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Topological Quantum Computing on a Conventional Quantum Computer
Xiao Xiao,1 J. K. Freericks,2 and A. F. Kemper1, ∗
1Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA2Department of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057 USA
Majorana fermions are self-adjoint fermionic particles that are believed to exist as the elemen-tary excitations in nanoengineered devices with superconductors and ferromagnets. They can beemployed in topological quantum computation and can aid the solution of a class of spin modelscreated by Kitaev. In both cases, the braiding of Majorana fermions plays a critical role in theseapplications. We explicitly construct quantum circuits for Majorana braiding and run them on IBMquantum computers to prepare the ground states of strongly correlated Kitaev-inspired models. Theentanglement entropy for these models is then measured to determine quantum phase transitions.We show how maintaining particle-hole symmetry is critical to carrying out this work.
INTRODUCTION
Quantum computers are fragile and susceptible torapid decoherence [1]. There are two propositions tomake quantum computation practical for deep circuits:(i) fault-tolerant quantum computers, which use logi-cal qubits and error correcting codes (like the surfacecode) to correct all errors that appear during compu-tation [2–11] and (ii) topological quantum computers,which use topological qubits, by nature protected fromthe environment, to perform intrinsically error-free com-putations [12–14]. Topological qubits are made fromanyons; their quantum algorithms create, braid and fusethese anyons in a controlled fashion. Unfortunately, ithas proved to be extremely difficult to create topologi-cal qubits in real systems [15]. In this work, we lever-age the intrinsic protection of topological quantum algo-rithms and states by simulating them on a conventionalnoisy intermediate-scale quantum computer (NISQ) withtopological braiding operations encoded in conventionalquantum gates. This allows us to use the discrete natureof topological properties to directly determine the phasediagram of Kitaev-inspired models. These topologicalquantum simulations are carried out on IBM’s transmon-based conventional quantum computing hardware.
Kitaev’s spin model on the honeycomb lattice [16] is aparadigmatic model for topological quantum computing.It is solved exactly by mapping to free Majorana fermions(a type of anyon) coupled to a static Z2 gauge fieldand exploiting an infinite number of conserved quanti-ties in the thermodynamic limit. After its introduction, alarge class of related, exactly solvable, strongly correlatedmodels with similar properties have been created [17–42]. When simulating Kitaev-inspired models that mapto Majorana fermions, one must explicitly incorporatethe anyonic braidings. In a quantum circuit, we createthis braiding with low-depth circuits and then use themto prepare ground states of Kitaev-inspired models. Thedetermination of the phase boundaries in these modelsis challenging because they lack a local order parameter.
The standard approach is to calculate the entanglemententropy [43], but this normally requires a large systemsize, beyond what is currently available in NISQ ma-chines. Instead, we use particle-hole symmetry-enforcedcircuits to explore throughout the Brillouin zone. Dueto the topological nature of the quantum phase transi-tion, these circuits can identify the different phases withonly a small number of qubits allowing for efficient NISQimplementation.
1
32 4
5 76
8z
xy
87654321
M
M
M
M
gaugefermions
bond fermionsin real space
Fy
Fy
Fx
Fx
bond fermionsin k-space
B
BBogoliubons
M1
2
3
4
z
3
2
1
4
Majorana braiding
U1(2 )
U2(2 , 2 )
U2( 2 , 2 )
U1(2 ) U2( 2 , 2 )
U2(2 , 2 )
inter-fermion braiding
intra-fermionbraiding
Fj U1( ) H
Z
B
X Rx X
FSWAP
Z
(a) (b)
(c)
FIG. 1. Quantum circuit for the Kitaev honeycombmodel with braiding. (a) The lattice structure of the Ki-taev honeycomb model with the numbers denoting the order-ing along the Jordan-Wigner string; (b) the quantum circuitused to prepare the ground-state wave function; (c) detail ofthe elements of the quantum circuit shown in (b). In (a) thex-, y- and z-bonds are shown in black, green and red respec-tively.
RESULTS
Kitaev’s model has spins distributed on a honeycomblattice with nearest-neighbor spin couplings Jασαi σ
αj that
depend on the bond direction [16]. The model is solved
arX
iv:2
006.
0552
4v1
[qu
ant-
ph]
9 J
un 2
020
2
by fermionization; we use Jordan-Wigner strings that areoriented along the x, y bonds (see Fig.1). Many Kitaev-inspired models generalize this geometry to include struc-tures such as chains and 2-leg ladders and generalize theinteractions to include non-Ising-like couplings such asσxi σ
yj . For an exact solution to exist, the lattice geometry
and interactions must conform to a broad set of rules[41].We consider two particular models: the original honey-comb lattice and a one-dimensional variant (called theKitaev spin chain), which is a chain extracted from theoriginal Kitaev honeycomb model by turning off the Jycouplings.
After the Jordan-Wigner transformation, z-z spin cou-plings map to fermion-fermion interactions, making theeffective fermion model interacting. But in Kitaev-inspired models, these terms can be simplified by Ma-jorana braiding: the fermions at each site are split intotwo Majorana fermions, braided both internally and witha neighbor connected by a z-bond (see Sec. I and III in SI[44] for details) and recombined into two new fermions.The first is an itinerant bond-centered fermion (on thez-bonds) and the second is a set of conserved gaugefermions (one per z-bond). Thus, the original spin modelis fermionized into bond fermions and gauge fermionsthat act as local chemical potentials. As long as we areinterested in the ground state, the gauge fermions donot play any further role [16, 43]. The eigenstates inthe free fermion sector are found by Fourier transfor-mation followed by Bogoliubov transformation [45, 46].A schematic of the quantum circuit that performs thesesteps is given in Fig. 1(b) and (c).
