (5.45)

where the fermionic operator c†i is used, and i, j, p denote the numbers ofthe sites in the chain. We will apply the convention of writing the state withi < j < · · · < p. Note that because of the fermionic nature of the creationoperators the following equality holds |j, i〉 = − |i, j〉. For completeness wewrite down all possible basis states in the system.

Nf = 0 : |0〉 , (5.46)

Nf = 1 : |1〉 , |2〉 , |3〉 , |4〉 , |5〉 , |6〉 , (5.47)

Nf = 2 : |1, 3〉 , |2, 4〉 , |3, 5〉 , |4, 6〉 , |1, 5〉 , |2, 6〉 , (5.48)

|1, 4〉 , |2, 5〉 , |3, 6〉 , (5.49)

Nf = 3 : |1, 3, 5〉 , |2, 4, 6〉 . (5.50)

We find that there is one state with zero fermions on the chain, the emptychain. Six states with one fermion, the fermion can be one each of the ver-tices. There are nine states with two fermions, six states with the fermionstwo sites apart and three states with the fermion three sites apart. Thereare also two states with three fermions. The M1 condition gives half fillingby three fermions as the maximal occupation. We can now find the Wittenindex

W = Tr(−1)Nf = 1− 6 + 9− 2 = 2, (5.51)

which shows that there are two supersymmetric ground states in the system.

80

Site 1 Site 2

Site 3

Site 4Site 5

Site 6

Figure 5.3: Schematic view of the hexagon lattice, a periodic chain of sixsites.

We can derive that the two supersymmetric ground states are two-fermionstates. Remember the definition of the supersymmetric charges as intro-duced in section 2.2. In the next section we will give the definition of stag-gering in the hexagon. By the interpretation of the supersymmetric chargeswe can divide the set of states into doublets and singlets. There is one one-fermion state

∣∣ψ1⟩

which forms a doublet with the empty chain |0〉 so that itsatisfies

∣∣ψ1⟩

= Q+ |0〉 ⇔ |0〉 = Q−∣∣ψ1⟩. The other five one-fermion states∣∣φ1

⟩i

must satisfy Q−∣∣φ1⟩i

= 0. However, for these five one-fermion states∣∣φ1⟩

1,...,5there exist five two-fermion states such that

∣∣ψ2⟩i

= Q+∣∣φ1⟩i⇔∣∣φ1

⟩i

= Q−∣∣ψ2⟩i. These five two-fermion states also satisfy Q+

∣∣ψ2⟩i

= 0.

Two two-fermion states∣∣φ2⟩

1,2form a doublet with the two three-fermion

states∣∣ψ3⟩

1,2so they satisfy

∣∣ψ3⟩i

= Q+∣∣φ2⟩i⇔∣∣φ2⟩i

= Q−∣∣ψ3⟩i. By

counting the number of states in doublets we find that there must be twotwo-fermion singlet states. This confirms the prediction by the Witten index,there are two two-fermion supersymmetric ground states. We will denotethem by

∣∣χ2⟩

1,2. For clarity we have written the states in the form

∣∣ANf ⟩i.

Nf denotes the fermion number of the state. A contains extra informationabout the state: A = 0 denotes the empty state, A = ψ the state withthe higher fermion number in the doublet, A = φ the state with the lowerfermion number in the doublet and A = χ a singlet state. The subscript iis the index in the space of a certain A and Nf . The zero-fermion sectoronly contains the empty state |0〉. The states

∣∣ψ1⟩

and∣∣φ1⟩

1,2,3,4,5are linear

combinations of the states in eq. (5.47). The states∣∣ψ2⟩

1,2,3,4,5,∣∣φ2⟩

1,2and∣∣χ2

⟩1,2

are linear combinations of the states in eq. (5.48) and eq. (5.49) and

81

λ1 λ2

λ3

λ4 = λ1λ5 = λ2

λ6 = λ3

Figure 5.4: The hexagon lattice with the staggering parameter ~λ showingits reduction by translational symmetry.

the states∣∣ψ3⟩

1,2of the states in eq. (5.50).

5.3.1 Staggering

We choose to introduce staggering parameters λi with a translational in-variance: staggering modulo 3 so (λ1, λ2, λ3, λ1, λ2, λ3). We choose the stag-gering ~λ = (λ1, λ2, λ3) in the same manner as in the triangle: We let themdepend on θ and φ by the standard choice of the coordinates on the surfaceof the sphere.

λ1 = sin θ cosφ, λ2 = sin θ sinφ, λ3 = cos θ (5.52)

Note that ~λ ∈ R3, that λ∗i = λi and that∣∣∣~λ∣∣∣2 = 1. The staggering parameter

imposed on the hexagon lattice is shown in figure 5.4.

The choice to define the staggering parameter modulo three sites introducesa symmetry into the hexagon system. Whereas the introduction of stagger-ing breaks general translational symmetry (‘translation over one site’), forthis choice of staggering a translational symmetry over three site remains.Let us consider the translation operator T3, corresponding to the transla-tion over three sites of a fermion chain. In our notation of the states onthe six-chain, the action of the translation operator T3 is T3 |i〉 = |i+ 3〉,where i mod 6 and we take i = 1, . . . , 6 (i = 6 instead of i = 0). Generallyfor a chain of 3n sites T3 satisfies Tn3 = 1. So for our hexagon we havethat T 2

3 = 1. The property allows us to find the eigenvalues. Assume that

82

T3~v = Λ~v, then

T3(T3~v) = T3(Λ~v)⇒ T 23~v = Λ2~v, (5.53)

~v = Λ2~v. (5.54)

As ~v 6= 0, Λ2 = 1, which is easily solved by Λ1 = 1 and Λ2 = −1. The eigen-state equation of T3 therefore has the form T3 |Ψ〉 = Λ |Ψ〉 where Λ = ±1.

The zero-energy ground states are two-fermion states that satisfyQ±∣∣χ2⟩

= 0.We define a two-fermion state by its coefficients with respect to a certainbasis states vector:

∣∣χ2⟩

= ~C · ~X. Here the basis ~X is defined by

~X = (|1, 3〉 , |2, 4〉 , |3, 5〉 , |4, 6〉 , |1, 5〉 , |2, 6〉 , |1, 4〉 , |2, 5〉 , |3, 6〉). (5.55)

In this basis we can define the action of the translation operator T3. Twoexamples are T3 |1, 3〉 = |4, 6〉 and T3 |2, 4〉 = |5, 7〉 = |5, 1〉 = − |1, 5〉. Nowwe can write down the eigenstates of the translation operator T3.

Λ = +1 : 1√2

(|1, 3〉+ |4, 6〉) , 1√2

(|2, 4〉 − |1, 5〉) , 1√2

(|3, 5〉 − |2, 6〉) ,(5.56)

Λ = −1 : |1, 4〉 , |2, 5〉 , |3, 6〉 , (5.57)1√2

(|2, 4〉+ |1, 5〉) , 1√2

(|3, 5〉+ |2, 6〉) , 1√2

(|1, 3〉 − |4, 6〉)(5.58)

We are looking for states which are simultaneous eigenstates of both theHamiltonian and T3. The reason is the fact that the Hamiltonian of thesupersymmetric lattice model commutes with the translation operator.The eigenstates of T3 reduce the set of possible coefficients of the zero-energystates. We find that both zero-energy ground states have T3 eigenvalueΛ = −1 by dividing linear combinations of the basis vectors in eq. (5.46)-eq. (5.50) by their eigenvalues of the translation operator. We define ~Y asbasis vector for all states with Λ = −1.

~Y =

|2, 4〉+ |1, 5〉√

2,|3, 5〉+ |2, 6〉√

2,|1, 3〉 − |4, 6〉√

2, |1, 4〉 , |2, 5〉 , |3, 6〉

(5.59)

The matrix form of the two-fermion sector of the HamiltonianH = Q+, Q−in this new basis with translational eigenvalue −1 for general staggering λ isgiven below. It can be found by considering the action of the supersymmetriccharges on each of the basis states.

λ21 + λ2

2 + λ23 0 0

√2λ1λ2

√2λ1λ2 0

0 λ21 + λ2

2 + λ23 0 0

√2λ2λ3

√2λ2λ3

0 0 λ21 + λ2

2 + λ23

√2λ1λ3 0 −

√2λ1λ3√

2λ1λ2 0√

2λ1λ3 2λ21 0 0√

2λ1λ2

√2λ2λ3 0 0 2λ2

2 0

0√

2λ2λ3 −√

2λ1λ3 0 0 2λ23

(5.60)

83

5.3.2 Ground states on the hexagon

We want to find explicit expressions for the zero-energy ground states onthe hexagon. We will again make use of a notation in terms of a coeffi-cient vector with respect to the basis ~Y defined in eq. (5.59), such that theground states are

∣∣χ2⟩

= ~C · ~Y . Just as in the triangle, it is our preferenceto calculate the coefficient vectors by construction, instead of numerically.The zero-energy ground states must satisfy the conditions Q

∣∣χ2⟩

= 0 andQ†∣∣χ2⟩

= 0 as the Hamiltonian is defined by H = Q,Q†.

Staggering choice ~λ

The operatorQ is dependent on the staggering parameter ~λ = (sin θ cosφ, sin θ sinφ, cos θ),used on the hexagon modulo three sites. We construct two states which givezero after applying operator Q, analogous to our choice in the triangle

eR = (0,− sinφ, cosφ, 0, 0, 0), (5.61)

eB = (sin θ, cos θ cosφ, cos θ sinφ, 0, 0, 0). (5.62)

The next construction step is choosing the last three entries of each co-efficient vector such that applying operator Q† also results in zero. Thefollowing vectors satisfy both the conditions Q |gR,B〉 = 0 and Q† |gR,B〉 = 0,

gR = (0,− sinφ, cosφ,− 1√2

cos θ

sin θ,

1√2

cos θ

sin θ,

1√2

sin θ

cos θ), (5.63)

gB = (sin θ, cos θ cosφ, cos θ sinφ,− 1√2

sinφ

sin θ cosφ,− 1√

2

cosφ

sin θ sinφ, 0).

(5.64)

To remove the singular behavior at θ = 0, π/2 and φ = 0, π/2 we rede-fine the coefficient vectors by multiplying by the expressions present in thedenominator. We find that

hR = sin θ cos θgR, (5.65)

= (0,− sin θ cos θ sinφ, sin θ cos θ cosφ,− 1√2

cos2 θ,1√2

cos2 θ,1√2

sin2 θ),

hB = sin θ cosφ sinφgB, (5.66)

= (sin2 θ cosφ sinφ, cos θ sin θ cos2 φ sinφ, cos θ sin θ cosφ sin2 φ,

− 1√2

sin2 φ,− 1√2

cos2 φ, 0),

satisfy the conditions Q |gR,B〉 = 0 and Q† |gR,B〉 = 0, while also being de-fined everywhere on the sphere, for all 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. Sucha multiplication used in the last step is allowed as we are interested in two

84

vectors in the null space of the Hamiltonian. The following holds: if A~v = 0then A(c~v) = c(A~v) = c(0) = 0. Unfortunately, the vectors found here sat-isfy neither orthogonality nor normalization. So we construct an updated hBby a single step of the Gram-Schmidt algorithm [28], which is now orthogo-nal to hR and normalize both coefficient vectors. The first coefficient vector,unaltered by the Gram-Schmidt algorithm, gets divided by a normalizationfactor 1

4

√11 + 4 cos 2θ + cos 4θ. The second, orthogonal, coefficient vector

is found by a rather difficult expression, as a result of the orthogonalizationand normalization performed in this last step.

Staggering choice ~λΩ

Let us also consider the second staggering choice, introduced on the trianglein eq. (5.6). Note that the definitions need to be corrected for the com-plex valued staggering parameter. We find that following two vectors canserve as coefficient vectors for the two ground states on the hexagon for thisstaggering

eΩ,1 = (0, 0, 0, 1, 0, 0), (5.67)

eΩ,2 = (0,1√2

sin θ exp(−iφ), 0, 0, cos2(θ/2) exp(−2iφ), sin2(θ/2)).

These coefficient vectors are both orthogonal to each other and normalized.

5.3.3 Berry connections associated to the hexagon lattice

The Berry connections are found by the same procedure as used for thetriangle lattice. Starting from the explicit expressions for the coefficientvectors of the ground states, first we find their derivatives to parametersθ, φ and second we calculate their projections (inner products) on the groundstates. The result is the following: The matrices of Berry connections Aθ

and Aφ are of the form (0 −αα 0

)(5.68)

which we could write as −iασy in terms of the Pauli-y matrix, σy. Theexpression of α for Aθ is equal to

αθ =32√

2 cos 2φ(21 sin 2θ + 4 sin 4θ + sin 6θ)

(11 + 4 cos 2θ + cos 4θ)2√f(θ, φ)

(5.69)

where the expression in the square root in the denominator is equal to

f(θ, φ) = cos 6θ(cos 4φ− 1) + 2(63 + cos 4φ+ cos 2θ sin2 2φ+ 2 cos 4θ sin2 2φ)(5.70)

85

and for Aφ

αφ =−2√

2 sin2 2θ sin 2φ√cos 6θ(cos 4φ− 1) + 2(63 + cos 4φ+ cos 2θ sin2 2φ+ 2 cos 4θ sin2 2φ)

(5.71)We see that the expressions αθ and αφ determine the Berry connections Aθand Aφ completely.

For the second choice of staggering parameter (eq. (5.6)) and correspondingcoefficient vectors (eq. (5.67)) the derivatives and Berry connections have asimpler form. The Berry connection AΩ,θ is equal to the zero matrix as allinner products are zero, and AΩ,φ has the following expression

AΩ,φ =

(0 00 −2i cos2(θ/2)

)(5.72)

5.3.4 Berry curvature and Chern number associated to thehexagon lattice

We can now construct the Berry curvature from the Berry connections ineq. (5.69) and eq. (5.71) using its definition in eq. (4.52). For visibility let usgive its components separately: we find Fθφ[1, 1] = 0, Fθφ[2, 2] = 0 and

Fθφ[1, 1] = 0,

Fθφ[1, 2] =1

g(θ, φ)(4√

2 cos θ sin3 θ(−3(510 + cos 2θ + 2 cos 4θ − cos 6θ) sin 2φ

− 32 cos2 θ sin4 θ sin 6φ)),

Fθφ[2, 1] =1

g(θ, φ)(2√

2 sin3 θ(3(1021 cos θ + 3 cos 3θ + cos 5θ − cos 7θ) sin 2φ

+ 8 sin θ sin3 2θ sin 6φ)),

Fθφ[2, 2] = 0,

(5.73)

where the denominator is equal to

g(θ, φ) = (126 + cos 2θ+ 2 cos 4θ− cos 6θ+ 32 cos2 θ cos 4φ sin4 θ)3/2 (5.74)

We can simply see that taking the trace of Fθφ will result in zero, thereforewe conclude that the first Chern number is zero, the trivial value.We can also find the Berry curvature (Fµν , eq. (4.52)) for the explicit Berryconnections found with the second staggering choice. For AΩ,θ equal to thezero matrix, and AΩ,φ as given in eq. (5.72) the Berry curvature is equal to

FΩ,θφ =

(0 00 i sin θ

)86

Now we can find the first Chern number using its definition from section5.2.4. The result is

c1 =i

2π

∫P

TrF =i

2π

∫ π

θ=0

∫ 2π

φ=0dφ dθTrFΩ,θφ

= −∫ π

0sin θ dθ = −2. (5.75)

The first Chern number for the second choice of staggering parameter in thehexagon is non-trivial.

5.4 The Berry phase associated with the triangleand hexagon lattices

In the previous sections we have found the explicit forms of the Berry con-nection matrices for two different staggering choices in the triangle and thehexagon. In this section we will take the next step by calculating the Berryphase of a path in (θ, φ)-parameter space using the found Berry connections.

We will see that in the cases considered in this chapter, the expression forthe non-Abelian Berry phase can be simplified. We will explain the reasonand simplification for each case separately. Recall that the most generalexpression for the non-Abelian Berry phase is given by eq. (4.34) in section4.4,

U(t) = P exp

[∫ t

0A(t′)dt′

]. (5.76)

This formula includes several components which can make general evalua-tion difficult: It includes a matrix exponential and a path ordering operator,both are necessary to correctly work with the non-commutativity of the con-nection matrices.

We are working in the degenerate subspace of supersymmetric ground states,so we introduced the non-Abelian Berry connection. The term ‘non-Abelian’or non-commuting is used as the Berry connection A is matrix-valued whichdoes not commute in general. We see that the Berry phase is defined by amatrix exponential of the integral of the Berry connection. In general weshould take care with the matrix exponential as for a sum of matrices in theexponent we have to keep the Baker-Campbell-Hausdorff formula in mind.For two matrices A,B it states that exp(A) exp(B) = exp(C) holds, if

C = A+B+1

2[A,B] +

1

12([A, [A,B]] + [B, [B,A]])− 1

24[B, [A, [A,B]]] + . . .

(5.77)where [., .] denotes the commutator.

87

(a) θb = π4 and φb = π

2 (b) θb = π2 and φb = 2π

Figure 5.5: Plots of a path on the sphere; triangular loop and a circle aroundthe equator.

We will consider writing the integral of the Berry connection in terms of theparameters θ and φ, and only integrating over one of the parameters at thetime. So the integral of the Berry connection is split and we need to considera sum of matrices in the exponent. This is precisely where the commutativityof the matrices comes in. A related operation which is included in thedefinition of the non-Abelian Berry phase is the path ordering operator.This operator ensures the matrix exponential is calculated correctly. Itasks to multiply the matrices in the correct order of the path, where therightmost matrix corresponds to the first part of the path. The path entersthe expression by the boundaries of the integrals over the Berry connectionmatrices.

5.4.1 Closed path in parameter space

The Berry phase is calculated by integrating the Berry connection over apath through parameter space, in this section we will define the path wewill use. We choose a simple path starting in the North Pole, then movingdown on the sphere to a height θbound while keeping φ = 0 constant, nextwe move in lateral direction to φ = φbound keeping θ = θbound constant. Inthe last part we move back up to the North Pole (θ = 0), completing thetriangular cycle on the sphere. This path is shown in figure 5.5a for theboundary values θbound = π

4 and φbound = π2 . Note that we end with the

same parameters as we started with (θ = 0), because the coefficient vectorsare unambiguously defined on the North Pole, without dependence on φ. Sothis qualifies as closed loop in parameter space, where the initial and finalpoints are equal.

Note that we choose on each part of the path to take only one of the para-meters θ, φ variable and keep the other constant. This enables us to evaluateeach part separately by integrating one of connection matrices Aθ or Aφ, de-

88

(a) θb = π and φb = π (b) θb = π and φb = π4

Figure 5.6: Plots of a path on the sphere; a circle through both North andSouth Pole and an orange-sliced-shaped loop.

pending on which parameter varies. We will use this path for evaluating boththe Berry phase in the triangle and in the hexagon, and we will comparetheir results. Note that by the definition of the path above, it is defined bythe boundary values θbound and φbound. We can take 0 ≤ θbound ≤ π and0 ≤ φbound ≤ 2π.