The braiding circuit in Fig. 1(c) accounts for the non-trivial phase changes due to the braiding of two Majoranafermions [47, 48]:
B±i,j =1√2
(1± γiγj) , (1)
where ± denotes the clockwise and counterclockwisebraidings of the two Majorana fermions γi and γj . Thereare two different types of braidings, intrafermion and in-terfermion. The intrafermion braiding operator relatesto the Jordan-Wigner fermions via
Bin =1√2
[1± i
(c†c− cc†
)], (2)
which is a single-particle operation [see Fig. 1(c)] andcan be realized by a phase gate in quantum circuits. Theinterfermion braiding is more complex, and can be ex-pressed as (see Sec. I in SI [44] for details):
Bex =1√2
[1± i
(c†1c†2 − c
†1c2 + c1c
†2 − c1c2
)]. (3)
From the matrix representation of Bex, we con-struct a quantum circuit realizing the braiding by twoCNOT gates in combination with single-qubit gates [49](Fig. 1(c)). In these operations, we must abide by the lo-cal constraints imposed by the Hamiltonian (see Sec. IIIin SI [44] for details).
With the braiding circuit in hand, we apply them tothe 8-site Kitaev honeycomb lattice and the 4-site Ki-taev spin chain and obtain the entanglement entropy fora subsystem (A) by measuring the density matrix. Whilethis approach clearly does not scale, it is feasible for thesesize systems. Fig. 2 shows the entanglement entropy forthe subsystem A illustrated in green in Figs. 2(a) and(b) as we vary one of the exchange constants. The re-sults from the quantum circuit simulator and the exactresults agree up to statistical noise (see Sec. IV in SI[44] for the ground-state energies). The results from thequantum hardware, although they show similar trends,exhibit significant deviations from the exact results dueto machine errors. Nevertheless, the circuits do correctlycapture the braiding operation that is fundamental to thesolution of the Kitaev-inspired models and to the simu-lation of a topological quantum computer.
The ranges of exchange parameters in Fig. 2 cover aregime where both models have quantum phase transi-tions (in the thermodynamic limit). We observe no sig-nature of these transitions due to finite-size effects (seeSec. V in SI [44] for details); however this limitation ofNISQ machines can be overcome for the Kitaev-inspiredmodels by employing a grid of shifted momenta.
The Kitaev-inspired models reduce to a block-diagonalform in momentum space after they have been con-verted to the Majorana-Fermion representation. But,because a Bogoliubov transformation is still needed tofully diagonalize the problem, one must be careful topreserve particle-hole (PH) symmetry when construct-ing the quantum circuit. Consider a system contain-
0.0
0.2
0.4
0.6
S A
x z x z xA
(a)
exactsimulator
0.0 0.5 1.0 1.5 2.0Jx
1.1
1.2
S A
Almaden
1.4
1.5
1.6S A
(b) exactsimulator
A xz
y
0.6 0.8 1.0 1.2 1.4Jy
2.0
2.3
2.6
S A
Almaden
FIG. 2. Quantum simulation of the entanglemententropy of Kitaev-inspired models: (a) the 1D Kitaevspin chain; (b) the 2D Kitaev spin model on the honeycomblattice. In both (a) and (b), the lattice structure and the defi-nition of the subsystem A (the green shaded parts) are shownin the insets of the upper panels, the comparison of the exactresults and the results from the simulator are shown in theupper panel, and the results from IBMQ-Almaden are shownin the lower panel. The data from Almaden were obtained byperforming 5 independent experiments, and for each of themN = 8196 shots were used. The simulator data for the 1DKitaev spin chain was obtained by N = 8196 shots, while thesimulator data for the 2D Kitaev spin model was obtained byN = 81960 shots.
3
z1
z0
z1
z0ai
M M M M
f1
g1f0
g0f1
g1f0
g0
c1,
c1,
c0,
c0,
c1,
c1,
c0,
c0,
ei ( k) ei ( k)
ei ( k)ei ( k)
kk
Brillouin zone
B
B
F
F
k
k
c1,
c0,
c1,
c0,
{k{k
Key: z z-dimer along a primitive vector aigi: gauge fermion at the ith dimerfi: bond fermion at the ith dimerci, : -related bond fermion at the ith dimerci, : -related bond fermion at the ith dimermomentum points constraint by discrete FTand PH symmetry
(a)
(b)
0.0 0.2 0.4 0.6 0.8 1.0k/2
0.0
0.2
0.4
0.6
0.8
1.0(c)
c0, c1,
c0, c1,
AexactsimulatorAlmaden
0.0 0.2 0.4 0.6 0.8 1.0k/2
0.2
0.3
0.4
0.5
0.6
0.7
0.8(d)
FIG. 3. The symmetry-enforced methodology: (a)symmetry-enforced particle-hole processes in real and momen-tum space; (b) the symmetry-enforced circuit; (c) the entan-glement spectra of the gapped phase for the 1D Kitaev spinchain with Jx = 0.5 < Jz = 1; and (d) the entanglement en-tropy of the gapless phase for the 1D Kitaev spin chain withJx = 1.5 > Jz = 1.
ing N unit cells. The discrete Fourier transformationto an arbitrary set of N momentum points in the BZ,K = {2πn/N | n ∈ [0, N)}+ δk can be written as:
cn,K =1√N
∑k∈K
eiknck, (4)
where cn,K denotes real-space operators obtained byFourier transformation of the set K. Here, δk is intro-duced to shift the momentum points. In the Fouriertransformation circuits, the phase shift requires a phasecorrection [50]. PH symmetry requires that the informa-tion from the set −K must also be included. Hence, thesymmetry-enforced Fourier transformation is given by:
cn =1√2
(cn,K + cn,−K) , (5)
which requires a more complex circuit with twice thequbits. The demonstration of the symmetry-enforcedprocesses under the particle-hole symmetry for the N = 2case is shown schematically in Fig. 3(a). We maintaintwo copies of each bond fermion fi (blue), one for eachset ±K, which are shifted by the corresponding ±δk.The corresponding quantum circuit is shown in Fig. 3(b).Since the Bogoliubov transformation mixes ±K, the cor-rect momentum set is swapped before the transformation,and swapped back after.
By using this circuit, we can obtain the entanglementspectra of the 2-site Kitaev spin chain as we sweep acrossthe Brillouin zone by measuring a fermionic correlationfunction [51, 52] (see Sec. V in SI [44] for details). When
0.0 0.5 1.0 1.5 2.0Jx
0.0
0.2
0.4
0.6
S A
Kitaev spin chain(a)
exactsimulatorAlmaden
0.0 0.5 1.0 1.5 2.0Jz
0.0
0.5
1.0
1.5
2.0
J x
(b)
0.4
0.5
0.6
0.6 0.8 1.0 1.2 1.4Jy
0.0
0.5
1.0
S A
Kitaev honeycomb model(c) exact
simulatorAlmaden
0.8 0.9 1.0 1.1 1.2Jy
0.0
0.1
0.2
J x
(d)
0.6
0.7
0.8
0.9
1
FIG. 4. Quantum phase transitions in Kitaev-inspiredmodels determined by entanglement entropy at highsymmetry points: (a/c) The entanglement entropy of bondfermions in the subsystem A defined in Fig. 2 for the Ki-taev spin chain/honeycomb model by measuring the contri-bution from high-symmetry points to the correlation matrix.(b/d) IBMQ-Almaden determined phase diagram of Kitaevspin chain/honeycomb model. The gold dashed line denotesthe exact phase boundary.
Jx < Jz, the entanglement spectra is always gapped [seeFig. 3(c)], while when Jx > Jz, the entanglement spec-trum is gapless; the gap closes at δk = π [see Fig. 3(d)].
The high symmetry points with δk = π in the BZ arethe PH symmetric points, and we expect the quantumphase transition to be dominated by the behavior nearthese points. Hence, we must ensure the simulation in-cludes the effects of these high-symmetry points, in or-der to efficiently determine the phase diagram, which wedo by the phase-shifting method discussed above. Usingthese specific momenta for the larger systems discussedin Fig. 2, we obtain the entanglement entropy (from thesame correlation function as above) as a function of ex-change parameter, shown in Fig. 4. Although the bondfermion entanglement entropy does not fully capture thesystem entanglement entropy, the portion that comesfrom the gauge fermions is featureless [43], and thus sharptransitions may be found by studying the bond fermionsalone. The bond-fermion entanglement entropy for theKitaev spin chain and honeycomb models are shown inpanels (a) and (c). The quantum phase transition isidentified by the discrete jump in the entanglement en-tropy. Using the phase-shifted momenta and the correla-tion function measurement, the results from the quantumcomputer now clearly show the phase boundary, even ifthe magnitude of the jump is reduced. Using this stepas the identifying feature, we are able to reconstruct thephase diagram of the two models as a function of both ex-change parameters on the IBM-Almaden quantum com-puter, as shown in Fig. 4(c) and (f).
4
CONCLUSION
We used a conventional quantum computer to simu-late a topological quantum computer. This is achievedthrough low-depth braiding circuits, which run on NISQmachines. By using a PH symmetry-preserving method-ology that includes high-symmetry points in the BZ, thesimulation determined the quantum phase diagram ofboth the Kitaev spin-chain and the original Kitaev hon-eycomb model. These results vividly demonstrate theuse of topological quantities and methods on NISQ hard-ware, as the effects of noise and other run-time errorswere naturally mitigated by the topological properties ofthe system. This allowed for the phase diagram to befound by finding discontinuities in the entanglement en-tropy, which are reduced in magnitude but remained easyto see. This work shows that the benefits of topologicalquantum computation can be realized on NISQ-era con-ventional quantum computers.
METHODS
The entanglement entropy of the ground states shownin Fig. 2 were obtained by projectively measuring the fullreduced density matrix, after the states were prepared bythe quantum circuit in Fig. 1(b) or its modifications. Themaximum likelihood estimator [53] was used to obtainthe closest physical density matrix based on the measuredone. For the ground state of the Kitaev-inspired model,as long as the system size is fixed, the gauge fermion givesa trivial contribution, so the nontrivial information ofdifferent phases can be extracted from the entanglementinformation of just the bond-fermions. The states usedto obtain entanglement information of the bond-fermions
were prepared by the symmetry-enforced circuit detailedin Fig. 3(b), and the the entanglement information ofthe bond-fermions was obtained by projectively measur-ing the correlation matrix [51, 52] (see Sec. V in SI [44]for details). A possible simplification of the symmetry-enforced circuit was found and detailed in Sec.VI in SI[44]. The entanglement entropy of the bond-fermions ofthe Kitaev spin model was obtained by measuring thecorrelation matrix via the simplified symmetry-enforcedcircuit.