Note that this definition of a triangular path on the sphere includes casessuch as a standard triangular loop (figure 5.5a), a circle around the equatorof the sphere (figure 5.5b), a circle through the North Pole (figure 5.6a)and the South Pole and a orange-slice-shaped path (figure 5.6b), which issuggested by Zhao et al. [56]. However, it is fundamentally different from aconical path, which is for example suggested in Jones et al. [26]. We havediscussed this ansatz in section 4.7, with the corresponding figure 4.4.

5.4.2 Non-Abelian Berry phase: mixing and non-mixing kinds

In this section we will calculate the Berry phase. We will consider boththe triangle and the hexagon and both the choices of staggering on thelattices. We have seen that the staggering choice ~λ in eq. (5.4) results inBerry connections of the form(

0 q(θ, φ)−q(θ, φ) 0

)(5.78)

in both the triangle for Aφ in eq. (5.30) and the hexagon for Aθ and Aφ ineq. (5.69) and eq. (5.71). In the triangle Aθ = 0 holds. The second staggeringchoice ~λΩ in eq. (5.6) results in Berry connections of the form(

0 00 ip(θ)

)(5.79)

89

in both the triangle for Aφ in eq. (5.43) and the hexagon for Aφ in eq. (5.72).In the triangle and hexagon Aθ = 0 holds for this staggering choice. Becauseof the diagonal form of eq. (5.79) we see that the corresponding Berry phasewill act as a Abelian Berry phase on the ground states, in other words thereis no mixing between the ground states. We will see this in more detailwhen we have calculated the matrix exponential. The form of eq. (5.78)equals iqσy, a constant times the Pauli-y matrix. Note that this is the casefor both connection matrices Aθ and Aφ. As σy trivially commutes withitself, we can see that the (integrated) connection matrices commute. Thisshows that in this case we can loosen the constraint of the path ordering op-erator for the matrix exponential. Trivially for the second staggering choiceresulting in connection matrix eq. (5.79), the connections Aθ and Aφ alsocommute.

Let us consider the general expression for the matrix exponential of theBerry connection matrices in eq. (5.78) and eq. (5.79). We can find that

exp

[−(

0 Q−Q 0

)]=

(cosQ − sinQsinQ cosQ

), (5.80)

where we have included a minus sign for convenience, which comes from twofactors i in the definition of Berry connection and Berry phase. The actionof eq. (5.80) includes mixing of the basis states in their two-dimensionalsubspace. Moreover, we find that the matrix exponentiation of eq. (5.79)results in

exp

[−(

0 00 iP

)]=

(1 00 exp(−iP )

). (5.81)

So the action does not mix the basis states, which gives rise to the termi-nology non-mixing. In these expressions we have used uppercase to stresswe have performed integrations on the entries denoted by lowercase p andq. The expressions for p, q, P,Q are real. Using the general form introducedabove we can calculate the Berry phase for each part of the path separately,and multiply the parts in the correct order. Note that if a connection matrixis equal to the zero matrix, its matrix exponential equals the identity ma-trix, so we can effectively neglect these parts of the path in parameter space.

Therefore, to find the explicit expressions for the non-Abelian Berry phasewe have to calculate the integrals. To be more precise we need to calculatethe integrals of the (non-zero) entries in the connection matrices.

Mixing non-Abelian Berry phase in the triangle lattice

Let us first look at the triangle lattice, which is the simpler case. Considerthe staggering parameter ~λ, as given in eq. (5.4). Remember we considera triangular closed path on the sphere, where θ changes in the first and

90

third part and φ in the second part. However, in the triangle we found thatthe Berry connection matrix Aθ is the zero matrix, so the first and thirdparts of the path do not contribute, we would multiply by the identity aftermatrix exponentiating. The integral over the second part we can evaluatedexplicitly: we integrate φ from 0 to φbound while keeping θ = θbound constant.The result is∫ φbound

φ=0dφ (1− cos θ)|θ=θbound = φbound(1− cos θbound). (5.82)

So we conclude that the unitary operator of the non-Abelian Berry phasefor the triangle with the staggering from eq. (5.4) is given by(

cos [φbound(1− cos θbound)] − sin [φbound(1− cos θbound)]sin [φbound(1− cos θbound)] cos [φbound(1− cos θbound)]

), (5.83)

a result conform the general discussion in eq. (5.78) and eq. (5.80).

Non-mixing non-Abelian Berry phase in the triangle lattice

The staggering choice ~λΩ, given ineq. (5.6), results for the triangle lattice inthe connection matrices Aθ = 0 and Aφ in eq. (5.43). The Berry connectionmatrix Aθ is the zero matrix, so the first and third parts of the path do notcontribute, we multiply by the identity after matrix exponentiating. Theintegral over the second part we can evaluated explicitly: we integrate φfrom 0 to φbound while keeping θ = θbound constant.∫ φbound

φ=0dφ i sin2(θ/2)

∣∣θ=θbound

= iφbound sin2(θbound/2) (5.84)

So we conclude that the non-mixing Berry phase for the triangle with thestaggering from eq. (5.6) is given by(

1 00 exp

[−iφbound sin2(θbound/2)

]) =

(1 00 exp

[12 iφbound(cos(θbound)− 1)

]) .(5.85)

Mixing non-Abelian Berry phase in the hexagon lattice

The Berry connections matrices, Aθ and Aφ, for the hexagon are both non-

trivial for the staggering ~λ in eq. (5.4), but both have the form in eq. (5.78),that is iqσy. Using the general form of the matrix exponential of iqσy,given in eq. (5.80), it is sufficient to calculate the integrals of the entriesof the connection matrices at this point. The non-Abelian Berry phase isthen found by multiplying the expressions for the matrix exponentials in thecorrect order of the path.

91

First part In the first part of the path we move starting from the NorthPole (θ = 0) to a height θbound. So we integrate Aθ from θ = 0 to θ = θbound,to be precise we integrate its off-diagonal element αθ, given by eq. (5.69).The result of this integration is

Q1 =π

4− arctan

(sec2(θbound)

), (5.86)

where we need to manually continue the expression at its apparent sin-gularity θbound = π/2 by defining it equal to −π/4 there. To be precisearctan

(sec2(θ)

)∣∣θ=π/2

is undefined, but limθ→π/2 arctan(sec2(θ)

)= π

2 .

Second part In the second part of the path we move starting in (θ, φ) =(θbound, 0) to (θ, φ) = (θbound, φbound). So we need to integrate Aφ from φ = 0to φ = φbound. This corresponds to evaluating the integral of αφ (eq. (5.71))from φ = 0 to φ = φbound. We find the following expression for the integral,denoting θb = θbound and φb = φbound,

Q2 =1√2

cos θb

[log(

cos θb sin2 θb(4√

2 + cos θb − cos 3θb))

− log(

cos θb sin2 θb(4 cos θb cos 2φb sin2 θb

+ cos2 φb

√32 + 2(30 + cos 2θb + 2 cos 4θb − cos 6θb) tan2 φb + 32 tan4 φb)

)].

(5.87)

This expression holds for the parameter values θb 6= 0, π2 , π and φb 6= π2 ,

3π2 .

We define the integral of αφ as follows to manually continue the expressionat these values for θb, φb. These expressions are found by calculating theappropriate limits.

Q2 = 0 for θb = 0,π

2, π

Q2 =1√2

cos θb

[log(

cos θb sin2 θb(4√

2 + cos θb − cos 3θb))

(5.88)

− log(

4 cos θb sin2 θb(√

2− cos θb sin2 θb))]

for φb =π

2,3π

2

Third part The third part of the path through parameter space is evalu-ated using Aθ again, by integrating the expression back from θ = θbound toθ = 0 while now setting φ = φbound. On the North Pole are the coefficientvectors independent of φ so we have returned to the same point in para-meter space as we started with, concluding the loop. Unfortunately, it doesnot seem possible to integrate the expression analytically, as a result of thegeneral parameter φ = φbound present. The numerical integral of eq. (5.69)provides us with the value Q3 for each choice of parameter bounds θboundand φbound. We choose an accuracy goal of eight digits for the numericalintegration.

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Complete path The results for each part of the path stated above are theexplicit expressions for placeholder Q in eq. (5.80). The total non-AbelianBerry phase of the triangular path is found by multiplying the matrix ex-ponentials for each part of the path. Here we have to keep the conventionof matrix multiplication is mind. We find that

Utot = U3U2U1 (5.89)

=

(cosQ3 − sinQ3

sinQ3 cosQ3

)·(

cosQ2 − sinQ2

sinQ2 cosQ2

)·(

cosQ1 − sinQ1

sinQ1 cosQ1

)=

(cos(Q1 +Q2 +Q3) − sin(Q1 +Q2 +Q3)sin(Q1 +Q2 +Q3) cos(Q1 +Q2 +Q3)

). (5.90)

So we see that Berry phase matrices from different parts of the triangularpath commute, as the sum Q1 + Q2 + Q3 is commutative. While commu-tativity is not the general case, here the connections are of the form iqσy,which (trivially) commute. This means that, while in general caution isneeded, in this case we are allowed to sum the Berry phase matrices beforematrix exponentiating them.

Non-mixing non-Abelian Berry phase in the hexagon lattice

The staggering choice ~λΩ in eq. (5.6) results for the hexagon lattice in theconnection matrices Aθ = 0 and Aφ in eq. (5.72). The Berry connectionmatrix Aθ is the zero matrix, so the first and third parts of the path do notcontribute, we multiply by the identity after matrix exponentiating. Theintegral over the second part we can evaluated explicitly: we integrate φfrom 0 to φbound while keeping θ = θbound constant.∫ φbound

φ=0dφ (−2i cos2(θ/2))

∣∣θ=θbound

= −2iφbound cos2(θbound/2) (5.91)

So we conclude that the non-mixing Berry phase for the hexagon latticewith the staggering from eq. (5.6) is given by(

1 00 exp

[2iφbound cos2(θbound/2)

]) =

(1 00 exp[iφbound(cos(θbound) + 1)]

).

(5.92)

5.4.3 Interpretation as a solid angle

A solid angle is a measure of the field of view covered by an object fromsome particular point. A solid angle can be expressed in the dimension-less SI unit called the steradian (sr) or square radian. Let us consider thesolid angle on the sphere. This is related to our work as we choose thestaggering parameter space to be a sphere S2, by the staggering choice

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~λ = (sin θ cosφ, sin θ sinφ, cos θ) (eq. (5.4)). The solid angle describes thearea on the surface of a sphere, for example when enclosed by a path on thesphere.As noticed in for example Solinas et al. [48] the geometrical parameter inthe expression for the (holonomic) operator is precisely equal to the solidangle. Solinas et al. [48] construct the unitary operator for a parameterchoice equivalent to ~λ

U = eiΦσy (5.93)

where the geometrical parameter Φ is the solid angle and σy the Pauli-ymatrix. The solid angle is defined by

Φ =

∫∫sin θ dθ dφ, (5.94)

which is also known as the surface integral of the standard volume form onthe sphere [9]. It is spanned by the parameter vector (~λ) on the parametermanifold (sphere). In other words this is the enclosed solid angle Ω =

∮dΩ

swept by the vector always pointing to the (θ, φ) direction.

We calculate the expression for the solid angle for the triangular path inparameter space we have proposed. We find

Φ =

∫ θb

0dθ

∫ φb

0dφ sin θ (5.95)

= φb(1− cos θb) (5.96)

for the triangular path determined by the angular boundaries θb = θbound

and φb = φbound. We have shown a schematic view of the triangular path inthe ~λ staggering parameter space in figure 5.7. So we see that the expres-sion of the solid angle is equal to the expressions found for the geometricparameter in eq. (5.82) for the triangle lattice with staggering parameter~λ = (sin θ cosφ, sin θ sinφ, cos θ) (eq. (5.4)).

The equivalence to the solid angle exemplifies the geometric nature of theprocedure, leading to the terminology of geometric quantum computation.The action of the constructed unitary is dependent on the enclosed solidangle in parameter space and not dependent on precise shape of the pathor the speed over the path. This fact also allows us to choose a simpletriangular path without an explicit time dependence.

5.5 The one-qubit quantum gates in the triangleand hexagon lattices

In the previous section we have found the explicit expressions for the Berryphase in the triangle and the hexagon after a closed triangular path. We

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λ2

λ3

λ1

Figure 5.7: Triangular path in the staggering parameter space

have considered two choices of the staggering parameter on the lattice, whichresult in different forms of the Berry phase matrix. In this section we willinterpret these results in the context of quantum gate matrices. Rememberthat our goal is to define the procedures for geometric quantum computingon the supersymmetric lattice. In this section we will take an importantstep towards this goal: the definition of the general one-qubit gate. We willshow that the Berry phase matrices found by the two different staggeringchoices are sufficient. As stated in section 3.7 a general one-qubit gatecan be constructed by multiplication of phase shift gates and rotation gates(eq. (3.12) and eq. (3.14), respectively). Therefore in this section we will firstshow the construction of the phase shift gate, second the construction of therotation gate and last the construction of a general one-qubit gate.

5.5.1 Phase shift quantum gate

The phase shift gate was introduced in eq. (3.12) in section 3.5. Its matrixform is given by

Phase(τ) =

(1 00 exp(iτ)

)(5.97)

or in different notation as(exp(−iτ/2) 0

0 exp(iτ/2)

). (5.98)

The difference between both notations is a global phase factor of exp(−iτ/2).We will choose the first notation in this section, but the second notation isalso used in literature. Its action consists of adding a phase factor exp(iτ)(of unit norm) to the computational basis state |1〉. As the basis state |0〉stays unchanged, it is clearly different from the operation of adding a (non-observable) global phase factor. If we compare the expression of the phaseshift gate in eq. (5.97) with our result for the Berry phase matrices eq. (5.85)and eq. (5.92) found for the staggering choice in eq. (5.6), we can clearly seethe similarity. For the triangle lattice we find the Berry phase given in

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eq. (5.85), which corresponds to a phase shift gate with a phase

τ3 =1

2φbound(cos(θbound)− 1). (5.99)

For completeness let us look at which values the phase can take. The expres-sion for τ3 in eq. (5.99) takes values between −2π and 0 for 0 ≤ θbound ≤ πand 0 ≤ φbound ≤ 2π. In other words we can construct a path (parametrizedby θbound and φbound) for any value of the phase τ3 of the phase shift gate.

For the hexagon lattice the Berry phase for staggering choice in eq. (5.6) isgiven by eq. (5.92). This forms corresponds directly to a phase shift gatewith a phase

τ6 = φbound(cos(θbound) + 1). (5.100)

The expression for τ6 in eq. (5.100) takes values between 0 and 4π for 0 ≤θbound ≤ π and 0 ≤ φbound ≤ 2π, so we can construct a path (parametrizedby θbound and φbound) for any value of the phase τ6 of the phase shift gate.Let us consider an example of the phase shift quantum gate. The matrixform in eq. (5.97) is equal to the Pauli-Z matrix, σz, for τ = π. Workingback to a path in θ, φ-parameter space, we can see that the following choicessatisfy τ = π. In the triangle we can take θbound = π and φbound = π, whilein the hexagon we can take θbound = 0 and φbound = π/2 to find σz. A moregeneral analysis is given in subsection 5.5.4.

5.5.2 Rotation quantum gate

The matrix form of the rotation quantum gate corresponds to the form of ageneric element of SO(2), the special orthogonal group (or rotation group)in two dimensions. The general form is

Rotation(ω) =

(cosω − sinωsinω cosω

)(5.101)

for a rotation over an angle ω. This matrix resembles the form of eq. (5.80),with the simple relation ω = Q. We conclude that the non-Abelian Berryphase matrices found in the triangle and hexagon for the staggering fromeq. (5.6), equal the matrix form of the rotation quantum gate precisely.Thus on the triangle lattice we can define a rotation quantum gate eq. (5.83),for a rotation over an angle φbound(1 − cos θbound). This direct parameterconnection allows us to tune the rotation angle in the rotation quantum gate(eq. (5.101)) by the boundary angles of the path. We find that in the tri-angle for a path defined by θbound and φbound as introduced in section 5.4.1,the minimum value of φbound(1 − cos θbound) is zero, for example for θboundor φbound equal to zero. The maximum value of φbound(1− cos θbound) is 4πfor parameter values θbound = π and φbound = 2π.

96

For the hexagon lattice we have found the same matrix form for the non-Abelian Berry phase, eq. (5.80), which agrees with the form of the rotationquantum gate in eq. (5.101). The expression for the rotation angle ω = Qis a lot more involved for the hexagon than for the triangle. Its expressionconsists of the sum of the integrated connections of the three parts of thetriangular path. The sum Q1 +Q2 +Q3 in eq. (5.90) is defined by eq. (5.86),eq. (5.87) and the numerical integral over eq. (5.69) (for the third part ofthe path). We will not include the complete expression here. By tuning thevalues of the bounds of the path, θbound and φbound, we can achieve rotationangles between −π/2 and 0. So if we can maximally achieve a (absolute) ro-tation angle of π/2 per path in parameter space, this means that to achievethe total range of angles 0 ≤ α ≤ 2π we have to repeat the closed path ifneeded.

Let us consider examples of the rotation gate for special angles. For clarity,consider the definition of the quantum rotation gate in eq. (5.101). Forω = π/2 we find (

0 −11 0

), (5.102)

which is closely related to the Pauli-Y gate, introduced in eq. (3.7) (in section3.5), we have Rotation(π/2) = −iσy. Combined with the Pauli-Z gate,which can be constructed as a special case of the phase shift gate, we canalso construct the Pauli-X gate by using that σx = iσzσy. In other wordsσz = Phase(π), σy = iRotation(π/2) and σx = −Phase(π)Rotation(π/2).Another example of the rotation gate we can consider is the case ω = π/4where we find

1√2

(1 −11 1

). (5.103)

This matrix form can be found with the hexagon lattice for parameterbounds θbound = π/2 and φbound = π/4, for example. From this expres-sion we can construct the Hadamard gate H, introduced in section 3.5, asfollows

1√2

(1 11 −1

)=

(0 11 0

)1√2

(1 −11 1

)(5.104)

using the Pauli-X matrix σx. In other words the Hadamard gate equals

H = σxRotation(π/4) = −Phase(π)Rotation(π/2)Rotation(π/4)

= −Phase(π)Rotation(3π/4).