ACKNOWLEDGEMENTS
This work was supported by the Department of Energy,Office of Basic Energy Sciences, Division of Materials Sci-ences and Engineering under Grant No. DE-SC0019469.J.K.F. was also supported by the McDevitt bequest atGeorgetown. We acknowledge use of the IBM Q for thiswork. The views expressed are those of the authors anddo not reflect the official policy or position of IBM or theIBM Q team. Access to the IBM Q Network was obtainedthrough the IBM Q Hub at NC State. We acknowledgethe use of the QISKIT software package [54] for perform-ing the quantum simulations. The authors also thankSonika Johri for insightful comments and Akhil Francisfor some technical expertise.
AUTHOR CONTRIBUTIONS
X.X. and A.F.K initialized the idea. X.X. conductedall the calculations with the help of A.F.K. and preparedthe figures. All authors discussed the results and wrotethe manuscript.
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6
Supplemental Materials: Topological Quantum Computing on a ConventionalQuantum Computer
I. CONSTRUCTION OF BRAIDING CIRCUITS
When the positions of Majorana fermions change, they braid, and must be described by a braiding circuit. Thesimplest braiding is the braiding of two Majorana fermions to construct a conventional fermion. A conventionalfermion, which can be written as f = (γ2 + iγ1)/2, is connected to another conventional fermion defined by f̃ =(γ1 + iγ2)/2 via a braiding operation on the Majorana fermions. Recall that the braiding operator (in terms ofMajorana fermions) is given by
B±i,j =1√2
(1± γiγj) . (I.1)
In this first example, the two Majorana fermions relate to the same conventional fermion. After expressing theMajorana fermions in terms of the conventional fermions, we find that
B±1,2 =1√2
[]1± i(f† − f)(f + f†)
]=
1√2
[]1± i(f†f − ff†)
], (I.2)
where ff and f†f† vanish due to the Pauli principle. To express this operator as a matrix in the qubit representation,we map the computational basis (|0〉, |1〉) onto the conventional fermion number basis yielding
B±1,2 = B±in =1√2
(1∓ i 0
0 1± i
). (I.3)
In the second example, we consider the case of the braiding of two Majorana fermions that belong to two differentconventional fermions. The conventional fermions are{
f1 = (γ2 + iγ1)/2,
f2 = (γ4 + iγ3)/2,(I.4)
when expressed in terms of the different Majorana fermions. There are four different ways to braid these Majoranafermions. Because they are noncommuting operators, we have to carefully specify the ordering scheme. We choose theascending order for the conventional fermions and for the Majorana fermions within the same conventional fermion.With this specification, we find that the different inter-fermion braidings can be expressed as the product of thebraidings within the same fermions and and one more braiding, denoted B±2,3. This is the fundamental braidingbetween two conventional fermions and is denoted Bex. Replacing γ2 and γ3 by f1 and f2, we find that
B±ex =1√2
(1± i(f1 + f†1 )(f†2 − f2)
),
=1√2
(1± i(f†1f
†2 − f
†1f2 + f1f
†2 − f1f2)
). (I.5)
Again, we use a conventional number basis (|0f10f2〉, |1f10f2〉, |0f11f2〉, |1f11f2〉), yielding the matrix representation forthe inter-fermion braiding given by
B±ex =1√2
1 0 0 ±i0 1 ∓i 00 ∓i 1 0±i 0 0 1
. (I.6)
Next, we decompose this braiding operation in terms of conventional quantum gates. Since the clockwise braidingis the Hermitian conjugate of the anti-clockwise braiding, we focus on the clockwise braiding. We first consider thebraiding within the same fermion, and we notice that
B+in =
1√2
(1− i 0
0 1 + i
)= e−iπ/4
(1 00 i
), (I.7)
7
so Bin can be realized by the phase gate U1(π/2). Similarly, we find:
B+ex = e−i
π4 σy⊗σy . (I.8)
So, the quantum circuit can be immediately constructed by via the general scheme proposed by Vidal et al. [49].