For completeness, let us state that the found quantum gates denoted byPhase(τ) and Rotation(ω) do not commute, apart from the trivial cases τ =0, 2π or ω = 0 where one of the matrices equals the identity. This can easily

97

seen by

Phase(τ)Rotation(ω) =

(1 00 exp(iτ)

)(cosω − sinωsinω cosω

)(5.105)

=

(cosω − sinω

exp(iτ) sinω exp(iτ) cosω

)6=

(cosω − exp(iτ) sinωsinω exp(iτ) cosω

)= Rotation(ω)Phase(τ)

5.5.3 General one-qubit quantum gate

In the previous section we have shown that we can define the quantum gatesknown as the phase shift gate and the rotation gate using the Berry phaseafter a geometric path through staggering parameter space. We will showthat we can choose the necessary boundary values of the path to performany angle. In particular, we have already shown the three Pauli matricesand the Hadamard gate.

As introduced in section 3.7, a general one-qubit gate can be written interms of a product of a global phase factor, phase shift gates and a rotationgate. In more detail

Ugeneral = exp(iα)I2 Phase(β)Rotation(δ)Phase(ε) (5.106)

as introduced in eq. (3.28). Note that the phase shift gate may be defined byeq. (5.97) or eq. (5.98), both are related by a phase which can be includedin the global phase factor exp(iα). We state that the general one-qubitquantum gate, or the general element of U(2), can be written as eq. (5.106).Note that for a general element of SU(2) a similar expression holds whichequals eq. (5.106) with α = 0. A different notation in terms of matrixexponentials is given by

Ugeneral = exp(iα)I2 exp(iβσz) exp(iδσy) exp(iεσz) (5.107)

where σi denote the Pauli matrices. This form is based on the equivalentnotation of the phase shift quantum gate given in eq. (5.98). The equiv-alence of the expressions in eq. (5.106) and eq. (5.107) can be shown byredefinition of the global phase factor α. We can clearly see the resemblancewith our calculations of the non-Abelian Berry phase, where we use such anequality during the construction of the unitary matrix associated with thenon-Abelian Berry phase.This definition proves that our work towards a phase shift gate and a rotationgate is sufficient to construct any one-qubit gate. This is an importantstep towards quantum control for geometric quantum computation on thesupersymmetric lattice.

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5.5.4 Construction of general angles for a general one-qubitgate

In this section we have shown that the matrices based on the non-AbelianBerry phase can be interpreted as a phase shift gate (eq. (5.97) in subsection5.5.1 and a rotation gate (eq. (5.101) in subsection 5.5.2). These two one-qubit gates can construct any general one-qubit gate by the definition givenin eq. (5.106). However, the prerequisite is that we can produce the necessaryangles (denoted by β, δ, ε in eq. (5.106)) by means of a geometric path. Sucha triangular path through (θ, φ) parameter space is defined by the boundaryvalues θbound and φbound.

General phase shift angle

Let us first consider the phase shift gate eq. (5.97) with angle τ . The expres-sions for the angle τ in terms of the boundary parameters θbound and φboundare given by eq. (5.99) and eq. (5.100)

τ3 = 12φbound(cos(θbound)− 1)

τ6 = φbound(cos(θbound) + 1)(5.108)

for the the triangle and the hexagon lattices respectively. To find a generalphase shift over an angle β (or ε) with boundaries 0 ≤ β, ε ≤ 2π, we need tochoose θbound and φbound such that τ = β. Keep in mind that the followingconditions hold: 0 ≤ θbound ≤ π and 0 ≤ φbound ≤ 2π. For clarity let uswrite θb = θbound and φb = φbound.

For the triangle lattice we suggest to take θb = π and φb = 2π − β, to findτ3 = β−2π. As a phase of ±2π is trivial, exp(±2πi) = 1, with this choice wecan construct a general phase shift of any angle β, 0 ≤ β ≤ 2π. A differentsuggested path would be to take θb = π/2 and φb = 2(π−β), which producesτ3 = β − π for possible 0 ≤ β ≤ π. While this is not the wanted equalityτ3 = β, the minus sign of exp(−iπ) can be compensated for in two ways.We can apply the Pauli-Z matrix, σz, for example produced with the firstmethod by θb = π and φb = π. On the other hand we can find τ3 = β byapplying the second method twice, that is with θb = π/2 and φb = 2π − β.

Note that in general the precision of the produced phase shift angle requiresthe same precision in the tuning of the parameters of the geometric path. Weimagine that tuning to rational multiples of π might be easier to perform.Of course the relations in eq. (5.108) may be inverted to find the neededvalue of φb for any choice of θb. Or vice versa the needed value of θb for anychoice of φb. For example in the triangle lattice we can calculate the neededφb (for θb 6= 0) by

φb =2β

cos θb − 1.

99

This requires precise tuning of φb for nontrivial choices of θb.

For the hexagon lattice the most convenient choice is to take θb = π/2 andφb = β, which results in τ6 = β. The choice of θb = π/2 is convenient in boththe expressions of the triangle and hexagon, as cos(π/2) = 0, although in thetriangle some more work is required. So we find that the relation between thepath and the phase shift angle can be directly stated for the one-qubit phaseshift gate constructed by the non-Abelian Berry phase. Choosing the SouthPole for the triangle and the equator for the hexagon with the boundaryparameter θb, allows us to relate the boundary parameter φb directly to theneeded phase shift angle β (or ε).

General rotation angle

Let us consider the rotation gate eq. (5.101) with angle ω. The expressionfor the angle in terms of the boundary parameters θbound and φbound is givenby

ω = φbound(1− cos θbound) (5.109)

for the triangle lattice. This expression is found by the integration over thesecond part of the triangular path (Q2), while the first and third parts resultin zero (Q1, Q3 = 0). The rotation angle found in the triangle is thereforegiven by a simple relation. To find a general rotation angle δ, for the generalone-qubit quantum gate in eq. (5.106), we can make the same choice as forthe phase shift gate in the hexagon. Consider a path parametrized by theboundary values θb = π/2 and φb = δ, then we find ω = δ. With this choicewe can tune to any angle 0 ≤ δ ≤ 2π.

In the case of the hexagon lattice the expressions for the integrated con-nections are more involved. Their equations are given in eq. (5.86) (Q1),eq. (5.87) (Q2) and the numerical integral over eq. (5.69) (Q3). The sumω = Q1 + Q2 + Q3 corresponds to the total acquired angle after the geo-metric triangular loop. Summing the expressions from each part is allowedas the (integrated) connection matrices commute. The total angle ω satisfiesthe bounds −π/2 ≤ ω ≤ 0, this means that to achieve a general rotationangle of 0 ≤ δ ≤ 2π, we have to repeat some geometric loop a maximum offour times. In such a procedure of repeating the geometric loop we suggesttwo options.Say one would like to achieve a general angle of δ, with 0 ≤ δ ≤ 2π, thenequivalently one can look for a δ, with −2π ≤ δ ≤ 0 and δ = δ + 2π.Constructing the angle δ fits the scheme using the geometric loops. A firstpossible procedure is to perform a geometric loop resulting in an angle ofδ/4 four times. A second possibility is to perform geometric loops achievingthe maximum value of −π/2 as many times as needed and finishing with afourth (non-maximal) loop to find the remaining angle to complete the total

100

angle δ. The procedure presented here fits both possible ways to finding ageneral rotation angle δ.

Consider taking the parameter value θb = π/2 which simplifies the expres-sions for Q1, Q2 and Q3 to

Q1 = −π/4,Q2 = 0,

Q3 =∫ 0π/2 αθ(θ, φ) dφ,

(5.110)

where αθ denotes the matrix element of the Aθ connection matrix givenin eq. (5.69). We are not able to find a general expression evaluating theintegral.The procedure to find a general rotation angle δ, with 0 ≤ δ ≤ 2π, followstwo steps. First we need to achieve the maximum value of Q = −π/2 upto three times. This maximum can be achieved by choosing the geometricpath parameters θb = π/2 and φb = π/2. This maximal path needs to be

repeated 3 −⌊

δπ/2

⌋times. After this first step of repeating the maximal

path, we are left with the following angle

δ = δ − 2π +π

2

(3−

⌊δ

π/2

⌋), (5.111)

where b.c denotes the floor function. The expression for δ is moved to thedomain −π

2 ≤ δ ≤ 0 by including the factor −2π. The second path is

choosing the geometric path parameter φb such that Q = δ. So we takeagain θb = π/2 and now approximate the expression for Q = Q1 + Q3 (seeeq. (5.110)) by

Q ≈ −π2

sin2(φb). (5.112)

We can now find the necessary path parameter φb by inverting this relation,so by choosing

θb =π

2, φb = sin−1

√− δ

π/2

(5.113)

we can find the remaining angle δ to complete the total rotation angle δ. Wecould also propose to use the second step (achieving an angle −π

2 ≤ δ ≤ 0)four times in succession. However, we imagine that constructing the pathof the maximal angle (where φb = π

2 ) is more error resistant than to find φbwith eq. (5.113).

The following method is more reliant on approximations. We can choosethe geometric path parameter φb = π/4 first, such that the expression Q3

gives zero contribution. Then the total Q can be roughly approximated by

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only Q1, the approximation is better where Q2 is small at θb = 0, π/2, π.The expression for Q ≈ Q1 takes values −π/4 ≤ Q1 ≤ 0. Let us considerimplementing this ansatz instead of the second step presented before. Toachieve a angle of −π

2 ≤ δ ≤ 0 we would have to repeat the geometric pathused here with φb = π/4 once or twice. We can again choose to do themaximal contribution first and the remaining angle in the second loop, orwe could do the procedure for δ/2 twice. The maximum value of Q1 = −π

4is attained for θb = π/2. Note that in our definition of Q1 we needed tomanually continue the expression for this θb, we assume this does not changeits applicability here. Otherwise the other suggested method is preferred.Recall the expression for the angle

Q1 =π

4− tan−1

(sec2(θb)

), (5.114)

and take δ = δ/2 or δ = δ+ π/4 depending on your choice of procedure. Sowe have that

δ =π

4− tan−1

(sec2(θb)

)(5.115)

which can be inverted to find the formula for the needed value of the pathparameter θb

θb = sec−1

(√tan(π

4− δ))

. (5.116)

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Chapter 6

Bow tie lattice

6.1 Introduction

In this chapter we will consider a third geometry for the supersymmetriclattice model. We coined it the ‘bow tie’ because of their resemblance inshape. The lattice is constructed by connecting two triangular lattices bya single edge. The bow tie lattice is shown in figure 6.1. We consider asupersymmetric lattice model on this lattice, while making new definitionsfor the specific Mk model and staggering on the lattice. The goal of thischapter is to investigate a non-trivial two-qubit gate on the bow tie lattice.To this end we will calculate the non-Abelian Berry phase after a closedpath in the (new) staggering parameter space. The construction of the bowtie lattice by coupling two triangle lattices allows us to use the definitionof the qubit on the triangle lattice and the results for the one-qubit gatefound in the previous chapter 5. This chapter is structured in the sameway as the previous chapter: First, we introduce the supersymmetric modelon the bow tie with the chosen staggering parameters. We construct thecoefficient vectors of the supersymmetric ground states and work out theBerry connections. Second, we construct the expression for the non-AbelianBerry phase and discuss its form in detail. Third, we investigate using theBerry phase to define an two-qubit gate which creates entanglement betweenthe two qubits defined on the bow tie lattice.1

6.2 The bow tie lattice

The bow tie lattice is constructed by connecting two triangular lattices. Weadd an extra edge to the model between vertices of the triangles. We havediscussed the lattice model based on a single triangle, with three latticesites. In the bow tie we number the lattice sites with 1, . . . , 6, using 1, 2, 3in the first triangle and 4, 5, 6 in the second triangle, and we connect sites

1The Mathematica files can be obtained by an email to the author.

103

Site 1

Site 2

Site 3

Site 4

Site 5

Site 6

Figure 6.1: Schematic view of the bow tie lattice configuration.

3 and 6. This is also shown in figure 6.1. As discussed earlier, we use theM1 (nearest-neighbor exclusion) model on a triangle lattice, such that onefermion is the maximal occupation. However, on the edge connecting thetriangles we impose the M2 model. This means that we allow the adjacentsites 3 and 6 both to be occupied at once, but the nearest-neighbor exclusionon the triangle forbids to occupy sites 2 and 3 or sites 2, 3 and 6 for thatmatter. This choice of the supersymmetric model results in the fact thatall states on each of the triangles are allowed, while introducing a couplingbetween them. The mixed M1/M2 rules result in a maximum occupation oftwo particles on the bow tie lattice, in other words one particle on each of thetriangles. This is important for the interpretation of the bow tie as a systemdescribing two qubits. Each of the qubits (triangles) is in principle allowedto be in any state, but the coupling adds interaction between the qubits.We choose to consider the simplest geometry describing two coupled qubits.However, the definition could be extended to include more connections be-tween the triangles. The model can easily be extended to three connectionsbetween lattice sites i and i+3, creating a lattice we have coined the ‘prism’.

We start the analysis by counting the number of possible states on the bowtie lattice. Trivially, there exist one state with zero particles; the emptylattice. There are six one-particle states, this can easily be seen as the pos-sibility of a particle occupying each of the six lattice sites. The combinedM1/M2 model allows all nine possible two-particle states. For complete-ness, if we consider only the M1 model, then adjacent occupation of twosites on different triangles is not allowed, which subtracts a possible statefrom the total number of states. This would be one state in the bow tieand three states in the prism. These states are still allowed if we imposethe M2 model on each connection we add. As a result of the M1 model ineach of the triangles, three-particle states are not allowed on the bow tielattice. We can now easily find the Witten index from the number of states:W = 1− 6 + 9 = 4, which implies (at least) four zero-energy ground states.The counting can also be seen by considering that the zero-particle stateforms a doublet with one one-particle state and the other five one-particle

104

states form doublets with five two-particle states. This leaves 9 − 5 = 4two-particle singlets which are supersymmetric zero-energy ground states.As expected this corresponds to the two supersymmetric ground states oneach of the triangle lattices, resulting in four ground states on the bow tielattice.

In the next section we will introduce the staggering parameters on the bowtie lattice; we will need two types, ~λ and µ, as introduced in section 2.2. Wewill make use of mirror symmetry for convenience. Then we will discuss theconstruction of the coefficient vectors of the ground states on the bow tielattice and the work to find the Berry connections.

6.2.1 Staggering

Let us first introduce notation for the lattice states on the bow tie, which isequivalent to the notation for the triangle and the hexagon lattices. We areparticularly interested in the two-fermion sector, where the ground statesare present. We take the following basis of the two-fermion sector (Nf =2), to be able to construct a matrix representation of the mixed M1/M2

supersymmetric Hamiltonian.

|1, 4〉 , |1, 5〉 , |1, 6〉 , |2, 4〉 , |2, 5〉 , |2, 6〉 , |3, 4〉 , |3, 5〉 , |3, 6〉 (6.1)

Note that as a result of the fermionic character of the particles on the lattice,for example |1, 4〉 = − |4, 1〉. We will choose the convention, as used in thebasis, |i, j〉 where i < j.

Consider the definition of the supersymmetric Hamiltonian on the bowtielattice. The supersymmetric operator Q+ is found starting from its generaldefinition for the Mk supersymmetric lattice model

Q+ =6∑j=1

∑a,b=1,2;a≤b

λ[a,b],jd†[a,b],j . (6.2)

as introduced in eq. (2.13) in section 2.2. The operator d†[a,b],j is defined asadding a particle at site j, which then lies on position b in a chain of length a.The staggering parameter λ[a,b],j shows corresponding labeling. In this casewe consider the M2 model, for k = 2 we can rewrite the general staggeringparameter λ[a,b],j in terms of λj and µj . This works as follows: λ[1,1],j = λj ,

λ[2,1],j = λjµj and λ[2,2],j = λjµj−1. This definition can be seen as taking ~λas the ‘M1 staggering’ parameter and ~µ as the ‘M2 staggering’ parameter.The result in the definition of the supersymmetric charge is

Q+ =

6∑j=1

[λjd†[1,1],j + λjµjd

†[2,1],j + λjµj−1d

†[2,2],j

]. (6.3)

105

Note that the notation used here was in the first place defined in a latticechain equipped with theM2 model. In the last two terms the indices can onlytake values of sites which are next to each other, i.e. ‘j−1’ is connected to ‘j’,but does not necessarily have a site number increased by one in the bow tielattice. We must remember that the d†[2,b],j terms concern adjacent sites on

different triangles, following from our definition of the mixed M1/M2 model.Visually one could imagine the staggering parameters λj to be associatedto a lattice site, while the µj are associated to a connecting edges betweentriangles. We have shown the bow tie lattice with the staggering parameters~lambda and µ in figure 6.2. We take λj and µj real-valued. The general

case presented so far holds for both the bow tie and the prism. In the prismthere are three parameters µ1, µ2, µ3, one for each connection. If we considerthe bow tie, with only one connection between the triangles, to be precisebetween sites 3 and 6, we substitute µ1 = µ2 = 1 and keep µ3 variable, suchthat

Q+ =6∑j=1

(λjd†[1,1],j

)+ λ3µ3d

†[2,1],3 + λ6µ3d

†[2,2],6. (6.4)

Remember that the M1 creation operator d†[1,1],j is defined by d†[1,1],j = d†j =

Pi−1c†iPi+1 where c†i is the fermionic creation operator on site i and Pi =

1 − c†ici is the projection operator on site i. Note again that the indicesi−1, i+1 must be evaluated cyclic in each of the triangles. The M2 creationoperators, after Fokkema [19], are now defined by

d†[2,1],j = Pj−1c†jc†j+1cj+1Pj+2, (6.5)

d†[2,2],j = Pj−2c†j−1cj−1c

†jPj+1. (6.6)

The supersymmetric charge Q− with annihilation operators is defined equiv-alently, Q− = (Q+)†. Now we can construct the HamiltonianH = Q+, Q−.It is convenient to use its matrix representation in the basis stated ineq. (6.1). The action of the Hamiltonian on a general two-fermion stateis given by

H |i, j〉 = Q+, Q− |i, j〉 = Q+Q− |i, j〉 , (6.7)

where we used that two fermions equals the maximal occupation, by theprojection operators in the definition of Q+.

6.2.2 Mirror symmetry

We will analyze the (ground) states on the bow tie in the context of themirror symmetry. The use of symmetries of supersymmetric lattices wasintroduced in section 2.4. We consider the bow tie with a mirror placed inthe middle of the coupling edge. Therefore we identify lattice sites i (left

106

λ1

λ2

λ3

λ4

λ5

λ6

µ3

Figure 6.2: The bow tie lattice with the definition of the staggering para-meter on this lattice.

triangle) and i + 3 (right triangle) with each other. Accordingly, the stag-gering parameter ~λ is simplified by λ1 = λ4, etc. This leaves us with atotal of four staggering parameters λ1, λ2, λ3, µ3 to describe the system.For staggering parameter ~λ = (λ1, λ2, λ3) we choose the definition alsoused in the triangle (and hexagon) of the vector to a point on the unitsphere parametrized by angles θ and φ, introduced in eq. (2.15) in section2.2: ~λ = (sin θ cosφ, sin θ sinφ, cos θ). So effectively by imposing the mirrorsymmetry we take the angular parameters θ and φ equal in both triangles:θL = θR and φL = φR.

The staggering parameter µ ≡ µ3 which parametrizes the coupling betweenthe triangles, is taken to be a real scalar parameter larger than zero. Wenote that the uncoupled situation, two separate triangles, corresponds tosetting the M2 staggering parameter to one, µ = 1. This follows from directcalculations with the explicit expressions of the (staggered) supersymmetriccharges. Two uncoupled triangles will not impose any condition on the sitesthat would be coupled, so it is natural that the dependence on µ needs tovanish. In other words without coupling the state |3, 6〉 is equivalent to anyother two-qubit product state.

The mirror symmetry will be used in the next section to simplify the con-struction of the ground states. We would like to reduce the number ofcoefficient vectors we need to find. Therefore we will separate the coefficientvectors based on their eigenvalue of mirror matrix S. Mirror matrix S is de-fined in the basis of the two-particle sector, eq. (6.1), and performs the swapi↔ i+3 in the kets: |i, i+ 3〉 ↔ |i+ 3, i〉 with i mod 6 (using i = 6 insteadof i = 0). Its matrix representation is a sparse matrix with the non-zero value−1 at the indices (1, 1), (2, 4), (4, 2), (3, 7), (5, 5), (7, 3), (6, 8), (8, 6), (9, 9) ofthe matrix. Note that the fermionic character of the states requires that|i, j〉 = − |j, i〉, which comes in effect when we rewrite the transformed statesback into their form as stated in the basis in eq. (6.1). Recall that there arefour zero-energy ground states on the bow tie. Now we find that with re-

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spect to the mirror operator there are three states with eigenvalue s = −1and one state with s = +1. In the next section we will discuss their explicitforms, where their eigenvalues will be apparent.

6.2.3 Ground states on the bow tie

In this section we construct the coefficient vectors of the zero-energy groundstates on the bow tie lattice. The coefficient vectors are defined relativeto the basis given in eq. (6.1) equivalently to the definition in the triangleand hexagon. For better insight, we choose to start the construction ofthe ground states on the bow tie by the construction of the ground stateson two uncoupled triangles. Therefore we start from our findings for thetriangle lattice in subsection 5.2.2. This set of two coefficient vector for thesupersymmetric ground states on the triangle is constructed such that theylie in the null space of the Hamiltonian, they are orthogonal to each otherand independent of φ at θ = 0, the North Pole. Let us repeat these resultsfrom eq. (5.11). The coefficient vectors of the ground states in the triangleare defined as

f1 = (cos θ cos2 φ+ sin2 φ, (cos θ − 1) cosφ sinφ,− sin θ cosφ), (6.8)

f2 = ((cos θ − 1) cosφ sinφ, cos2 φ+ cos θ sin2 φ,− sin θ sinφ). (6.9)

Here we used the Hamiltonian dependent on the staggering parameter ~λ =(sin θ cosφ, sin θ sinφ, cos θ). Note again that on the North pole (θ = 0)the states are equal to (1, 0, 0) and (0, 1, 0), whereas the staggering para-meter equals ~λ = (0, 0, 1). This provides an interpretation of the stateson the triangle as qubit states as introduced before: |0〉qubit = (1, 0, 0) and|1〉qubit = (0, 1, 0). For the bow tie lattice we will extend this definition toa qubit on each of the two triangles. In the first triangle, with sites num-bered 1, 2, 3 we have that |0〉qubit = |1〉 and |1〉qubit = |2〉, equivalent to thetriangle lattice. In the second triangle, with sites numbered 4, 5, 6 we havethat |0〉qubit = |4〉 and |1〉qubit = |5〉. The two-qubit basis, generally givenby |00〉 , |01〉 , |10〉 , |11〉, is then on the bow tie lattice defined by the latticestates |1, 4〉 , |1, 5〉 , |2, 4〉 , |2, 5〉.

Starting from the previously defined ground states for each of the triangles,we construct the ground states on two triangles (without extra added edges)by using the Kronecker product, denoted ⊗. Denoting the ground states onthe triangles (left=L, right=R) as f1,L, f2,L, f1,R, f2,R, with angles θL, φLand θR, φR, we find

F1 = f1,L ⊗ f1,R, (6.10)

F2 = f1,L ⊗ f2,R, (6.11)

F3 = f2,L ⊗ f1,R, (6.12)

F4 = f2,L ⊗ f2,R. (6.13)

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Let us write out F1 as an example.

F1 = f1,L ⊗ f1,R = (f1,L,1f1,R,1, f1,L,1f1,R,2, f1,L,1f1,R,3,

f1,L,2f1,R,1, f1,L,2f1,R,2, f1,L,2f1,R,3,

f1,L,3f1,R,1, f1,L,3f1,R,2, f1,L,3f1,R,3) (6.14)

= ((cos θL cos2 φL + sin2 φL)(cos θR cos2 φR + sin2 φR),

(cos θL cos2 φL + sin2 φL)(cos θR − 1) cosφR sinφR,

− (cos θL cos2 φL + sin2 φL) sin θR cosφR,

(cos θL − 1) cosφL sinφLf1,R,

− sin θL cosφLf1,R) (6.15)

The states based on coefficient vectors F1, F2, F3, F4 satisfy the wanted con-ditions: they are orthogonal to each other, normalized and they lie in thenull space of the Hamiltonian describing the supersymmetric model on twotriangles.

The next step is to derive the coefficient vectors of the ground states onthe bow tie, with the geometry of coupled triangles. We need to includethe coupling, parametrized by staggering parameter µ, into the form of theground states, as we are looking for the expressions corresponding to theground states for any value of coupling µ. By looking into the matrix repre-sentation of the Hamiltonian describing the two triangles with a connectionbetween sites 3 and 6, we can see that we can adjust the coefficient vectorsF1, F2, F3, F4 to lie in the null space of the new bow tie Hamiltonian. Wemultiply the last component of each coefficient vector by 1/µ. As the uncou-pled triangles correspond to µ = 1, it will be convenient to write µ = 1 + dµand take the deviation dµ from the uncoupled case as parameter. Let usdenote this set of vectors by F1, F2, F3, F4. This simple manipulation ofthe coefficient vectors is enough to construct zero-energy eigenvectors of theHamiltonian describing the bow tie. Unfortunately, this action does neitherconserve orthogonality nor normalization. We have to do work to regainthese features necessary for the calculation of the Berry phase.

At this stage we use our considerations of the mirror symmetry in the bowtie to simplify the orthogonalization and normalization procedures. Firstof all, the set of parameters reduces to θ ≡ θL = θR, φ ≡ φL = φR anddµ ≡ dµ3 = µ3 − 1. We rewrite the coefficient vectors to eigenvectors ofthe mirror operator S, such that S |ψ〉 = s |ψ〉. There are three states witheigenvalue s = −1 and one state with s = +1. Their eigenvalue of S ismost clearly seen on the North Pole θ = θL = θR = 0, but is unchanged fornonzero θ, φ. To be precise, we have the following vectors with eigenvalues = −1: F1, F4 and 1√

2(F2 + F3), evaluated with θL = θR and φL = φR. The

vector with eigenvalue s = +1 is 1√2(F2 − F3). This vector is normalized

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and also orthogonal to the other three vectors. The explicit form of thecoefficient vector is

1√2

(F2 − F3) =1√2

(0, cos θ,− sin θ sinφ,− cos θ, 0,

sin θ cosφ, sin θ sinφ,− sin θ cosφ, 0) (6.16)

This orthogonality property will remain during the re-orthogonalization ofthe three vectors with eigenvalue s = −1. So we are able to reduce thenumber of coefficient vectors we need to orthonormalize by separating offthe s = +1 subspace. This result is not presented here, but we will work witha set of orthonormal coefficient vectors in the next subsection. Moreover,this procedure will not reduce the possibilities of the two-qubit gate in theend, because a two-qubit gate can be supplemented by a one-qubit gate toperform a rotation out of the s = −1 subspace of the total qubit space.

6.2.4 Berry connections

In the previous subsection we have found expressions for the coefficient vec-tors of the four zero-energy ground states on the bow tie. By a comparableprocedure used for the triangle and hexagon lattices, we will derive the Berryconnections in this subsection. Following from our choice of staggering para-meters ~λ(θ, φ) and µ, we need to construct three connections: Aθ, Aφ andAdµ. As stated in the previous section we split the space of ground statesinto subspaces of eigenstates of the mirror operator.

Recall the definition of the Berry connection, eq. (5.14), in terms of a pro-jection of the derivative of the basis states back on the basis states. Weconsider derivatives with respect to θ, φ and dµ. The connections are four-dimensional square matrices, where we simplify its form by considering thesubspaces of the mirror operator. The matrix forms are found to be anti-symmetric. As the ground state with eigenvalue s = +1 lies in a differentsubspace its projection on states from the s = −1 subspace is zero. Theconnection matrices have the general form

A4×4 =

0

A3×3 00

0 0 0 0

(6.17)

so we will concentrate on the form of A3×3 in the remainder of this subsec-tion.

As the exact expressions for the orthonormalized set of the coefficient vec-tor are large, we will not show them here. The procedure to find the Berry

110

connections is the same as in the triangle and hexagon. For the three con-nection matrices,Aθ, Aφ and Adµ, we find the following general form, withdifferent expressions for matrix entries a, b, c which are functions of θ, φ anddµ,

A =

0 a b 0−a 0 c 0−b −c 0 00 0 0 0

(6.18)

Let us discuss the connection matrices further by considering some specialparameter values. These findings will be used in the next section to simplifythe calculation of the Berry phase. For example we can find the zero matrixfor the connections in the following cases: For Adµ if θ = 0 (‘on the NorthPole’) or φ = 0 and for Aθ if φ = 0.Let us consider what happens for the matrix form of the connections foruncoupled triangles, that is for µ = 1 ↔ dµ = 0. In that case we find thata = 0 and also b = −c.

There is an important difference between the connection matrices intro-duced above, compared to the connection matrices found in the triangleand the hexagon. In subsection 5.4.2 we have stated that for the trian-gle and hexagon lattices both Aθ and Aφ equal a constant times σy, thePauli-Y matrix. We concluded that the connection matrices commute andpath ordering does not need to be strictly enforced. However for the bowtie case, the connections are of the form of eq. (6.18) and these matricesdo not commute for general a, b, c. This feature can be constructed fromthe theory of Gell-Mann matrices. This set of three-dimensional matricesdefines the generators Ta = λa/2 of the Lie algebra su(3) of SU(3) in thedefining representation. The Gell-Mann matrices λa are a set of eight lin-early independent traceless Hermitian square matrices, which correspond tounitary group elements after exponentiation. Their matrix forms are givenby (where only non-zero entries are shown)

λ1 =

11

, λ2 =

−ii

, λ3 =

1−1

, λ4 =

1

1

,

(6.19)

λ5 =

−i

i

, λ6 =

11

, λ7 =

−ii

, λ8 =1√3

11−2

.

So we can clearly see that the three dimensional block in the matrix formof the connections, eq. (6.18), can be written in terms of the Gell-Mannmatrices by

A3×3 = iaλ2 + ibλ5 + icλ7. (6.20)

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Now, from their interpretation as generators of SU(3) the Gell-Mann ma-trices have known commutation relations. For Ta = λa/2 we have that

[Ta, Tb] = ifabcTc, (6.21)

where summation is implied by Einstein convention. The factor fabc is calleda structure constant. As we are interested in the set of matrices λ2, λ5, λ7,the only relevant structure constant is f257 = 1/2. So for example, we have

[T2, T5] = if257T7 =i

2T7.

Their commutation relations are important in relation to the Baker-Campbell-Hausdorff formula for matrix exponentials. This structure of the Gell-Mannmatrices shows us that the BCH formula will never terminate. Therefore weare not allowed in our case of the Berry phase to first sum the integratedconnections before matrix exponentiating. We have to take path orderinginto account by exponentiating first and multiplying in the correct order ofthe path.

6.3 The Berry phase associated to the bow tie lat-tice

In this section we will consider the calculation of the non-Abelian Berryphase based on the ground states on the bow tie lattice. The path in para-meter space is very similar to that considered for the triangle and hexagonlattices. For staggering parameters θ and φ we consider the same path inparameter space. In this case we need to add the change of µ into thescheme. We choose to turn ‘on’ coupling µ before the triangular loop andturn it ‘off’ after. Here the ‘on’ value is µ 6= 1 or dµ 6= 0 and ‘off’ corre-sponds to µ = 1 or dµ = 0. The total scheme is therefore as follows. Theparameters start on the North Pole (θ = 0, φ = 0) with the coupling off(dµ = 0). We change dµ adiabatically to a non-zero value dµbound. Thenwe perform the three parts of the triangular loop of θ to θbound, φ to φboundand θ back to 0. Last, we change dµ back to zero, which closes the loop in(θ, φ, dµ)-parameter space. The parameter constraints, as mentioned earlier,are given by 0 ≤ θbound ≤ π, 0 ≤ φbound ≤ 2π and dµ > −1.

The extended triangular path introduced here defines the parameter inter-vals over which we integrate the appropriate Berry connections. For thischoice the connection matrices are trivial on some parts, which reduces theneeded calculations. As the connection Adµ = 0 for θ = 0 the parts whereµ is changed do not contribute. Just as in the case of the hexagon we alsofind that Aθ = 0 for φ = 0, therefore the first part of the triangular pathon the sphere does not contribute. Note that during the integration of the

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triangular path the expressions are dependent on the value of µ, althoughthe integration over dµ itself does not contribute.

The two integrations that contribute to the expression of the non-AbelianBerry phase for the bow tie lattice are the integration of Aφ from φ = 0to φ = φbound with θ = θbound and dµ = dµbound constant and of Aθ fromθ = θbound to θ = 0 with φ = φbound and dµ = dµbound constant. As shownin the previous section we have to take the path ordering into account. Wewill matrix exponentiated the integrated connections separately and multi-ply them in the correct order of the path. If we number the parts of the pathby 1, . . . , 5, then parts 3 and 4 are non-trivial. The total unitary is thereforegiven by Utot = U4U3, as U1 = U2 = U5 = I4. Matrix-vector multiplicationis defined from the right, so the unitary corresponding to part 3 is appliedbefore the unitary of part 4.

The integration of connection Aθ is trivial for φbound = π2 , π,

3π2 , 2π. More-

over, this integrated connection is trivial if dµbound = 0, as expected becausethis corresponds to two uncoupled triangles and we found the same trivialconnection matrix in the triangle in subsection 5.2.3. The integrated con-nection Aφ has a zero for the entry denoted by a in the case that dµbound = 0and also for θbound = π.

Let us introduce the general form of the matrix exponent of the integratedconnections. We start from the form of the Berry connections in eq. (6.18).After (numerical) integration the matrix takes the same form, with entriesa, b, c now respresenting the resulting numerical values of the integration.Unfortunately, we were not able to find the analytical expressions for theintegration of the connections for general boundary values. Recall the defi-nition of the non-Abelian Berry phase introduced in subsection 5.2.3, wheresymbolically

γ = i

∫A (6.22)

and therefore the corresponding unitary is equal to

Uγ = exp(iγ). (6.23)

The result for the four-dimensional unitary matrix Uγ is given by Uγ =exp(i2∫A), where A (and

∫A) is a matrix of the form of eq. (6.18). As the

matrix entries a, b, c are real, the most convenient form is in terms of sinesand cosines of the argument

√a2 + b2 + c2. For clarity let us introduce short

forms; let us denote D = a2 + b2 + c2, C = cos(√

a2 + b2 + c2)

= cos√D

and S = sin(√

a2 + b2 + c2)

= sin√D.

113

In terms of these short forms Uγ is given by

c2+(a2+b2)C

D−bc+bcC−a

√DS

Dac−acC−b

√DS

D 0

−bc+bcC+a√DS

D

b2+(a2+c2)CD

−ab+abC−c√DS

D 0

ac−acC+b√DS

D−ab+abC+c

√DS

D

a2+(b2+c2)CD 0

0 0 0 1

(6.24)

=1

D

c2 +

(a2 + b2

)C −bc+ bcC − a

√DS ac− acC − b

√DS 0

−bc+ bcC + a√DS b2 +

(a2 + c2

)C −ab+ abC − c

√DS 0

ac− acC + b√DS −ab+ abC + c

√DS a2 +

(b2 + c2

)C 0

0 0 0 D

This form is written in the basis of eigenvectors of the mirror operator, whichis the basis we used to calculate the connection matrices. For the correct im-plementation for all possible paths in parameter space, we have to continuethe functional forms of the matrix entries for the case that a = b = c = 0.Then the matrix form equals the four-dimensional identity matrix, whichcan be found by explicitly evaluating this limit.

Let us consider the transformation back to the qubit basis. This is mosteasily done by applying a transformation matrix. The two-qubit basis isgiven by |00〉 , |01〉 , |10〉 , |11〉, while the basis of eigenvectors of the mirroroperator is given by |00〉 , |11〉 , 1√

2(|01〉 + |01〉), 1√

2(|01〉 − |01〉). The qubit

basis is discussed in more detail in section 6.4. So the transformation matrixcan easily be constructed and results in

P =

1 0 0 00 0 1√

21√2

0 0 1√2− 1√

2

0 1 0 0

, (6.25)

which is orthogonal so its inverse P−1 equals its matrix transpose P>. Thismeans that we can produce the resulting unitary of the non-Abelian Berryphase in the qubit basis, while performing the calculation in the more con-venient ‘mirror’ basis. We need to consider Uqubit = PUmirrorP

>.

The unitary, using the introduced short forms, in the computational qubit

114

basis is then equal to

Uqubit =

c2+(a2+b2)CD −−ac+acC+b

√DS√

2D−−ac+acC+b

√DS√

2D− bc−bcC+a

√DS

D

ac−acC+b√DS√

2D12

(a2+(b2+c2)C

D + 1

)(b2+c2)(C−1)

2D−ab+abC+c

√DS√

2D

ac−acC+b√DS√

2D

(b2+c2)(C−1)

2D12

(a2+(b2+c2)C

D + 1

)−ab+abC+c

√DS√

2D

−bc+bcC+a√DS

D−ab+abC−c

√DS√

2D−ab+abC−c

√DS√

2D

b2+(a2+c2)CD

(6.26)

where D = a2 + b2 + c2, C = cos(√

a2 + b2 + c2)

= cos√D and S =

sin(√

a2 + b2 + c2)

= sin√D.

This matrix form it can be written in an even shorter form as

U =

k l l mn o p qn p o qr s s t

(6.27)

This expression contains ten different expressions, all functions of the a, b, c,which are in turn dependent on the integration of the connections over thepath, parametrized by the boundary values of the staggering parameters.Ten different expressions are expected as the matrix form found in eq. (6.24)contains the same number.