II. THE EXACT SOLUTION OF THE 1D KITAEV SPIN CHAIN
The Hamiltonian of the 1D Kitaev spin chain is given by
H1D = 4Jz
N/2∑i=1
σz2i−1σz2i + 4Jx
N/2∑i=1
σx2iσx2i+1, (II.1)
where the odd sites belong to the • sublattice while the even sites belong to the ◦ sublattice (see the configuration inFig.1 in the main text). The factor of 4 is added for convenience, as we will see below. To solve this model, we firstmap the spin model into a fermionic model by the Jordan-Wigner transformation:
σ+m →
∏j<m σ
zj cm,
σ−m → i∏j<m σ
zj c†m,
σzm → 12
(1− 2c†mcm
),
(II.2)
where we use the standard mapping |0〉 = | ↑〉 and |1〉 = | ↓〉. After fermionization, the Hamiltonian becomes
H̃1D = Jz
N/2∑i
(2n2i−1 − 1) (2n2i − 1)
+ Jx
N/2∑i=1
(c2i − c†2i
)(c2i+1 + c†2i+1
), (II.3)
which is identical to the fermionic expression of the Kitaev spin model in the limit Jy = 0 [16]. To solve this, we musttransform the four-fermion term. We introduce the Majorana fermions on each sublattice η◦/• and γ◦/• [19, 48]:{
c◦ = (η◦ + iγ◦) /2,
c• = (γ• + iη•) /2.(II.4)
The Hamiltonian is re-expressed as
H̃1D = −iJx∑i
γi,◦γi+1,• + iJz∑i
Diγi,•γi,◦, (II.5)
where Di = iηi,•ηi,◦ [16, 19] is a local gauge flux. Here i labels the unit cell, which have two sublattices • and ◦. Sincethe Majorana fermions do not have well-defined occupation numbers, we construct quantum circuits after mappingthe Majorana fermions back back to conventional fermions. We introduce bond fermions for the two sites connectedby the z-bonds as follows:
fi =1
2(γi,• + iγi,◦) . (II.6)
Finally, we re-express the Hamiltonian again and find
H̃1D = Jx∑i
(f†i − fi
)(f†i+1 + fi+1
)+ Jz
∑i
Di(2f†i fi − 1). (II.7)
Because of the translational symmetry, Lieb’s theorem indicates that the fluxes Di must be uniform [16]. Since Di
commutes with the Hamiltonian H̃A, it is a constant of motion. Note that η2◦ = η2• = 1, so Di = ±1.To diagonalize the Hamiltonian, we start with a Fourier transformation
fn =1√N
∑k
eiknf̃k, (II.8)
8
where the Hamiltonian becomes
H̃1D =∑k
Ψ†k
(JzD + Jx cos k iJx sin k−iJx sin k −JzD − Jx cos k
)Ψk, (II.9)
after dropping some irrelevant constants. Here, we introduce the ”spinor”
Ψk =
(f̃kf̃†−k
). (II.10)
This Hamiltonian in the momentum space can be easily diagonalized through a Bogoliubov transformation. Doing sogives us the final diagonal form
H̃1D =∑k
Ek
(b†kbk − b−kb
†−k
), (II.11)
where Ek =√
(JzD + Jx cos k)2 + J2x sin2 k, and bk (b†k) denotes the annihilation (creation) operator for Bogoliubov
fermions with momentum k.
III. CONSTRUCTION OF BRAIDING CIRCUITS FOR SPECIFIC MODELS
We first describe how different braiding operations affect quantum states. The braiding operations are used to changethe positions of Majorana fermions, allowing Majorana fermions belonging to two different conventional fermions torecombine to new fermions. This implkies that all braiding processes can be constructed with one inter-fermionbraiding and a few intra-fermion braidings. The inter-fermion braiding changes the original two-qubit states into asuperposition of their particle-hole partners with a ±π/2 phase difference. Intra-fermion braidings behave differentlydepending on whether they are performed before or after the inter-fermion braiding. If the intra-fermion braiding isperformed before the inter-fermion braiding, it introduces a ±π/4 phase to the fermion. However, if the intra-fermionbraiding is performed after the inter-fermion braiding, it introduces a superposition of the original state (before inter-fermion braiding) and its particle-hole partner (with a ±π/2 phase difference). These basic properties of braidinghelp us construct braiding circuits based on local constraints.
The first constraint imposed on the braiding operations comes from the local z-bonds. For Kitaev-inspired models,the local fermionic Hamiltonian coupling two sites connected by a z-bond is
Hz = Jz
(2c†1c1 − 1
)(2c†2c2 − 1
). (III.1)
The two-qubit states are |n1n2〉. This Hamiltonian is re-expressed in terms of bond and gauge fermions as
H̃z = JzD(2f†f − 1
), (III.2)
where D is the local gauge flux and f stands for the bond fermion. The two-qubit states here are |nfng〉 with ngdenoting the occupation of gauge fermion. The two Hamiltonian and their corresponding states are connected bybraiding operations, so the matrix elements of the Hamiltonian (before and after transformation) are consistent. Thisimplies that the states |nfng〉 are transformed to the states |n1n2〉 as follows:
|00〉f,g → α0000|00〉1,2 + α11
00|11〉1,2,|11〉f,g → α00
11|00〉1,2 + α1111|11〉1,2,
|10〉f,g → α1010|10〉1,2 + α01
10|01〉1,2,|01〉f,g → α10
01|10〉1,2 + α0101|01〉1,2.
(III.3)
A direct calculation yields
〈00|Hz|00〉1,2 = 〈11|Hz|11〉1,2=− 〈10|Hz|10〉1,2 = −〈01|Hz|01〉1,2 = Jz (III.4)
and
〈00|Hz|00〉f,g = −〈11|Hz|11〉f,g=− 〈10|Hz|10〉1,2 = 〈01|Hz|01〉1,2 = −JzD. (III.5)
9
This imposes the constraint D = 1 for ng = 1 and D = −1 for ng = 0.This local constraint imposed by the z-bond is general for Kitaev-inspired models. But, the local constraints
imposed by other bonds depend on the Jordan-Wigner strings, because those strings determine the ordering of bondfermions emerging from different braiding operations. The details of the braiding circuit of the Kitaev spin model isshown explicitly in the main text, so we demonstrate here how to construct the braiding circuit for the 1D Kitaevspin chain.