We have thought about different paths in parameter space, especially in thecontext of the commutation of different connections. For example let usconsider a square loop in parameter space with sides of θ and µ. To be moreprecise for a certain value of φ we consider dµ = 0 to dµ = dµbound, thenθ = 0 to θ = θbound, and returning to the North Pole by dµ = dµbound todµ = 0, then θ = θbound to θ = 0. This path investigates the commutationof the connection matrices of θ and dµ for a chosen value of φ. For the caseφ = 0 both connection matrices Aθ and Adµ are the zero matrix and thepath is completely trivial. However, also for nonzero φ we find the trivialresult, the four-dimensional identity matrix. The first and fourth partsof the square loop are trivial by definition, here the connection matricesequal to zero matrices, so only the second and third parts of the loop cancontribute. We see that found orthogonal (unitary) matrices for these partsare each others inverse; they result in (a, b, c, )2 = −(a, b, c)3 and thereforeU2 = U>3 = U−1

3 .We have also considered the need to implement an extra step in the schemeto turn on the coupling between the triangles parametrized by µ in the most

115

convenient way. This would need to be a point in (θ, φ) parameter spacewhere the connection matrix of dµ is zero. We are fortunate that the NorthPole (θ = 0) already supplies such a point, as Adµ is zero at θ = 0. So inthis case, the inclusion of a change in dµ can be neatly implemented at theNorth Pole, before performing the triangular loop in (θ, φ) parameter space.This also keeps the agreement of this triangular loop with the scheme in thetriangle and hexagon.

6.4 The two-qubit quantum gate in the bow tielattice

In the previous section we have described the construction of the non-Abelian Berry phase and given its general matrix form. In this sectionwe will consider its interpretation as two-qubit gate. We will take specialinterest in its possibilities to create entanglement between the qubits. Theentangling two-qubit gate is essential for defining universal quantum com-puting on the supersymmetric lattice. Let us reintroduce the definition ofthe qubit on the bow tie lattice. The bow tie lattice consists of two coupledtriangles; each triangle corresponds to a qubit. This is schematically shownin figure 6.3. The supersymmetric model on the triangle defines two zero-energy ground states, which we have taken to be the qubit states |0〉 , |1〉of the qubit defined on the triangle. In chapter 5 we have considered thetriangle lattice and we have defined one-qubit gates. These considerationsstill stand for the bow tie lattice without coupling; we can perform one-qubit gates on each of qubits constructed on the triangles. The coupling,introduced as the M2 staggering parameter µ, provides the possibility to con-struct a two-qubit gate which is fundamentally different from the productof two one-qubit gates. The difference leads to the creation of entanglementbetween the qubits.

The explicit correspondence between the qubit states and the lattice statesis given by the definition on each of the triangles and the construction of thestates on the bow tie by the Kronecker product ⊗. We have the followingequivalent notations

|00〉 = |0〉L ⊗ |0〉R = |1〉 ⊗ |4〉 = |1, 4〉 (6.28)

between the two-qubit state |00〉 and the lattice state |1, 4〉. Recall thatwe have defined this direct correspondence on the North Pole of stagger-ing parameter space. That is, at θ = 0, dµ = 0 one of the supersymmetricground states on the bow tie lattice equals |1, 4〉. The two-qubit space hasdimension four, with the computational basis |00〉 , |01〉 , |10〉 , |11〉 or thelattice basis |1, 4〉 , |1, 5〉 , |2, 4〉 , |2, 5〉. During the geometric path throughparameter space the lattice states explore the state space of dimension nine

116

Qubit 1

1 2

3

Qubit 2

4 5

6

1

2

3

4

5

6

Two-qubitsystem

Figure 6.3: Construction of the bow tie lattice by two triangle lattices andtherefore the creation of a two-qubit system by connecting two one-qubitsystems.

of the two-particle states on the bowtie. During the calculation of the non-Abelian Berry phase it was convenient to consider the eigenspaces of themirror operator. At the end we made sure to write the result of the unitaryfound in the qubit basis. This assumption in the calculation will not influ-ence the possibilities of the two-qubit gate, as we can apply a one-qubit gateto rotate away from an eigenspace of the mirror operator.

6.4.1 Two uncoupled triangles

We will show that the value of staggering parameter µ is directly relatedto the entangling nature of the found two-qubit gate. As a first step, letus consider some examples for dµ = 0 ↔ µ = 1, the case of two uncoupledtriangles. We will show that in this case we can write the four-dimensionalunitary matrices as a Kronecker product (or tensor product) of two two-dimensional matrices. This exemplifies their nature as trivial two-qubit gatematrices, as they can be written in terms of one-qubit gate matrices.

Consider the following unitary (in the computational basis), M1, which isproduced for parameter bounds θbound = π and φbound = π/4.

M1 =

0 0 0 10 0 −1 00 −1 0 01 0 0 0

(6.29)

which is equal to the Kronecker product iσy,1⊗ iσy,2, where σy,i denotes thePauli-Y gate on the first or second qubit. In view of our symmetric choice ofthe staggering parameters on both triangle, we can interpret this operationas performing the iσy gate on both qubits at the same time. The reason for

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the triviality here is the fact that θbound = π, as we will discussed at the endof this chapter. The value of dµ plays a minor role in this specific case.In this subsection we take dµ = 0 ↔ µ = 1 as mentioned. For parameterbounds θbound = π

2 and φbound = π4 unitary matrix M2 is produced, which

equals

M2 =

0.5 0.5 0.5 0.5−0.5 0.5 −0.5 0.5−0.5 −0.5 0.5 0.50.5 −0.5 −0.5 0.5

(6.30)

This matrix can be written as

I2 ⊗ (1

2I2 +

1

2iσy) + iσy ⊗ (

1

2I2 +

1

2iσy). (6.31)

The shown M1 and M2 are two examples of a general matrix form, given ineq. (6.34). The unitary matrices M3 and M4 are two more examples, whereM3 can be produced by parameter bounds θbound = π

2 and φbound = π6 and

M4 by θbound = π and φbound = π6 . For M3 and M4 we find that a = 0 and

b = −c.

M3 =

0.75 −0.433013 −0.433013 0.25

0.433013 0.75 −0.25 −0.4330130.433013 −0.25 0.75 −0.433013

0.25 0.433013 0.433013 0.75

(6.32)

M4 =

0.25 −0.433013 −0.433013 0.75

0.433013 0.25 −0.75 −0.4330130.433013 −0.75 0.25 −0.433013

0.75 0.433013 0.433013 0.25

(6.33)

The corresponding general form of all Mi, i = 1, . . . , 4, is given by

Mgen =

k l l m−l k −m l−l −m k lm −l −l k

(6.34)

which equals the expansion in two-dimensional matrices given by

I2 ⊗ (kI2 + liσy) + iσy ⊗ (lI2 +miσy). (6.35)

In comparison to the general form given in eq. (6.27), we find here thatk = o = t, n = −l = s = −q and r = m = −p. We see that the matrixform of this trivial two-qubit quantum gate is defined by only three differentexpressions. We find that l/k = m/l for all shown cases, so we can also write

(kI2 + liσy)⊗ (I2 + (l/k)iσy). (6.36)

118

Moreover, the general matrix form in eq. (6.34) can be written in terms ofone-qubit rotation quantum gates, introduced in eq. (3.14) in section 3.5.We have the following equality

Rotation(β) = cos(β)I2 − i sin(β)σy. (6.37)

We can now consider the construction of the matrices Mi. For M1 we alreadyseen that M1 = iσy,1 ⊗ iσy,2 so this corresponds to

M1 = Rotation(π

2

)⊗ Rotation

(π2

). (6.38)

For the matrixM2 we rewrite the expression (of the general form in eq. (6.36))to

M2 =

(1

2I2 +

1

2iσy

)⊗ (I2 + iσy) (6.39)

=

(1

2

√2I2 +

1

2

√2iσy

)⊗(

1

2

√2I2 +

1

2

√2iσy

)= Rotation

(−π

4

)⊗ Rotation

(−π

4

). (6.40)

The expressions in terms of rotation quantum gates for the matrices M3 andM4 can also be found. The results are

M3 =

(3

4I2 −

√3

4iσy

)⊗

(I2 −

√3

3iσy

)(6.41)

=

(1

2

√3I2 −

1

2iσy

)⊗(

1

2

√3I2 −

1

2

√2iσy

)= Rotation

(π6

)⊗ Rotation

(π6

)(6.42)

and

M4 =

(1

4I2 −

√3

4iσy

)⊗(I2 −

√3iσy

)(6.43)

=

(1

2I2 −

1

2

√3iσy

)⊗(

1

2I2 −

1

2

√3iσy

)= Rotation

(π3

)⊗ Rotation

(π3

). (6.44)

This rewritten form explicitly shows that that the quantum gates corre-sponding to the matrix forms Mi are trivial two-qubit gates. The sameoperation can be achieved by the application of rotation quantum gates toboth qubits separately.

119

6.4.2 Entanglement after application of two-qubit gate

We will investigate the possibilities for the two-qubit gate by the measure-ment of the entanglement entropy. The definition for the entanglemententropy was introduced in eq. (3.24) in section 3.6. To be more precise,we consider the application of the unitary constructed by the non-AbelianBerry phase (eq. (6.27)) on a simple input state. For example we take thetwo-qubit state |00〉 as input and consider the entanglement of the two qubitsin the produced two-qubit state |ψ〉 = U |00〉.

The most general analysis of the two-qubit gate is performed by the calcu-lation of the entanglement entropy after choosing the following parameters:the initial two-qubit state |ψ〉in, the boundary values of the geometric pathθb and φb, and the maximum value of the parameter µ (or dµ),dµb.

Let us start with making some observations about the behavior of the en-tanglement entropy as a function of the introduced parameters. The highlynon-trivial dependence on the parameters θb, φb, dµb make a general analysisdifficult. While the initial state has heavy influence on the size of the entan-glement entropy, we will often take |00〉 as initial state. In our preliminaryanalysis we found that this choice can produce the states with the highestentanglement entropy.

In (the first stage of) our analysis we have focused on two goals. Firstwe investigate if we are able to achieve the theoretical maximum possibleentanglement entropy S = ln(2) ≈ 0.693147. We add a comparison betweenthe produced state and a Bell state,

|ψ〉Bell =1√2|00〉+

1√2|11〉 (6.45)

which achieves this theoretical maximum. Second we will investigate if weare able to produce a quantum state with the same entanglement entropyas the quantum state

|ψ〉Pyth =3

5|00〉+

4

5|11〉 , (6.46)

which corresponds to S = 1625 ln

(2516

)+ 9

25 ln(

259

)≈ 0.653418.

Near-maximal entanglement entropy

Let us consider producing a quantum state with near-maximal entangle-ment entropy. We take the initial state to be |00〉 and focus on tuningthe parameters θb, φb, dµb. In general a viable strategy for finding a certainentanglement entropy in a state is taking the path boundary parametersθb = π

2 , φb = π2 . The maximally entangled state is generally produced close

120

to this (simple) choice. By precise tuning of the dµ parameter one can findthe wanted entanglement entropy.

We consider two parameter choices which can approximate the maximallypossible entanglement entropy. These choices will be investigated by theircapability of approximating well-known objects. The state |ψ〉 = Ubowtie |00〉is compared to the Bell state and the unitary Ubowtie is compared to the CNOTquantum gate. Remember that the CNOT quantum gate has been proven tobe an essential ingredient for universal quantum computation as a result ofits non-trivial two-qubit action.

Consider the parameter choice θb = 7π16 , φb = π

2 and dµb = 10. This set wasfound by first finding the maximum entanglement entropy for a set of valuesof dµb. The set we have used is equal to

1, 2, 3, 4, 5, 10, 50, 100, 200, 500, 1 · 103, 5 · 103, 1 · 104, 1 · 105, 1 · 106. (6.47)

The step size of the search in values for parameters θb and φb is equal to π16 .

From this investigation the highest entanglement entropy was found for thecase given above: θb = 7π

16 , φb = π2 and dµb = 10.

The entanglement entropy of the state∣∣∣ψ 7π

16,π2,10,00

⟩= Ubowtie[

7π16 ,

π2 , 10] |00〉

is equal to S ≈ 0.693121. For comparison, the values for the entanglemententropy after the action of Ubowtie on the other computational basis statesis given by S ≈ 0.0984473 for |01〉 and |10〉, and S ≈ 0.200405 for |11〉.This is the reason we focus on |00〉 as initial state. The deviation fromthe theoretical maximum entanglement entropy of S = ln(2) is equal to2.63 · 10−5.

When decreasing the step size to π32 , the maximal value of the entanglement

for any θb and φb for dµb = 10 stays the same. However, we find that thereexist parameter choices with higher entanglement entropy. For values of dµbin the list higher than dµb = 100 the entanglement entropy is higher thanfor dµb = 10. For example the choice θb = 15π

32 , φb = 11π16 and dµb = 100 or

dµb = 200 results in S ≈ 0.693146, which is the highest value that can bereached for the considered set in eq. (6.47).

The introduced choice results in the following unitary matrix, produced bythe non-Abelian Berry phase after a geometric path defined by the para-meters θb = 7π

16 , φb = π2 and dµb = 10,

Ubowtie[θb = 7π16 , φb = π

2 , dµb = 10] =0.209899 −0.140779 −0.140779 0.9572380.676022 0.623421 −0.376579 −0.1119320.676022 −0.376579 0.623421 −0.111932−0.204771 0.670608 0.670608 0.242152

(6.48)

121

The state∣∣∣ψ 7π

16,π2,10,00

⟩= Ubowtie[θb = 7π

16 , φb = π2 , dµb = 10] |00〉 is given by∣∣∣ψ 7π

16,π2,10,00

⟩= 0.209899 |00〉+0.676022(|01〉+|10〉)−0.204771 |11〉 . (6.49)

This quantum state is close to the following form

|ψ〉 = A |00〉+B(|01〉+|10〉)−A |11〉 = A(|00〉−|11〉)+B(|01〉+|10〉), (6.50)

which achieves the maximal entanglement entropy of

S = −2(A2 +B2) ln(A2 +B2

)= ln(2) (6.51)

as 2A2 +2B2 = 1 by normalization. We find that the maximal entanglemententropy is reached if the coefficients of |00〉 and |11〉 are equal. While this

is not precisely true for∣∣∣ψ 7π

16,π2,10,00

⟩, it is still what we would be looking

for. Unfortunately translating this condition back to the formulas for thematrix entries does not give more insight; we see that k = A and r = −Afrom eq. (6.27), so by eq. (6.26)

c2 + (a2 + b2)C

D= −

(−bc+ bcC + a

√DS

D

)

which simplifies to

c2 + (a2 + b2)C = −bc+ bcC + a√DS, (6.52)

where D = a2 + b2 + c2, C = cos√D and S = sin

√D.

As the maximum entanglement is also achieved by the Bell states, we canlook for a transformation between the two. The state |ψ〉 is a linear combi-nation of two Bell states, with coefficients equal to A and B. This can beseen as follows

|ψ〉 = A(|00〉 − |11〉) +B(|01〉+ |10〉) (6.53)

=√

2

[A|00〉 − |11〉√

2+B|01〉+ |10〉√

2

](6.54)

=√

2 [AUBell |10〉+BUBell |01〉] (6.55)

=√

2UBell [A |10〉+B |01〉] , (6.56)

where UBell denotes the operation from the computational basis to the Bellbasis, UBell = (H ⊗ I2)CNOT, introduced in figure 3.23 in section 3.6. Re-member that an essential part of the quantum circuit for the UBell operationis the non-trivial two-qubit gate CNOT. In the next section we will look intothis further.

122

The state∣∣∣ψ 7π

16,π2,10,00

⟩does not satisfy the precise form of |ψ〉. However, we

might say that the difference between the coefficients of |00〉 and |11〉 gives usan indication of how close the produced state is to a (transformed) Bell state.

We can also consider a second unitary, which is produced by the parametersθb = 15π

32 , φb = 11π16 and dµb = 200. This set was found by the procedure

described earlier.

Ubowtie[θb = 15π32 , φb = 11π

16 , dµ = 200] =0.340478 0.111188 0.111188 0.9270110.619406 0.652616 −0.347384 −0.2641090.619406 −0.347384 0.652616 −0.264109−0.341684 0.664119 0.664119 −0.0338171

(6.57)

The entanglement entropy of the state∣∣∣ψ 15π

32, 11π

16,200,00

⟩= Ubowtie[

15π32 ,

11π16 , 200] |00〉

is equal to S ≈ 0.693146. For comparison, the values for the entanglemententropy after the action of Ubowtie on the other computational basis statesare given by S ≈ 0.325991 for |01〉 and |10〉, and S ≈ 0.0575051 for |11〉. Thedeviation from the theoretical maximum (S = ln(2)) is equal to 1.46 · 10−6.

The state∣∣∣ψ 15π

32, 11π

16,200,00

⟩is given by∣∣∣ψ 15π

32, 11π

16,200,00

⟩= 0.340478 |00〉+ 0.619406(|01〉+ |10〉)− 0.341684 |11〉 .

(6.58)We can see that this unitary produces a state with even higher entanglemententropy. Strongly related is the fact that the coefficients of |00〉 and |11〉 areeven closer.

Non-trivial two-qubit quantum gate CNOT

In this part we will focus on investigating how well the unitary constructedby the non-Abelian Berry phase approximates the standard non-trivial two-qubit gate CNOT. We will again use the two parameter choices presentedabove.

A first sight the matrix expressions of the unitary matrices produced withthe described procedure of the non-Abelian Berry phase are different fromthe well-known expression of CNOT. The two-qubit quantum gate CNOT isintroduced in eq. (3.18) in section 3.5 and has been shown to be essential forquantum gate universality in section 3.7. We are looking into a way to trans-form the produced dense four-dimensional matrices to a form close to CNOT

in the computational basis. We will again consider the unitaries producedby the parameter choices θb = 7π

16 , φb = π2 and dµb = 10 and 15π

32 ,11π16 , 200, as

the entanglement entropy found is close to maximum when these matrices

123

work on the initial state |00〉. We expect that this is a sign that the unitarymight be close to equivalence with the CNOT. More quantitatively speaking,we might expect from approximating the maximal entanglement entropy,that we can transform the unitary to the form of CNOT with deviation onthe matrix entries of size 1 (which might be complex).

A two-qubit quantum gate is said to be equivalent to another quantum gate,such as CNOT, under the following condition. There must exist a transforma-tion between the two matrix representations, which can be written in termsof one-qubit quantum gates only. So we are looking for a transformationmatrix P such that

PUbowtieP−1 = CNOT (6.59)

where

P = U1 ⊗ U2. (6.60)

In this notation U1 denotes a unitary operation on the first qubit and U2

on the second qubit, and P is constructed by the tensor product ⊗ betweenthe two. If we consider a general form for a two-dimensional unitary matrixfor both U1,2 such as

Ui =1√

|a|2 + |b|2

(a b

−b∗eiα a∗eiα

), (6.61)

we can construct the general form of the product U1 ⊗ U2:

1√(|a|2 + |b|2)(|c|2 + |d|2)

ac ad bc bd

−ad∗eiγ ac∗eiγ −bd∗eiγ bc∗eiγ

−b∗ceiα −b∗deiα a∗ceiα a∗deiα

b∗d∗ei(α+γ) −b∗c∗ei(α+γ) −a∗d∗ei(α+γ) a∗c∗ei(α+γ)

.

(6.62)We have undertaken an investigation of the structure of the unitary Ubowtie

by considering its diagonalization, i.e. finding the eigenvalues and eigenvec-tors of Ubowtie. We find that the eigenvalues λi satisfy the following generalform

λi = 1, 1, e±iβ (6.63)

for any value of the geometric parameters θb, φb, dµb. The correspondingeigenvectors of λ3,4 = e±iβ are complex conjugate, as expected, and theirfirst entry is real. The eigenvectors of λ1,2 = 1 are also real, as expected forthe real matrix Ubowtie.Unfortunately, we are not able to find a connection between the values of theeigenvalues (or eigenvectors) and the entangling nature of the matrix Ubowtie.We can consider the same example introduced before. For the parameterchoice 7π

16 ,π2 , 10 the eigenvalues are equal to eq. (6.63) with β ≈ 1.72192. This

gave the impression that the eigenvalues for β = π2 ≈ 1.5708, that is 1, 1,±i,

124

might produce a non-trivial matrix form. However, such a considerationof the eigenvalues is not enough. A (not so insightful) counterexample isgiven by the case of θb = 2.84765, φb = 3.59762, dµb = 191635. Here we findeigenvalues with a β ≈ 1.52766, 1, 1, 0.043 ± 0.999i, but an entanglemententropy of S ≈ 1.15535 · 10−5.

With the work presented in this section we can not proof the equivalenceor approximation of the matrix form of CNOT by the unitary matrix foundwith the proposed geometric scheme. The proof would ideally consist of theconstruction of a trivial two-qubit operation that performs the matrix trans-formation precisely. This trivial two-qubit operation would be a product ofone-qubit operations. Note that we have shown in this work that the generalone-qubit operation can be produced, which would enable us to define thetransformation with geometric paths on the separate triangles.

Approximating a non-maximal entanglement entropy

Besides our procedure to approximate the maximal entanglement entropy,we can also produce a certain entanglement entropy corresponding to aknown state. Given the state in eq. (6.46) we find that SPyth = 16

25 ln(

2516

)+

925 ln

(259

)≈ 0.653418. Let us consider procedures where we tune the value

of the parameter dµb to match the value of the entanglement entropy. Forthe convenient choice of angular path parameters θb = π

2 and φb = π2 we find

a good approximation by dµb = 51 (when the unitary is applied to initialstate |00〉): S −SPyth ≈ 1.22 · 10−5. By even more precise tuning of the dµbparameter, we can find for dµb = 50.975 a deviation of 7.47 · 10−7.

The parameter choices which can be used to create near-maximal entan-glement entropy can also be used to approximate the (lower) entanglemententropy of SPyth. The best results are found for precise tuning of the para-meter dµb. The choice θb = 7π

16 , φb = π2 and |ψini〉 = |00〉 with dµb = 3.6493

results in a deviation of 1.39 · 10−6. Moreover, the choice θb = 15π32 , φb = 11π

16and |ψini〉 = |00〉 with dµb = 10.6827 gives a deviation of 1.42 · 10−7.

We see that a wanted value of the entanglement entropy can be approxi-mated precisely by specific tuning of the parameter dµb. We also see thatthe angle parameter choices corresponding to the near-maximal entangle-ment entropy can be used to obtain this value for small values of the para-meter dµb.

Entanglement entropy as function of the geometric parameters

The entanglement entropy can be seen as a functional of the parameterswhich define the geometric path, denoted by θb, φb and dµb. We can alsoinclude the initial state to which we apply the unitary as a parameter. Thegeneral form of the initial state is given by eq. (3.6) in section 3.5. However,for simplicity we will consider the standard initial state |ψini〉 = |00〉. We

125

will show in this section that the dependence of the entanglement entropyon the parameters is highly non-linear.

Let us first consider the dependence on the M2 staggering parameter µ ≡ µ3,which is re-parametrized as µ = 1+dµ. This parameter can be interpreted asthe size of the coupling between the triangles. For µ = 1↔ dµ = 0 we thinkof two separate (decoupled) triangles. We find that as expected this choiceleads to a trivial two-qubit operation which can be decomposed as productof one-qubit operations, as shown in subsection 6.4.1. Here we performone-qubit geometric quantum gates on each of the triangles as there is nocoupling between them. We see that dµ = 0 leads to a numerically smalldeterminant of the reduced density matrix. The definition of the reduceddensity matrix is introduced in eq. (3.25).In our explicit calculations we find that a certain parameter choice leadsto zero entanglement (entropy) if the determinant of the reduced densitymatrix (of the left system) is zero, such that the matrix is not invertible. Inthat case the entanglement entropy can not be explicitly calculated by theformula given in eq. (3.24) as the matrix logarithm is undefined in that case.

Besides dµb = 0 other choices which do not produce entanglement entropyare trivially choices where the geometric path does not enclose any area. Ex-amples include θb = 0 and φb = 0. In addition, also the choice θb = π doesnot lead to the creation of entanglement. The first two cases (θb = 0, φb = 0)produce zero entanglement as the found unitary matrices are equal to thefour-dimensional unit matrix up to numerical errors. Note that this defi-nition was specially implemented for the case that a, b, c equal zero in theexpression for the matrix form of the non-Abelian Berry phase (eq. (6.24)).The third case (θb = π) for φb = π/2 gives also the identity matrix but forφb = π/4 it produces the unitary we have already considered in eq. (6.29) insubsection 6.4.1, which is a trivial two-qubit operation but not the identity.The value of dµb plays a minor role for parameter choices with θb = π, whichitself is enough for triviality of the two-qubit quantum gate.

One could also expect that the entanglement entropy will now scale with thesize of the ‘coupling’ parameter dµ3, but this is not the case. This can bestbe seen for choices of θb and φb which achieve high values of the entangle-ment entropy. From our general analysis of the behavior we see a increaseup to some maximum value and then a decrease with increasing values ofparameter dµ3. We show the different behavior of the entanglement entropyas a function of dµ3 in figure 6.4.

We choose to focus our analysis on the parameter domain dµb ≥ 0. Theparameter dµ is derived from µ = 1 + dµ ∈ [0,∞) so in principle dµ ≥ −1holds. However, the value dµ = −1 is a singularity in the expressions we

126

0 10 20 30 40 50dμ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) S(dµb) for θb = π3 , φb = π

3 .

0 10 20 30 40 50dμ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) S(dµb) for θb = π2 , φb = π

2 .

10 20 30 40 50dμ

0.688

0.689

0.690

0.691

0.692

0.693

(c) S(dµb) for θb = 7π16 , φb = π

2 . Notethe different scaling on the vertical axisin this figure.

0 10 20 30 40 50dμ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(d) S(dµb) for θb = π2 , φb = 13π

16 .

Figure 6.4: The entanglement entropy as a function of dµb generated by ageometric two-qubit quantum gate defined by path parameters θb and φb.

127

0.0 0.5 1.0 1.5 2.0 2.5 3.0θ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) S(θ) for φb = π3 , dµb = 10.

0.0 0.5 1.0 1.5 2.0 2.5 3.0θ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) S(θ) for φb = π2 , dµb = 10.

0 1 2 3 4 5 6ϕ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(c) S(φ) for θb = π3 , dµb = 10.

0 1 2 3 4 5 6ϕ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(d) S(φ) for θb = 7π16 , dµb = 10.

Figure 6.5: The entanglement entropy as a function of θb (a,b) or φb (c,d)generated by a geometric two-qubit quantum gate defined by path para-meters θb, φb and µb.

use because of the way we introduced this parameter in the definitions ofthe ground states in subsection 6.2.3. Our choice of domain for dµb is also(more) consistent with our interpretation of the influence of parameter µ inthe bow tie system. We think about it as the coupling between the triangles.Directly from the definition of the M2 staggering parameter, we find thatleaving this parameter out of the model corresponds to setting it to unity.This situation is therefore interpreted as a system of two uncoupled trian-gles. In other words this is the minimal value of the coupling, the couplingbeing ‘off’. Therefore parameter values 0 < dµ < 1 can not be given aninterpretation and are neglected in our analysis. By choice, we use positiveinteger values of dµb.

The structure of the expressions of the entanglement entropy as a functionof the angular parameter θb or parameter φb is difficult to analyze. Thebehavior is highly non-linear and depends on the specific values of otherconstant parameters chosen. We have shown examples in figure 6.5.

We can also look at the entanglement as a function of both θb and φb for acertain value of dµb. This visualization also exemplifies its highly non-linearbehavior. Unfortunately, the analysis is not simplified in this way. In figure

128

6.6 we show the entanglement entropy as a function of both θb and φb fordµb = 1, 10 and 1000.

We consider the behavior of the entanglement entropy for a large range ofvalues for dµb by means of a log-linear plot with the values for dµb as givenin eq. (6.47) on the horizontal logarithmic axis.It seems that closer to the choice which creates the near-maximal entangle-ment entropy, the decay of the entanglement entropy as a function of dµbis smaller. Let us compare the two values of the entanglement entropy fordifferent values of dµb. The decay between dµb = 103 and dµb = 106 forθb = 7π

16 , φb = π2 is from 0.690592 to 0.690591, but for θb = π

2 , φb = π2 the

entanglement entropy decays from 0.675709 to 0.674966. The plots for theseparameter choices are shown in figure 6.7.Another choice which creates near-maximal entanglement entropy θb = 15π

32and φb = 11π

16 for dµb = 100, 200, namely S ≈ 0.693146, creates somewhatcomparable entanglement entropy for dµb = 1 and dµb = 4. For thesespecific parameter choices we find S ≈ 0.199239 (dµb = 1) and S ≈ 0.227191(dµb = 4) so we conclude that the entanglement entropy does not havestrictly increasing behavior before the maximum value (for a certain choiceof θb and φb). This is shown in figure 6.8a.We can also find a parameter choice which show non-monotonous behaviorof the entanglement entropy for high values of dµb. As shown in figure 6.8bthis is the case for θb = π

2 and φb = 3π4 . Other examples of non-monotonous

behavior over the entire dµ domain are θb = 2π5 and φb = 6π

5 or θb = π3

and φb = 4π3 , although the latter only produces very small values of the

entanglement entropy.

129

(a) S(θ, φ) for dµb = 1.

(b) S(θ, φ) for dµb = 10.

(c) S(θ, φ) for dµb = 1000.

Figure 6.6: The entanglement entropy S as a function of parameters θb andφb for three values of the parameter dµb = 1, 10 and 1000.

130

1 10 100 1000 104 105 106dμ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) S(dµb) on a logarithmic scale forθb = π

2 and φb = π2 .

1 10 100 1000 104 105 106dμ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) S(dµb) on a logarithmic scale forθb = 7π

16 and φb = π2 .

Figure 6.7: The entanglement entropy as a function of dµb generated by ageometric two-qubit quantum gate defined by path parameters θb and φb.

1 10 100 1000 104 105 106dμ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) S(dµb) on a logarithmic scale forθb = 15π

32 and φb = 11π16 .

1 10 100 1000 104 105 106dμ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) S(dµb) on a logarithmic scale forθb = π

2 and φb = 3π4 .

Figure 6.8: The entanglement entropy as a function of dµb generated by ageometric two-qubit quantum gate defined by path parameters θb and φb.

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Chapter 7

Discussion, outlook, andconclusion

In the last two chapters we have presented the results of our work intogeometric quantum computation. In this chapter we discuss our results,suggest further directions and conclude this work.

7.1 Discussion

7.1.1 Parameter choices

As a first point, let us discuss the expressions used in the explicit calcula-tion of the non-Abelian Berry phase, with a focus on the parameters used.In chapter 5 we considered two definitions for the coefficient vectors of thesupersymmetric ground states, differing by a rotation in the two-dimensionalsubspace. We have chosen the vectors such that they are unambiguouslydefined in θ = 0 (the North Pole). However, we use this set of coefficientvectors to describe the ground states for all values of the staggering para-meter, including in θ = π, where they are ambiguously defined. This leadsto the fact that in our protocol we add an integration of the connection Aφin θb = π, but not when we are back in θ = 0. This comment is relatedto the fact that the sphere S2 can not be described by a single patch. Onecould say for our choice that the North Pole and South Pole can not be in-cluded in the same patch. Moreover, this would lead to the introduction ofdifferent connections on different patches [34, 9]. In the work presented herewe work with the connection of the ‘North Pole patch’ for paths over theentire sphere, so in theory we need to have extra care for geometric pathswhich go through the South Pole (θ = π).

For the expressions of the connections found for the hexagon lattice (chap-ter 5) with the staggering choice ~λ we find explicit singularities. When we

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integrate the connections over the geometric path, we have to resort to con-tinuation of the numerical expressions by their calculated limits. This is thecase for

∫Aθ dθ (first part of the triangular path) at θb = π

2 and also for∫Aφ dφ (second part of the triangular path) at θb = 0, π2 and φb = π

2 ,3π2 .

The introduction of the M2 staggering parameter µ also adds singularitiesfor the calculations in the bow tie. The way we introduce this parameterdirectly into the expression of the coefficient vectors of the bow tie leads toa singularity at µ = 0 ↔ dµ = −1. As discussed in the results (Chapter 6)we stay clear of this singularity in our analysis by considering µ ≥ 1.

An important part of the analysis in this work is the reduction of the de-grees of freedom. On the one hand we reduce the size of the spaces oflattice states by imposing a translational (hexagon) or mirror symmetry(bow tie). The same symmetry argument is then used during the imple-mentation of the staggering parameter. This means that our definition bya three-dimensional vector is sufficient. On the other hand we also simplifythe three-dimensional vector describing the staggering. For example on thetriangle lattice, in the general case the staggering parameter is describedby three complex-valued components. However, in the explicit forms of thestaggering parameters we used we allow for less freedom. In the first stag-gering choice ~λ we parametrize by two real angular parameters θ and φ. Wehave chosen to take the components real valued and imposed normalization.In the second staggering choice ~λΩ we set the first component to zero, choosethe second component complex an the third component real. In this case wealso imposed normalization. Therefore both staggering parameter choicesare described by only two independent variables, θ and φ.

In accordance to the definition proposed in Solinas et al. [46, 47] we choseto parametrize the second staggering choice with the argument θ/2. If weassume the standard angular boundaries 0 ≤ θ ≤ π, the staggering para-meter space does only consist of the upper half of a sphere, while the firststaggering choice clearly describes the whole sphere S2.

7.1.2 Noisy Intermediate Scale Quantum

Let us quickly consider two points from the context of Noisy Intermedi-ate Scale Quantum (NISQ) computation: the protection to noise and thescalability of the ideas presented in this work.

Our motivation for this work was investigating a new approach for quan-tum computation. By combining the supersymmetric model with geometry-based operation we expect to be able to decrease the influence of deco-herence. This is motivated by sources from literature such as Ekert et al.[14], Jones et al. [26], Tomka et al. [50]. However, we have not performed

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explicit calculation on this subject. It may be interesting to explicitly calcu-late the influence of decoherence in this system, to get a better notion of theprotection against decoherence provided by the approach considered here.

We have considered the bow tie lattice in detail in this work as a multi-qubitsystem (chapter 6). The lattice is constructed by connecting two trianglelattices, each describing a single qubit. So we can also consider the extensionto more connected qubits. The definition of multiple qubits is integral tothe criteria of scalability of the quantum computation implementation. Inour case we can imagine a couple possible extensions of the bow tie lattice.We can connect the triangle lattices (corresponding to qubits) by definingextra edges between triangles as we did for one edge in the construction ofthe bow tie lattice. Starting from the bow tie lattice we can connect thenext triangle in the same vertex, such that we construct a chain of triangleswith the edges to the neighboring triangles in the same vertex. We can alsoconnect the triangles to neighboring triangles on different vertices. However,this will change the definition of the qubit states on the unconnected verticesof each triangle. Both ideas produce a one-dimensional chain of triangles.Equivalently to the bow tie, we could define two-qubit operations on anypair of adjacent triangles. If this chain configuration does not give enoughfreedom to construct the wanted n-qubit operation, one could consider aquasi-one-dimensional configuration where some triangles are connected tothree (or even four) other triangles.

We have investigated quantum computation defined with supersymmetriclattice models, with a focus on constructing quantum control. As intro-duced, a special requirement to general quantum control is quantum gateuniversality. Following the discussion presented in Nielsen and Chuang [36]we have seen that a universal set of one- and two-qubit gate can be usedto approximate any n-qubit operation. In our work (chapter 5) we haveshown that we can construct any one-qubit unitary by decomposition intothe phase shift and rotation quantum gates. In chapter 6 we have investi-gated the construction of a non-trivial two-qubit quantum gate. We haveshown the creation of entanglement by application of the unitary, which isa characteristic for non-triviality. Unfortunately, we were not able to proveits equivalence to a known non-trivial two-qubit quantum gate. Thereforewe can not conclude that we have achieved gate universality for geometricquantum control.

7.1.3 Mathematical point of view

In this work we have also been looking at mathematical descriptions of geo-metric quantum computation (section 4.6). With our explicit calculationof the Berry connections, we have also calculated the expressions for the

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Berry curvature (or field strength). For completeness we have calculatedthe Chern numbers by integrating the trace of the Berry curvature over theentire parameter space S2. As shown in chapter 5 one staggering choice gavezero for both the triangle and hexagon expressions, but the other choice gavea more interesting result. For the triangle we found c1 = −1 and for thehexagon c1 = −2, which might suggest a linear relation to the length of thelattice chain, i.e. c1 = −n for a chain of length L = 3n. However, we arenot able to prove this. To our knowledge there is no mathematical reasonwhich would lead to this relation. The only relation that comes to mindis the standard relation between the first Chern number and gauge theory.The difference of the integrals of the curvature F over different patches of atwo-sphere S2 is equal to the first Chern number, 2πc1. In the outlook wewill discuss the periodic chain of length L = 3n more.