Example: The braiding circuit of the 1D Kitaev spin chain: the constraint imposed by the x-bond
To make the discussion explicit, we focus on the x-bond connecting sites 2 and 3. The original Hamiltonianconnecting across an x-bond is
Hx = Jx
(−c2 + c†2
)(c3 + c†3
), (III.6)
and it is re-expressed in terms of bond and gauge fermions as
H̃x = Jx
(f†1 − f1
)(f†2 + f2
), (III.7)
where the braiding operations on c1 and c2 give the bond fermion f1 and the braiding operations on c3 and c4 givethe bond fermion f2. Hence, the states considered here contain four qubits. Start with the ng1 = ng2 = 1 case. The
relevant states for H̃x are |nf1nf2ng1ng2〉 = |nf1nf211〉f . They are transformed under H̃x as
H̃x|0011〉f = Jx|1111〉f , H̃x|1111〉f = Jx|0011〉f ,H̃x|0111〉f = Jx|1011〉f , H̃x|1011〉f = Jx|0111〉f . (III.8)
From the general form of braiding provided by Eq. (III.3), we determine the corresponding states in terms of|n1n2n3n4〉c (relevant to Hx) as follows:
|0011〉 → (α0101)2|0101〉c + α10
01α0101|1001〉c
+ α0101α
1001|0110〉c + (α10
01)2|1010〉c,|1111〉 → −(α00
11)2|0000〉c − α0011α
1111|0011〉c
− α1111α
0011|1100〉c − (α00
11)2|1111〉c,|0111〉 → −α01
01α0011|0100〉c − α01
01α1111|0111〉c
− α1001α
0011|1000〉c − α10
01α1111|1011〉c,
|1011〉 → α0011α
0101|0001〉c + α00
11α1001|0010〉c
+ α1111α
0101|1101〉c + α11
11α1001|1110〉c. (III.9)
The action of hx on its states is
Hx|0101〉c = Jx|0011〉c, Hx|1001〉c = Jx|1111〉c,Hx|0110〉c = Jx|0000〉c, Hx|1010〉c = Jx|1100〉c,Hx|0100〉c = Jx|0010〉c, Hx|0111〉c = Jx|0001〉c,Hx|1000〉c = Jx|1110〉c, Hx|1011〉c = Jx|1101〉c. (III.10)
Consistency between Eq. (III.8) and (III.10) requires
(α0101)2 = (α10
01)2 = −α0011α
1111, (α00
11)2 = (α1111)2 = −α01
01α1001,
α0101 = −α10
01, α0011 = −α11
11. (III.11)
This yields α0101 = ± i√
2, α10
01 = ∓ i√2, α00
11 = ± i√2, and α11
11 = ∓ i√2. Since α01
01 and α1111 are imaginary, there are two intra-
fermion braidings acting on the same fermion to adjust the global phase by ±i. The inter-fermion braiding introducesa π/2 phase difference. To increase the phase difference to π, one of the intra-fermion braiding is performed after the
10
inter-fermion braiding. WSe still need to determine which fermion the intra-fermion braiding acts on. Eq. (III.11)suggests that: {
|01〉 → α0101|01〉 − α10
01|10〉,|11〉 → α11
11|11〉 − α0011|00〉.
(III.12)
But, the phases introduced by the inter-fermion braiding operations are opposite to each other for |01〉 and |11〉. Tokeep the form of Eq. (III.12) up to a global minus sign, the only choice is that the intra-fermion braiding operationis applied to the first qubit. This is different for the two states.
B +in
Bex
B +in
FIG. S1. The braiding circuit used for the 1D Kitaev spin chain.
From the analysis provided above, the braiding circuit must satisfy the following:1. the circuit is constructed by one inter-fermion braiding and two intra-fermion braiding;2. one of the intra-fermion braidings is performed after the inter-fermion braiding;3. the two intra-fermion braiding operations act on the first qubit.A circuit fulfilling these requirements is the one shown in Fig. S1. One can apply a similar analysis to the case with
ng = 0. One finds the same requirement in that case.
IV. THE CORRECTNESS OF THE GROUND STATE PREPARATION
0.0 0.5 1.0 1.5 2.0Jx
4.5
4.0
3.5
3.0
2.5
2.0
E min
(a)
ExactEDsimulator
0.0 0.2 0.4 0.6 0.8 1.0J
6.5
6.0
5.5
5.0
4.5
4.0
E min
(b)
ExactEDsimulator
FIG. S2. Verification of the ground-state preparation: (a) the 1D Kitaev spin chain and (b) the Kitaev honeycomb model. Forthe 1D Kitaev spin chain, we set Jz = 1 to determine the lowest energy as a function of Jx. For the Kitaev honeycomb model,we set Jz = 1 and Jx = Jy = J to determine the lowest energy as a function of J . Each data point from quantum simulatorsis obtained by performing N = 8196 experiments.
In this section, we prove that the quantum circuit proposed in the main text (Fig. 1) correctly prepares the groundstate of Kitaev-inspired models. The system sizes of the models considered here are the same as those shown in themain text. The lowest energy of the finite-size cluster is determined by exact diagonalization, and then compared to
11
the results obtained by exact solution and quantum circuit simulation. Results for the 1D Kitaev spin chain and theKitaev spin model are summarized in Fig. S2. It turns out that the lowest energies obtained from exact diagonalizationare identical to those obtained from the exact solution. Results from the quantum simulators have small deviationsdue finite-number sampling.
V. MEASURING ENTANGLEMENT ENTROPY BY SYMMETRY-ENFORCED CIRCUITS
For Kitaev-inspired models, ground states have uniform gauge configurations, which contribute trivially to theentanglement entropy. The non-trivial quantum information is stored in the Brillouin zone of the bond-fermions. Wedescribe here how to extract this information with a quantum circuit.
To begin, the original spin model is mapped to a conventional fermionic model by the Jordan-Wigner transformationfollowed by a Majorana braiding. In general, the fermionic Hamiltonian can be written as
Hf =∑i,j
Ai,jf†i fj +
∑i,j
Bi,j
(f†i f
†j + h.c.