The approach we have presented is based on using the non-Abelian Berryphase to explicitly construct unitary matrices. The Berry phase is basedon the adiabatic condition. Because of the theoretical nature of this work,we consider changes in the parameters and we do not implement a timevariable explicitly. We assume that the adiabatic condition is implicitly sat-isfied by the geometric path. We have not checked adiabaticity explicitly.Our calculation assumes that the space of the supersymmetric ground statesis isolated in correspondence to the adiabatic theorem, such that no statesoutside this two-dimensional subspace are occupied at any time. For an ex-perimental implementation it is important to be able to distinguish differenttime scales, so for future investigation in this direction we would like to addan explicit time variable. The next step can include a numerical analysis intothe adiabatic condition and the decoherence time of the approach proposedhere. The adiabatic condition can not be satisfied perfectly in practice asover very long timescales decoherence will dominate. To balance differentsources of error (experimentally) further work is necessary. The alternativederivation of the non-Abelian Berry phase included in the appendix fromRezakhani et al. [40] takes the adiabatic error into account.

In this work we have combined analytical and numerical calculations. Wehave calculated the expression for the Berry connections analytically, al-though it was necessary to use Mathematica. Integrals of the connectionsover a part of the geometric path are generally performed numerically, asan analytic expression could often not be found here. We have chosen abound for the accuracy of the numerical integration of 10−8 to achieve somemidpoint between speed and accuracy. This is motivated by the fact thatthis work is a first investigation into the subject and done from a theoreti-cal point of view. We have not focused on numerical precision or accuracy;we have investigated the possibilities of this approach, so shorter calcula-tion times were preferred. Moreover, for the calculations of the non-Abelian

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Berry phase in the bow tie lattice we have worked with the general formof the matrix exponent of the integrated connection. As the connectionsAθ, Aφ, Aµ had a standard form with only three independent expressions,we could write the matrix exponentiated form as a function of these threeexpressions. Our combined approach has enabled us to construct the generalone-qubit gate and investigate the entanglement entropy produced by thetwo-qubit unitary matrix.

7.1.4 Entanglement entropy

Let us discuss using the entanglement entropy as a measure of the non-triviality of the two-qubit gate unitary. It is a well-known measure of en-tanglement, a curiosity of quantum mechanics, which is precisely what de-scribes the non-triviality of a two-qubit quantum gate. A trivial two-qubitgate, which is defined by a product of one-qubit gates, can not produceentanglement. We have seen that in our approach we can easily find theentanglement entropy of the produced state to show that the used unitarymatrix corresponds to a non-trivial two-qubit operation. However, we alsohave to make some remarks. The calculation of the entanglement entropyonly uses a single initial state and produces the entanglement entropy of asingle output state. We would be interested in the form of the entire four-dimensional unitary matrix. Moreover, the calculation of the lattice states,Berry connections and Berry phase are all written down in bra-ket notation.Therefore it may feel unnatural that we need density matrix language forthe analysis in the last step. The reason is that the definition of the entan-glement entropy is stated in terms of the reduced density matrix.

The procedure presented has shown a highly non-trivial dependence of theentanglement entropy on the geometric parameters. It seems that thereis precise tuning necessary to achieve high values of the entanglement en-tropy. Therefore it is near impossible to find a optimization procedure of theentanglement entropy. We expect that the unitary producing the highestentanglement entropy (closest to the maximal value) is the most interestingfrom the point of view of quantum gate universality. But we were not ableto construct a maximally entangling unitary, only some approximations. Wehave shown some work on procedures to approximate some specific value ofthe entanglement entropy. On the one hand, we realize that achieving themaximal entanglement entropy may be of the most interest. On the otherhand, we have shown precisely tuned parameters, which can approximate anon-maximal entanglement entropy better than any shown unitary approx-imated the maximal entanglement entropy.

We have seen that the two-qubit unitary can produce entanglement in chap-ter 6. For simplicity we have focused on the entanglement entropy of the

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state produced by action of the unitary on the initial state |00〉. If we com-pare using the computational basis states |00〉 , |01〉 , |10〉 and |11〉, we findthat they do not produce the same entanglement entropy as initial states.In some preliminary analysis of these four basis states the initial state |00〉resulted in the highest values. However, we expect that this is not the op-timal choice for an initial state. The optimal choice for the geometricalparameters might also depend on the initial state. Consider the example ofproducing the Bell states, states with maximal entanglement. The combina-tion of the Hadamard gate, creating a equal superposition on the first qubit,and the non-trivial two-qubit CNOT gate creates these maximally entangledstates for any of the computational basis states. This well-known exampleincites the suggested connection between non-triviality of the two-qubit gateand the possibility of creating maximally entangled states. In this exampleeach computational basis state used as initial state produces the same en-tanglement entropy (the maximal value). This difference with the geometricunitary might be solved by a basis transformation on this unitary. Such atransformation can also be used to directly compare the produced geometricunitary to the known (simple) form of CNOT or the product (H ⊗ I)CNOT.

In the work presented here we were not able to prove the equivalence of thegeometrically produced unitary to a known non-trivial two-qubit quantumgate, such as CNOT. This proof would be based on constructing a transfor-mation by one-qubit quantum gates between the two unitary matrices. Thefound two-qubit unitary is given by a dense four-dimensional matrix withnumerical entries satisfying −1 ≤ u ≤ 1, as each column or row is normalizedto unity. The entries are numerical values instead of analytical expressionas the integration of the connection over the path needs to be performednumerically, because analytical forms do not exists. We can ponder if thisfact impedes finding an exact transformation.

7.1.5 The two-qubit system

Let us reflect on our approach for the construction of the two-qubit system,the bow tie lattice, in chapter 6. We have included a new edge connectingtwo triangle lattices. We have seen that we can interpret the two-dimensionalsubspace of supersymmetric ground states on the triangle lattice as thecomputational subspace of a single qubit. Naturally, the space of supersym-metric ground states on the bow tie corresponds to the computational spaceof two qubits. As each of the triangles represents a single qubit, we canuse our analysis of the triangle lattice also on the bow tie lattice. This isdone explicitly with the construction of the coefficient vectors of the groundstates using the mirror symmetry in the bow tie lattice. The constructionof the two one-qubit gates is still valid for each of the triangles in the bowtie. So this defines a general one-qubit operation on each of the qubit in the

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two-qubit system.

Moreover, the simplification with the mirror symmetry in the bow tie latticeidentifies the lattice sites on both of the triangle lattices with each other.In more detail, we choose the staggering parameter on both of the trianglesequal. We imagine that this identification is only introduced for the defini-tion of the action of the two-qubit quantum gate and that we are still ableto have separate parameters on both triangles for the action of one-qubitgates in one of the triangles. The choice to set the staggering parametersequal in both triangles can be interpreted as simultaneously controlling thestaggering parameter in both triangles in precisely the same manner. Thetwo-qubit gate is produced by a simultaneous geometric path of the stagger-ing parameter while two vertices are coupled by a connection controlled byan additional parameter. We find that for nonzero coupling we are able todefine non-trivial two-qubit operations, as expected. However, we are notsure if this choice enables us to construct the wanted non-trivial operationwhich is equivalent to CNOT for example.

One could think about different simplifications of the set of staggering para-meters of the bow tie lattice model. For example, one can take the staggeringparameter in one of the triangles constant, say (0, 0, 1) or 1√

3(1, 1, 1), and

perform a geometric path with the staggering parameter on the other trian-gle. An extension of the bow tie lattice is discussed in the next section, theoutlook.

7.1.6 Experimental discussion

As a last point let us discuss about the possibilities of an experimental im-plementation of the approach presented here. In this text we have alreadylooked into some experimental approaches for geometric quantum compu-tation in section 4.7. Especially the connection between the triangle latticeand the tripod system is very interesting. There is an equivalence betweenthe supersymmetric ground states on the lattice and the dark states in thetripod atomic level structure of a trapped ion. We have seen that all variantsof geometric quantum computation, Abelian or non-Abelian and adiabaticor non-adiabatic, are investigated as research directions. This shows theexperimental possibilities of geometric quantum computation.

An experimental implementation of the supersymmetric lattice model, whichforms the basis of our approach, will be a difficult endeavor. The model isbased on a very delicate balance of the kinetic and potential terms in theHamiltonian. Moreover, our approach is based on a certain (small) latticegeometry with well-defined lattice sites and neighbors. More importantly,the geometric approach to quantum operations is completely built on the as-

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sumption that we can control the staggering parameters on the lattice sitesat will. Such precise control will be difficult to achieve for such small latticeconfigurations as considered here. The staggering parameter corresponds toa site dependent change in occupation. We need to impose nearest-neighborexclusion inside the triangle or hexagon, but also we allow for the bow tielattice adjacent sites over the additional edge to be occupied. This singleadditional edge would need a separate tuning parameter, which can be inter-preted as the staggering parameter µ. Apart from the conceptual agreementbetween the triangle lattice and the tripod scheme, the implementations arevery different. The possibilities of geometric quantum computation for tri-pod related approaches do not translate to the supersymmetric lattice. Inthe proposed laser schemes the geometric path is defined and controlled withlaser parameters which are readily available, such as the Rabi frequencies.

We have defined the geometric path in staggering parameter space by atriangular closed path. We need to include a staggering parameter on thelattice and we want to control its value such that it performs this triangularclosed path. The triangular loop is written in terms of angular parametersθ and φ, where each part of the path leaves one of the angles constant.This choice is particularly convenient for our theoretical investigation. Thisparticular path will probably not be conveniently realized in a experimentalsetup. However, we have seen that the enclosed surface inside the closedpath determines the angle in the geometric quantum gate. Therefore anypath can be used and our analysis leading to the phase shift and rotationquantum gates still holds.

In this work we have focused on the quantum control aspect of quantum com-putation. In the context of an experimental implementation two other ele-ments are also very important: the initialization and measurements of qubitstates. We have defined the qubit states by the supersymmetric groundstates and our proposal for quantum control starts and ends in this two-dimensional manifold. We need to be able to initialize the qubit(s) in aknown initial state, such as the simple state |0〉 (or |00〉 for the bow tie).Only when starting from the same, known state each evaluation the com-putation is useful. Moreover, we also need to be able to measure the resultof the computation. As measurement is destructive on the quantum state,we have to be able to repeat the calculation several times to extract allthe wanted information, which requires the ability to start from the sameinitial state each time. The supersymmetric lattice forms the basis of ourapproach, so we want to define initialization and measurement in terms ofa lattice procedure. Intuitively, this would mean we want to accumulatethe fermion density on one lattice site for initialization. We expect thatwe would need to lift the degeneracy to achieve this localization. For mea-surement we need to find the occupation on each of the lattice sites, and

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translate this information to the corresponding qubit states. For the tripodscheme, closely related to the triangle lattice, the initialization consists ofthe creation of a superposition of the atomic states. The measurement con-sists of observing the population in the atomic states [51].

As we have shown in chapter 5 that we can define both the phase shiftquantum gate and the rotation quantum gate on the triangle and hexagonlattices. To achieve this we have considered two different staggering para-meters. This would mean that in a experimental implementation, we wouldlike to be able to switch the staggering imposed on the lattice sites. In ourprocedure each geometric path starts in the North Pole (θ = 0), which cor-responds to the same effective staggering for both choices, i.e. ~λ = (0, 0, 1).The form of the staggering parameter during a path in (θ, φ)-parameterspace is very different, but necessary to be able to perform the two (inde-pendent) one-qubit quantum gates.

7.2 Outlook

We have considered periodic lattice chains of length three and six (the tri-angle and the hexagon). We could extend our analysis to even longer latticechains, such as the nonagon shown in figure 7.1a or any chain with length3n, n ∈ N, as shown in figure 7.1b. These periodic chain will feature twodegenerate supersymmetric ground states, as shown in Fendley and Hagen-dorf [16], Fendley et al. [18]. Each lattice can therefore be interpreted asa single qubit, where the space of computational qubit states equal to thetwo-dimensional space of ground states. However, a longer lattice chain re-sults in ground states with a larger number of particles. So the space oflattice states of which the ground states form a subspace grows fast. Forexample, the ground states on the triangle lattice are one-fermion statesfrom a three-dimensional state space and on the hexagon the ground statesare two-fermion states from a nine-dimensional space. In general, the peri-odic chain of length 3n provides ground states of n fermions. We wonderif considering a longer lattice chain is beneficial for the geometrical quan-tum computation approach. If the computational subspace is embeddedin a larger state space it could have a negative influence on the protectionagainst decoherence, as there are more non-qubit states which can becomeoccupied. From an computational point of view it will be more difficult toconstruct analytical expressions representing the ground states as functionsof the staggering parameters.

We have already seen that our explicit calculations for the hexagon latticeresulted in a more difficult expression for the (integrated) Berry connec-tions. Also the bow tie lattice, constructed from two connected triangle

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(a) The nonagon (or nine-chain),the periodic lattice chain of ninesites.

Periodic chain of length L = 3N

(b) Schematic view of a periodiclattice chain with a length of 3Nsites.

Figure 7.1: Examples of longer lattice chains

lattices, resulted in even more involved expressions for the connections andthe integration could not be performed analytically. So we do not expectthat investigating a lattice constructed by connecting two hexagon latticesis feasible. The interpretation of a two-qubit system will naturally hold, butwe do not see an advantage over the bow tie. Maybe for a multi-qubit sys-tem using a hexagon as single qubit will extend the possibilities in possiblegeometries.

For the analysis in the bow tie lattice we have considered only one of thestaggering choices introduced for the triangle and hexagon lattices, namelythe normal vector on the two-sphere, denoted by ~λ in this work. It mightalso be interesting to calculate the four-dimensional unitary found for theother staggering parameter choice. As for the triangle and hexagon latticeswe might find a second independent two-qubit gate with a simpler form thanthe four-dimensional unitary found in this work. The reason we have notconsidered this option yet is the fact that the quantum gate universalitytheorem requires only one non-trivial two-qubit quantum gate.

The bow tie lattice is constructed by connecting two triangles by a singleedge. We could also consider connecting all three vertices by edges betweenthe two triangles. The geometry is named the prism and shown in figure7.2a. This will in general introduce two more M2 staggering parameters onthe new edges. This extension of the parameter space will naturally increasethe number of possible geometric paths. We can think about parameteriza-tions of the staggering ~λ ∈ C6 on the lattice sites as well as the staggering~µ ∈ C3 on the connecting edges. To simplify the analysis we can also thinkabout reducing the parameter space. Again we could make use of mirrorsymmetry between the triangles, but we could also reduce the ~µ by trans-

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µ1

µ2

µ3

(a) The prism lattice, an exten-sion of the bow tie lattice.

(b) The trident lattice, a differentansatz for a single qubit configu-ration.

Figure 7.2: Two new lattice configurations as ansatz of two qubits (a) orone qubit (b).

lational symmetry, i.e. choosing its components equal. By considering theprism we lose the fact that in the bow tie the qubit (on the North Pole ofstaggering space of ~λ) is defined by the lattice states on the sites which arenot connected. So we expect that turning on µ parameters might have aninfluence before we have performed an adiabatic path in ~λ-space.

We could also consider different lattices equipped with a supersymmetriclattice model. We already did some work on the trident lattice presentedin the appendix, which is shown in figure 7.2b. With the M2 supersym-metric model on this lattice geometry, we also find two degenerate groundstates. When looking at lattice geometries, we are specifically looking forthis feature: two (degenerate) ground states which can be interpreted asqubit states. Moreover, we could also think about lattice configurationswhich produce even more ground states as for example a ‘qutrit’, wherethree ground states could correspond to three qubit states.

In our analysis we have assumed could the periodic path in parameter spacethat is traversed adiabatically. Inspired by Sjoqvist et al. [45] we couldalso consider a non-adiabatic approach. Non-adiabatic geometric quantumcomputation might have a preference over a adiabatic approach because ofreduction of the evaluation time required. We believe that special care hasto be taken when applying non-adiabatic geometric quantum computationto a supersymmetric lattice model. The physical implementation is very dif-ferent compared to for example laser-based approaches and the calculationtime might be less important.

From a condensed matter physics point of view it may be interesting to (the-oretically) investigate if other models producing two (degenerate) groundstates can be interpreted as a qubit system. The supersymmetric lattice

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model may be quite special in this regard, as a result of its precise tuningof the kinetic and potential terms. As also discussed here, we have to con-sider if such a new approach will be experimentally feasible, though. Forgeometric quantum computation we would probably need to implement asite dependent parameter, over which we need enough control to reliablyperform a closed (adiabatic) path. In such an ansatz the scalability willprobably also prove to be a difficult problem.

7.3 Conclusion

In this thesis we have investigated quantum computation based on geometricquantum gates. We have considered qubits defined by the two-level systemof the two degenerate ground states of a supersymmetric lattice model. Twodegenerate zero-energy ground states is a special feature of this lattice modelon periodic chains with a length of a multiple of three. These ground statesof the supersymmetric model can be explicitly constructed for any value ofthe staggering parameter on the lattice chain. The staggering parametergives us a way to adiabatically influence the lattice states. By their connec-tion to the qubit states we can investigate performing quantum operations.The focus of the work presented is achieving quantum control for qubit sys-tems defined with a supersymmetric lattice model.

With periodic chains of length three and six, the triangle and hexagon, wehave investigated constructing one-qubit quantum gates. By two differentstaggering choices on the supersymmetric lattice we are able to constructboth the phase shift gate and the rotation quantum gate. The main ingre-dient is the non-Abelian Berry phase found after a closed adiabatic pathin staggering parameter space. The non-Abelian Berry phase provides theunitary expression for the quantum gate directly. We have explicitly foundthe relation between the quantities parameterizing the path and the corre-sponding phase and rotation angles.

For the construction of a two-qubit gate we have looked into a third lattice.We couple two triangle lattices in one of their vertices and include a newstaggering parameter µ in the supersymmetric model, creating the bow tielattice. This structure allows us to interpret the bow tie lattice as a two-qubit system. Our analysis for the triangle lattice holds in both the trianglestructures of the bow tie lattice, including the construction of one-qubitgates. From the discussed quantum gate universality theorem we see thenecessity of constructing a non-trivial two-qubit quantum gate, in additionto the found one-qubit quantum gates.

We investigate non-trivial two-qubit gates with the bow tie lattice. We have

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again considered an adiabatic closed path in staggering parameter space,which results in a non-Abelian Berry phase. With the entanglement entropyas measure we consider the non-triviality of the constructed two-qubit uni-tary. We have found that the entanglement entropy is dependent on the pathin staggering parameter space in a highly non-linear fashion. We have shownexamples of a near-maximal entanglement entropy and we constructed pro-cedures to approach a specific value. The form of the constructed unitaryis investigated in the context of (non-)triviality of the two-qubit quantumgate and the construction of a maximally entangled Bell state. Further workis necessary to compare the results with known entangling two-qubit gates,such as the CNOT quantum gate.

Concluding, we have shown that specific lattice configurations equipped witha staggered supersymmetric lattice model can be used for geometric quan-tum computation. The simple chains with three and six sites can inter-preted as qubit systems, which can be controlled by adiabatic change of thestaggering parameter. This enables us to construct the unitary matricescorresponding to one-qubit quantum gates, which are sufficient to constructa general one-qubit quantum gate. In the bow tie lattice system we inves-tigate the construction of a non-trivial two-qubit quantum gate, which isa necessary condition for general quantum gate universality. The entangle-ment entropy gives us an indication for the non-triviality of the two-qubitquantum gate. This gives hope that the work presented here can provide acontribution to the research into geometric quantum computation.