), (V.1)
where Ai,j is the hopping matrix and Bi,j is the pairing matrix. The explicit form of the two matrices depends onthe details of models. The entanglement entropy of this Hamiltonian is easily obtained by introducing Majoranafermions fi = γ2i−1 + iγ2i for each physical fermion. In terms of the Majorana fermions, the entanglement entropyof subsystem A is calculated from the eigenvalues of the correlation matrix C [51, 52], whose matrix elements satisfyCm,n = 1
2 [ργmγn] = 12 〈γmγn〉. The entanglement entropy is then
SA = −1
2
∑n
(1− λn) ln(1− λn) + λn lnλn, (V.2)
with λn the eigenvalues of Cm,n.To measure the correlation matrix Cm,n on a quantum computer, we note that:
C2m−1,2n−1 = 〈f†mf†n〉+ 〈fmfn〉+ 〈fmf†n〉+ 〈f†mfn〉,C2m,2n = −〈f†mf†n〉 − 〈fmfn〉+ 〈fmf†n〉+ 〈f†mfn〉,C2m−1,2n = i〈f†mf†n〉 − i〈fmfn〉+ i〈fmf†n〉 − i〈f†mfn〉,C2m,2n−1 = i〈f†mf†n〉 − i〈fmfn〉 − i〈fmf†n〉+ i〈f†mfn〉.
(V.3)
This suggests that we need to measure four expectation values: 〈f†mf†n〉, 〈fmfn〉, 〈fmf†n〉 and 〈f†mfn〉. From theJordan-Wigner transformation, we have: {
fn =∑n−1j=1 σ
zjσ
+n ,
f†n =∑n−1j=1 σ
zjσ−n ,
(V.4)
where σ+n = 1
2 (σxn + iσyn) and σ−n = 12 (σxn − iσyn). Then the four different expectation values become〈f†mf†n〉 = 1
4
(Cxxm,n − Cyym,n − iCxym,n − iCyxn,m
),
〈fmfn〉 = 14
(Cxxm,n − Cyym,n + iCxym,n + iCyxn,m
),
〈f†mfn〉 = 14
(Cxxm,n + Cyym,n + iCxym,n − iCyxn,m
),
〈fmf†n〉 = 14
(Cxxm,n + Cyym,n − iCxym,n + iCyxn,m
),
(V.5)
where:
Cαβm,n = 〈σαmn−1∏j=m
σzjσβn〉, (V.6)
with α, β = x, y.Now we derive how to explicitly calculate the entanglement entropy of bond fermions of the 1D Kitaev spin chain,
using the circuits in the main text. We begin the discussion with Eq. (II.9), which can be rewritten as
H̃1D =∑k
Ψ†kEk
(cos θk i sin θk−i sin θk − cos θk
)Ψk, (V.7)
12
where Ψk is defined as Eq. (II.8), and:Ek =
√(JzD + Jx cos k)
2+ J2
x sin2 k,
cos θk = (JzD + Jx cos k)/Ek,
sin θk = Jx sin k/Ek.
(V.8)
The Bogoliubov transformation is(bkb†−k
)=
(cos θk2 e
−iϕk/2 sin θk2 e
iϕk/2
sin θk2 e−iϕk/2 − cos θk2 e
iϕk/2
)(f̃kf̃†−k
), (V.9)
where eiϕk = i, independent of k. This means that ϕk = π/2, independent of k. The correlations 〈fnf†m〉, 〈f†nfm〉,〈f†nf†m〉, and 〈fnfm〉 are then determined from the ground state expectations of Bogoliubov fermions
〈bkb†k′〉 = δk,k′ , 〈b†kbk′〉 = 〈b†kb†k′〉 = 〈bkbk′〉 = 0. (V.10)
Straightforward calculations show that:
〈f†nf†m〉 =1
Nk
∑k
e−ik(n−m) sin θk2
e−i(ϕk+ϕ−k)/2
= −i 1
Nk
∑k
e−ik(n−m) sin θk2
,
〈fnfm〉 =1
Nk
∑k
e−ik(n−m) sin θk2
ei(ϕk+ϕ−k)/2
= i1
Nk
∑k
e−ik(n−m) sin θk2
,
〈fnf†m〉 =1
Nk
∑k
e−ik(n−m) cos2θk2,
〈f†nfm〉 =1
Nk
∑k
e−ik(n−m) sin2 θk2, (V.11)
where Nk is the number of momentum points in the Brillouin zone. Note that C2m−1,2n−1 and C2m,2n are thecorrelations of the real and imaginary parts of the fermions, respectively. As expected, we find C2m−1,2n−1 = δm,nand C2m,2n = δm,n. We also find that
C2m−1,2n = i∑k
e−ik(m−n) (cos θk + sin θk) = igm−n, (V.12)
and
C2m,2n−1 = −C2n−1,2m. (V.13)
The correlation between the two sites labeled by m and n is expressed by the following block matrix:(C2m−1,2n−1 C2m−1,2nC2m,2n−1 C2m,2n
)= δm,nI2×2 + iΞm−n, (V.14)
where:
Ξm−n =
(0 gm−n
−g−(m−n) 0
). (V.15)
This means that for a subsystem containing N sites, the correlation matrix C is given by
C = I2N×2N + i
Ξ0 Ξ1 · · · ΞN−1−ΞT1 Ξ0 · · · ΞN−2
... · · ·. . .