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152

Appendix

In this appendix we have included three sections with some additions tothe work in this thesis, which did not fit the main narrative. First, wepresent our calculations with supersymmetric model on the trident lattice,which is a different ansatz for a single qubit system. Second, we showan alternative derivation of the non-Abelian Berry phase after Rezakhaniet al. [40]. Third, we give our calculations of the unitary operator found forgeometric quantum computation in a experimental tripod level structure,inspired by Leroux et al. [29].

Trident lattice

We consider a third ansatz for a qubit system based on a supersymmetriclattice model. The ‘trident’ lattice is defined as follows: site 1, 2, 3 are allconnected to site 4, but not to each other, see also figure A.1a. This latticedoes not follow the criteria of the triangle and hexagon, both being periodicchains of a length a multiple of three sites. However, the trident lattice canbe shown to also possess two degenerate ground states. We impose the M2

supersymmetric model on the trident, allowing up to two sites next to eachother to be occupied. In other words, we allow two-fermion states with themiddle site occupied. The number of ground states follows again from theWitten index eq. (2.18), introduced in section 2.3, by just the number ofpossible states on the trident. We find one zero-fermion state (the emptylattice) and also one three-fermion state, which corresponds to the maximumoccupation (figure A.1a). As the trident lattice consists of four sites, thenumber of one-fermion states is also equal to four. Importantly, we findsix two-fermion states. The zero-energy ground states lie in the manifoldof two-fermion states. This follows from the counting of the number ofdoublets possible: A zero-one doublet, three one-two doublets and a two-three doublet, this leaves two singlets in the two-fermion sector.

Hamiltonian and rotational symmetry

For the definition of the Hamiltonian on the trident, we make use of theM2 definition of the supersymmetric charges, introduced in eq. (2.13) and

153

Site 1

Site 2Site 3

Site 4

(a) The trident lattice showingthe only possible three-fermionstate.

µ1

µ2µ3

λ1

λ2λ3

λ4

(b) The trident lattice with the

staggering parameters ~λ and ~µ.

Figure A.1: Two perspectives at the trident lattice configuration.

eq. (2.14) in section 2.2. We have to keep in mind that the indices of pro-jectors Pi go over all adjacent sites. The Hamiltonian is dependent on thestaggering parameters λi, i = 1, . . . , 4, on the lattice sites and staggeringparameters µj , j = 1, . . . , 3, on the lattice edges.

The Hamiltonian can be constructed explicitly by considering the action ofthe M2 supersymmetric charges on the two-fermion section. We choose thefollowing basis

|1, 2〉 , |1, 3〉 , |2, 3〉 , |1, 4〉 , |2, 4〉 , |3, 4〉 (7.1)

where |i, j〉 = c†ic†j |0〉 denotes the occupation of the sites in the trident lat-

tice shown in figure A.1a (|0〉 denotes the empty trident lattice).

On the trident lattice we can consider a rotational symmetry of the threesites (sites 1, 2, 3) connected to the middle lattice site (site 4). We willconsider the construction of the two supersymmetric ground states bothwith and without taking this symmetry into consideration. The rotationalsymmetry can be seen as a rotation of the trident over 2π

3 which acts on thelattice sites as 1→ 2→ 3→ 1 and 4→ 4. This action translates directly tothe lattice states; |1, 2〉 → |2, 3〉, |1, 4〉 → |2, 4〉, etc.

The rotation operator R related to this action satisfies

R |ψ〉 = Λ |ψ〉 (7.2)

R3 |ψ〉 = I |ψ〉 , (7.3)

so its eigenvalues are the unit roots Λ = 1,−12 ±

i2

√3. In the basis of two-

fermion states (eq. (7.1)) the eigenvalues are all doubly degenerate and therotation operator R has a sparse matrix form with values 1 on (3, 1), (4, 6),(5, 4), (6, 5) and value −1 on (1, 2), (2, 3) and zero otherwise.

154

Imposing the rotational symmetry on the trident lattice severely decreasesthe degrees of freedom of the staggering parameter space. The staggeringparameters on the trident lattice are shown in figure A.1b. With the rota-tional symmetry the following must hold for λi and µj : λ1 = λ2 = λ3 andµ1 = µ2 = µ3. So the set of staggering parameters is effectively reduced toλ, λ4, µ, denoting λi, µi i = 1, 2, 3 by λ, µ.

Non-Abelian Berry phase

Let us consider the explicit calculation of the non-Abelian Berry phase forboth the cases with and without using the rotational symmetry. The ro-tational symmetry influences two properties; the ground states must beeigenvectors of the rotation operator and the space of staggering parametersis reduced.

In the case with rotational symmetry we have found the following expressionsfor (the coefficient vectors of) the ground states |ψ1,2〉. The analysis isperformed using Mathematica. They are zero-energy states H |ψ1,2〉 = 0and have complex conjugate rotation eigenvalues Λ1,2 = −1

2 ±i2

√3. Their

normalized expressions are

ψ1 =1√

36λ2 + 12λ24µ

2

((−1 + i

√3)µλ4, (1 + i

√3)µλ4, 2µλ4,

−2i√

3λ, (−3 + i√

3)λ, (3 + i√

3)λ),

ψ2 =1√

36λ2 + 12λ24µ

2

((−1− i

√3)µλ4, (1− i

√3)µλ4, 2µλ4,

2i√

3λ, (−3− i√

3)λ, (3− i√

3)λ).

Now we will calculate the connection matrices for these two coefficient vec-tors. We calculate the derivatives of the expressions above with respect tothe reduced set of staggering parameters λ, λ4, µ. We find that the connec-tion matrices Aλ, Aλ4 and Aµ are all three equal to the two-dimensionalzero-matrix as all components are zero. The non-Abelian Berry phase iscompletely trivial for the presented system with rotational symmetry.

Let us consider the trident system without the inclusion of rotational symme-try. Now all staggering parameters λi, i = 1, . . . , 4 and µj , j = 1, . . . , 3 areindependent. We will not show the expressions for the ground states here.They are found with Mathematica as the null space of the Hamiltonian;they are the zero-energy eigenstates. We ensure their orthogonalization andnormalization.

We calculate the derivatives to all staggering parameters of both groundstate coefficient vectors using Mathematica. This allows us to explicitly

155

calculate the connection matrices; Aλ1 , Aλ2 , Aλ3 , Aλ4 , Aµ1 , Aµ2 , Aµ3 .These matrices all have the form of Γiσy with different expressions forΓ(λ1, λ2, λ3, λ4, µ1, µ2, µ3). We find a special case for Aλ2 which is the zeromatrix. It is interesting that we find a matrix form equal to what we foundfor the triangle and hexagon lattices with staggering parameter ~λ. Equiva-lently, these connection matrices are matrix exponentiated to the standardrotation matrix, with an rotation angle Γ which depends non-trivially onthe staggering parameters.

Alternative derivation of the non-Abelian Berry phase

In section 4.4 we have included the derivation of the non-Abelian Berryphase as given by Wilczek and Zee [53]. A second derivation is providedhere as given in Appendix A of Rezakhani et al. [40]. This calculationproceeds differently from Wilczek and Zee [53] and is also more elaborate.The Hamiltonian family H(x) depends on M real parameters x, x ∈M. Let us introduce the total evolution time T and the rescaled times = t/T , then a path x : s ∈ [0, 1] → M represents the dynamics. Weconsider a g0(x)-fold degenerate ground state eigensubspace of H(x), witheigenstates |Φα

0 (x)〉. This subspace can be identified by the projector

P0(x) =

g0(x)∑α=1

|Φα0 (x)〉 〈Φα

0 (x)| .

We assume the ground state energy is separated by a non-vanishing gapfrom the rest of the spectrum. Consider the initialization

|ψ(0)〉 =

g0(x)∑α=1

aα |Φα0 (x0)〉 , (7.4)

using the notation xs ≡ x(s) and |ψ(s)〉 ≡ |ψ(xs)〉.

The state of the system at any rescaled time s is given (in units ~ = 1) interms of the propagator V (s) as follows

|ψ(s)〉 = V (s) |ψ(0)〉 , (7.5)

where propagator V (s) is a solution of the time-dependent Schrodinger equa-tion

i∂

∂sV (s) = TH(s)V (s). (7.6)

As the goal in Rezakhani et al. [40] is to study the adiabatic error, let usdefine the propagator and the Hamiltonian corresponding to ideal adiabaticevolution, in the same manner

i∂

∂sVad(s) = THad(s)Vad(s). (7.7)

156

The adiabatic propagator is defined by the intertwining property such thatVad(s) preserves the band structure of the eigensubspace of ground states ofH(s), this property states

Vad(s)P0(0)V †ad(s) = P0(s). (7.8)

If we take the derivative on both sides of the intertwining property, we findthat i ∂∂sP0(s) = T [Had(s), P0(s)], when it holds we have that

|ψad(s)〉 = Vad(s) |ψ(0)〉 =

g0∑α,α′=0

aαV[0]αα′(s) |Φ

α′0 (s)〉 (7.9)

where V[0]αα′(s) = 〈Φα

0 (s)|Vad(s) |Φα′0 (0)〉 is the (non-Abelian) Wilczek-Zee

holonomy. This holonomy is often expressed as the path-ordered exponential

V [0](s) = P exp

[−∫ s

0A(s′)ds′

](7.10)

with the gauge connection

Aαα′ ≡ 〈Φα0 |

∂

∂s|Φα′

0 〉 . (7.11)

The expression in eq. (7.10) can be obtained from the following differentialequation

∂

∂sV

[0]αα′ =

∂

∂s[ 〈Φα

0 (s)|Vad(s) |Φα′0 (0)〉] (7.12)

= 〈 ∂∂s

Φα0 (s)|Vad(s) |Φα′

0 (0)〉+ 〈Φα0 (s)| ∂

∂sVad(s) |Φα′

0 (0)〉

In Rezakhani et al. [40] the adiabatic Hamiltonian is expressed in terms ofthe original Hamiltonian plus a correction term

Had(s) = H(s) + i

[∂

∂sP0(s), P0(s)

]/T. (7.13)

Its action equals

Had(s) |Φα0 (s)〉 =

(H(s) + i

[∂

∂sP0(s), P0(s)

]/T

)|Φα

0 (s)〉 (7.14)

=

(H(s) + 2i

∂

∂sP0(s)P0(s)/T − i ∂

∂sP0(s)/T

)|Φα

0 (s)〉

= E0(s) |Φα0 (s)〉+ i

∂

∂sP0(s) |Φα

0 (s)〉 /T (7.15)

157

Here we have used the property of the projector P0(s), P 20 = P0, such that

∂

∂sP0 =

∂

∂s(P0)P0 + P0

∂

∂s(P0) (7.16)[

∂

∂sP0(s), P0(s)

]=

∂

∂s(P0)P0 − P0

∂

∂s(P0) (7.17)

=∂

∂s(P0)P0 −

(∂

∂sP0 −

∂

∂s(P0)P0

)(7.18)

= 2∂

∂s(P0)P0 −

∂

∂sP0 (7.19)

Moreover, from the definition of the projector

P0(s) =

g0∑α=1

|Φα0 (s)〉 〈Φα

0 (s)|

we can find

∂

∂sP0(s) =

g0∑β=1

(| ∂∂s

Φβ0 (s)〉 〈Φβ

0 (s)|+ |Φβ0 (s)〉 〈 ∂

∂sΦβ

0 (s)|)

(7.20)

such that

∂

∂sP0(s) |Φα

0 (s)〉 = | ∂∂s

Φα0 (s)〉+

g0∑β=1

〈 ∂∂s

Φβ0 (s)|Φα

0 (s)〉 |Φβ0 (s)〉 (7.21)

Now, we can rewrite the expression in eq. (7.12), using the expressions wejust derived, as follows

∂

∂sV

[0]αα′ = 〈 ∂

∂sΦα

0 (s)|Vad(s) |Φα′0 (0)〉 − iT 〈Φα

0 (s)|Had(s)Vad(s) |Φα′0 (0)〉

= −iTE0(s) 〈Φα0 (s)|Vad(s) |Φα′

0 (0)〉

−g0∑β=1

〈Φα0 (s)| ∂

∂sΦβ

0 (s)〉 〈Φβ0 (s)|Vad(s) |Φα′

0 (0)〉 . (7.22)

Without loss of generality we can assume that E0(s) = 0, which corresponds

to neglecting the dynamical phase. So V[0]αα′ must satisfy the following dif-

ferential equation

∂

∂sV

[0]αα′ = −

∑g0

β=1

⟨Φα

0 (s)∣∣∣ ∂∂sΦβ

0 (s)⟩⟨

Φβ0 (s)

∣∣∣Vad(s) ∣∣∣Φα′0 (0)

⟩(7.23)

= −∑g0

β=1Aαβ(s)V[0]βα′(s). (7.24)

The solution is given by the path-ordered exponential defined in terms ofthe gauge potential Aαβ ≡ 〈Φα

0 | ∂∂s |Φβ0 〉

V [0](s) = P exp

[−∫ s

0A(s′)ds′

]. (7.25)

158

σ+ π σ−

mg = 7/2mg = 5/2 mg = 9/2

me = 7/2

Fg = 9/2

Fe = 9/2

Figure A.2: Tripod atomic level scheme as investigated by Leroux et al. [29].

Unitary operator found by an experimental tripodscheme

The most recent article we have introduced in the literature review in sec-tion 4.7 is Leroux et al. [29]. They consider a experimental implementationof the tripod scheme in a cold Strontium gas. Equivalent to our approachLeroux et al. are looking to find non-Abelian adiabatic SU(2) geometrictransformations. Here we will consider their explicit parameter choice anddiscuss their results.

The four-level tripod scheme includes three coplanar coupling laser beamswhich are set on resonance with their common excited state. Their relativephases can be varied using two electro-optic modulators, which enable us toapply a non-Abelian transformation. The equivalent objects to the groundstates in the supersymmetric lattice model for the tripod scheme are itsdark states. In more detail, for any value of the amplitude and phase ofthe laser beams, the effective Hilbert space contains two degenerate darkstates which do not couple to the excited state. The effective Hilbert spaceis defined by four coupled bare levels of the 1S0, Fg = 9/2→ 3P1, Fe = 9/2intercombination line (see figure A.2).Let us consider the explicit expressions for the two degenerate dark-statesas given by [29]

|D1〉 =e−iΦ13(r)|1〉−e−iΦ23(r)|2〉

√2

, (7.26)

|D2〉 =e−iΦ13(r)|1〉+e−iΦ23(r)|2〉−2|3〉

√6

where |i〉 ≡ |mg = i+ 3/2〉 (i = 1, 2, 3) and Φij = Φi − Φj . The space-dependent laser phases read Φi(r) = ki · r + ϑi, where ki is the wavevectorof the beam coupling state |i〉 to |e〉 and ϑi its phase at the origin. The twoindependent offset phases tuned by the electro-optic modulators are defined

159

φ0

φ0

φ2

φ1

a

bc

Figure A.3: Path in (φ1, φ2) phase parameter space as implemented byLeroux et al. [29].

by φi = ϑi − ϑ3 (i = 1, 2).

Leroux et al. [29] investigate the non-Abelian unitary operator acting on thedark-state manifold when the relative phases of the tripod are adiabaticallyswept along a closed loop C in parameter space. The calculation is performedusing the loop integral of the two-dimensional Berry one-form ω ≡ [ωjk] ≡[i~ 〈Dj |dDk〉] and results in

U = B exp

(i

~

∮Cω

)(7.27)

The path in parameter space corresponds to a phase loop in (φ1, φ2)-spaceas shown in figure A.3. The path is cycled counterclockwise, with a phaseexcursion of φ0.As already given in subsection 4.7.9 Leroux et al. [29] construct a full geo-metric unitary operator U of the form(

α β−β∗ α∗

)(7.28)

with |α|2 + |β|2 = 1.

Let us try to reconstruct this result with the method we use in this work forthe triangle and hexagon lattices. We will explicitly construct the Berry one-form and calculate the geometric unitary. We will start from the definitionof the dark-states in eq. (7.26), which form an orthonormal set. From theexpressions of the dark states in the basis |1〉 , |2〉 , |3〉 we calculate thederivatives with respect to the phase parameters explicitly. The expandedform of the coefficient vectors are given by

|D1〉 =1√2

(e−i((k1−k3)·r+φ1) |1〉 − e−i((k2−k3)·r+φ2) |2〉 , (7.29)

|D2〉 =1√6

(e−i((k1−k3)·r+φ1) |1〉+ e−i((k2−k3)·r+φ2) |2〉 − 2 |3〉).(7.30)

160

The expressions for the derivatives with respect to φi (i = 1, 2) can now befound to be

∂

∂φ1|D1〉 =

−ie−i((k1−k3)·r+φ1) |1〉√2

, (7.31)

∂

∂φ2|D1〉 =

ie−i((k2−k3)·r+φ2) |2〉√2

, (7.32)

∂

∂φ1|D2〉 =

−ie−i((k1−k3)·r+φ1) |1〉√6

, (7.33)

∂

∂φ2|D2〉 =

−ie−i((k2−k3)·r+φ2) |2〉√6

. (7.34)

The explicit forms of the connections Aφ1 and Aφ2 can now be calculated,where (Aφ)jk = 〈Dj |∂φDk〉. Let us consider the form ωφi = i~Aφi , theresults are

ωφ1 = ~

(12

12√

31

2√

316

), (7.35)

ωφ2 = ~

(12 − 1

2√

3

− 12√

316

). (7.36)

The next step is the calculation of the Berry phase by integration of theBerry connections. For the path introduced before and shown in figure A.3we find the following

γa =

∫ φ0

0ωφ2(φ1 = 0, φ2) dφ2, (7.37)

γb =

∫ φ0

0ωφ1(φ1, φ2 = φ0) dφ1, (7.38)

γc =

∫ 0

φ0

(ωφ1(φ1 = α, φ2 = α) + ωφ2(φ1 = α, φ2 = α)) dα. (7.39)

The corresponding unitary matrices are found by matrix exponentiatingthese expressions, such that the geometric unitary of the closed path isgiven by the multiplication while taking path ordering into account. So forUp = exp

(i~γp), we find that U = UcUbUa is explicitly given by1

8e−iφ0

(3e

4iφ03 + 6e

2iφ03 − 1

)−√

38 e−iφ0

(e

2iφ03 − 1

)2

√3

8 eiφ0

(e−2iφ0

3 − 1)2

18eiφ0

(3e−4iφ0

3 + 6e−2iφ0

3 − 1).

(7.40)

This is indeed an unitary matrix of the form given in eq. (7.28).

161