...−ΞTN−1 −ΞTN−2 · · · Ξ0
. (V.16)
13
0.0 0.5 1.0 1.5 2.0Jx
0.0
0.2
0.4
0.6
0.8
1.0
S A
N=2N=4N=8N=20N=80
FIG. S3. The finite-size effect on the entanglement entropy of the 1D Kitaev spin chain: the signature of the quantum phasetransition appears when the size of the lattice formed by bond fermions N is significantly larger than 10. The position ofquantum phase transition point can be found accurately when N ≥ 80. We define the subsystem A to be the equal partitionof the lattice, and Jz is set to be 1.
We now focus on the two-site case. The correlation matrix becomes
C = I2×2 + iΞ0. (V.17)
Particle-hole symmetry requires sin θk = − sin θ−k. This implies that
g0 =1
Nk
∑k∈K∪−K
cos θk. (V.18)
Using this expression, we obtain the exact curves in black shown in Fig. 3(c) and (d).
0.0 0.2 0.4 0.6 0.8 1.0J
0.0
0.1
0.2
0.3
0.4
0.5
S A
Lx × Ly = 2 × 2Lx × Ly = 4 × 4Lx × Ly = 8 × 8Lx × Ly = 64 × 64
FIG. S4. The finite-size effects on the entanglement entropy of the Kitaev honeycomb model. The position of the quantumphase transition can be determined only when Lx × Ly ≥ 64 × 64. Subsystem A is defined to be the equal partition of thelattice, and Jz = 1 and Jx = Jy = J .
Another advantage of calculating entanglement entropy with correlation functions is that it allows us to studythe finite-size effects on the entanglement entropy. In Fig. S3, we show how the entanglement entropy of the bond-fermions changes with increasing lattice size. We note that when the bond-fermion lattice size N is smaller than 10,the entanglement entropy increases with the coupling strength Jx monotonically, and no obvious signature appearswhen Jx passes through the quantum phase transition point. If we further increase the lattice size (for example the
14
N = 20 case), the non-monotonic behavior of the entanglement entropy with increasing coupling strength starts toappear. However, an accurate determination of the quantum phase transition requires the bond-fermion lattice sizeN to be larger than 80.
Similar procedures can be used to find the entanglement entropy of the bond fermions of the Kitaev honeycombmodel. How the entanglement entropy changes with the increasing system size is shown in Fig. S4. It turns out thatthe accurate determination of the quantum phase transition requires a system size larger than Lx × Ly = 64× 64.
kkkkkkkk
1 2 3 41 + 2 + 3 + 4 +
1(2 )
2(3 )
3(4 )
4(1 )
1 +
42 +
33 +
24 +
1
1 +
3 +
2 +
3424 +
1
1 +
3 +
2 +
3424 +
1
1 +
3 +
2 +
4 +
4213
1 +
3 +
2 +
4 +
1324
1 +
3 +
2 +
4 +
Bog
F 1F 0
F 0F 0
3210
0 1 2 4
(a)
(b)
(c)
FIG. S5. Schematic illustration of the symmetry-enforced circuit using special momentum shifts: (a) momentum pointsbefore (the yellow dots) and after shifting (the orange and purple dots); (b) the simplification steps; (c) the quantum circuittransforming one of the copies from the momentum space to position space.
VI. CIRCUIT SIMPLIFICATION WITH SPECIAL MOMENTUM SHIFTS
Though in general, the symmetry-enforced circuit cannot be simplified, we show here that thecircuit can be simplifiedfor special momentum shifts. These special momentum shifts include the high symmetry points.
For a N -unit cell system, the discrete Fourier transformation and the particle-hole symmetry constraint requiresthe momentum points to be in the following set:
K =
{−N
2+
1
2,−N
2+
3
2, · · · , N
2− 1
2
}2π
N, (VI.1)
where N = 2n, with n a positive integer. As demonstrated in the main text, if we shift the momentum points byδk, their particle-hole partners must shift by −δk to maintain particle-hole symmetry. Based on this, we note thatδkN/2π = 1/2 is a special point, which makes the set of the momentum points by shifting δk to be identical to thosethat shift by −δk. Therefore, if the gap is vanishing at the high-symmetry points, the symmetry-enforced circuit canbe decomposed into two identical copies.
The case with N = 4 case is shown in Fig.S5. The simplification requires three steps, which are shown in Fig. S5(b)from left to right. In the first step, we identify the momentum points labeled by 3+ and 4−, which is obviousfrom Fig. S5(a); in the second step, due to the gap vanishing at the high-symmetry points, the two-qubit Bogoliubovtransformation becomes two separate single-qubit identity operations, so we can change the position of the momentumpoints 4+ and 3− at the cost of a trivial global phase; in the last step, we just renumber the −δk-shifted momentumpoints by using the numbers in red shownbelow the original labels (in purple) in Fig. S5(a). Then one can easilyfind that the result yields two identical copies. We emphasize that the vanishing gap at the high-symmetry points isessential in making this simplification possible.
It turns out that the Kitaev spin model fulfills this requirement, so the simplified circuit can be used to determinethe different phases of the model. The details of the simplified circuits are shown in Fig.S6.
15
(0, 0)
(1, 0)
(0, 1)
(1, 1)
F 0F 0
Proj
ectl
y m
easu
re
FIG. S6. Quantum circuit for measuring entanglement entropy of the Kitaev honeycomb model with the special momentumshift. The labels (kx, ky) in the unit of π denote the momentum points in the Brillouin zone.
