L. Hormozi, G. Zikos and N. E. Bonesteel- Topological Quantum Compiling

8/3/2019 L. Hormozi, G. Zikos and N. E. Bonesteel- Topological Quantum Compiling

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arXiv:quant-ph/0610111v2

23Apr2007

Topological Quantum Compiling

L. Hormozi, G. Zikos, N. E. BonesteelDepartment of Physics and National High Magnetic Field Laboratory,

Florida State University, Tallahassee, Florida 32310

S. H. SimonBell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974

A method for compiling quantum algorithms into specific braiding patterns for nonabelian quasi-particles described by the so-called Fibonacci anyon model is developed. The method is based onthe observation that a universal set of quantum gates acting on qubits encoded using triplets of thesequasiparticles can be built entirely out of three-stranded braids (three-braids). These three-braidscan then be efficiently compiled and improved to any required accuracy using the Solovay-Kitaevalgorithm.

I. INTRODUCTION

The requirements for realizing a fully functioningquantum computer are daunting. There must be a scal-able system of qubits which can be initialized and indi-

vidually measured. It must be possible to enact a uni-versal set of quantum gates on these qubits. And all thismust be done with sufficient accuracy so that quantumerror correction can be used to prevent decoherence fromspoiling any computation.

The problems of error and decoherence are particularlydifficult ones for any proposed quantum computer. Whilethe states of classical computers are typically stored inmacroscopic degrees of freedom which have a built-inredundancy and thus are resistant to errors, buildingsimilar redundancy into quantum states is less natural.To protect quantum information it is necessary to en-code it using quantum error correcting code states.1,2

These states are highly entangled, and have the prop-erty that code states corresponding to different logicalqubit states can be distinguished from one another onlyby global (topological) measurements. Unlike stateswhose macroscopic degrees of freedom are effectively clas-sical (think of the magnetic moment of a small part of ahard drive), such highly entangled topologically degen-erate states do not typically emerge as the ground statesof physical Hamiltonians. One route to fault-tolerantquantum computation is therefore to build the encod-ing and fault-tolerant gate protocols into the softwareof the quantum computer.3

A remarkable recent development in the theory ofquantum computation which directly addresses these is-sues has been the realization that certain exotic statesof matter in two space dimensions, so-called nonabelianstates, may provide a natural medium for storing and ma-nipulating quantum information.4,5,6,7 In these states, lo-calized quasiparticle excitations have quantum numberswhich are in some ways similar to ordinary spin quan-tum numbers. However, unlike ordinary spins, the quan-tum information associated with these quantum numbersis stored globally, throughout the entire system, and sois intrinsically protected against decoherence. Further-

more, these quasiparticles satisfy so-called nonabelianstatistics. This means that when two quasiparticles areadiabatically moved around one another, while beingkept sufficiently far apart, the action on the Hilbert spaceis represented by a unitary matrix which depends onlyon the topology of the path used to carry out the ex-change. Topological quantum computation can then becarried out by moving quasiparticles around one anotherin two space dimensions.4,5 The quasiparticle world-linesform topologically nontrivial braids in three (= 2 + 1) di-mensional space-time, and because these braids are topo-logically robust (i.e., they cannot be unbraided withoutcutting one of the strands) the resulting computation isprotected against error.

Nonabelian states are expected to arise in a va-riety of quantum many-body systems, including spinsystems,8,9,10 rotating Bose gases,11 and Josephson junc-tion arrays.12 Of those states which have actually beenexperimentally observed, the most likely to possess non-abelian quasiparticle excitations are certain fractionalquantum Hall states. Moore and Read13 were the first topropose that quasiparticle excitations which obey non-abelian statistics might exist in the fractional quantumHall effect. Their proposal was based on the observa-tion that the conformal blocks associated with correla-tion functions in the conformal field theory describingthe two-dimensional Ising model could be interpretedas quantum Hall wave functions. These wave functionsdescribe both the ground state of a half-filled Landaulevel of spin-polarized electrons, as well as states withsome number of fractionally charged quasihole excita-tions (charge = e/4). The particular ground state this

construction produces, the so-called Pfaffian, or Moore-Read state, is considered the most likely candidate for theobserved fractional quantum Hall state at Landau levelfilling fraction = 5/2 ( = 1/2 in the second Landaulevel).14,15

In this conformal field theory construction, states withfour or more quasiholes present correspond to finite-dimensional conformal blocks, and so the correspondingwave functions form a finite-dimensional Hilbert space.The monodromy or braiding properties of these
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conformal blocks are then assumed to describe the uni-tary transformations acting on the Hilbert space pro-duced by adiabatically braiding quasiholes around oneanother.13 Explicit wave functions for these states wereworked out in Ref. 16, and the nonabelian braiding prop-erties have been verified numerically in Ref. 17. In analternate approach, the Moore-Read state can be viewedas a composite fermion superconductor in a so-called

weak pairing px + ipy phase.18

In this description, thefinite-dimensional Hilbert space arises from zero energysolutions of the Bogoliubov-DeGennes equations in thepresence of vortices,18 and the vortices themselves arenonabelian quasiholes whose braiding properties havebeen shown to agree with the conformal field theoryresult.19,20 Recently, a number of experiments have beenproposed to directly probe the nonabelian nature of theseexcitations.21,22,23,24

Unfortunately, the braiding properties of quasihole ex-citations in the Moore-Read state are not sufficiently richto carry out purely topological quantum computation,although partially topological quantum computation

using a mixture of topological and non-topological gateshas been shown to be possible.25,26 However, Read andRezayi27 have shown that the Moore-Read state is justone of a sequence of states labeled by an index k corre-sponding to electrons at filling fractions = k/(2 + k),with k = 1 corresponding to the = 1/3 Laughlinstate and k = 2 to the Moore-Read state. The wave-functions for these states can be written as correlationfunctions in the Zk parafermion conformal field theory,27

and the braiding properties of the quasihole excitationswere worked out in detail in Ref. 28. There it was shownthat the quasiholes are described by the SU(2)k Chern-Simons-Witten (CSW) theories, up to overall abelianphase factors which are irrelevant for quantum compu-tation. More recently, explicit quasihole wave functionshave been worked out for the k = 3 Read-Reazyi state,29

with results consistent with the predicted SU(2)3 braid-ing properties. The elementary braiding matrices for theSU(2)k CSW theory for k = 3 and k 5 have beenshown to be sufficiently rich to carry out universal quan-tum computation, in the sense that any desired unitaryoperation on the Hilbert space of N quasiparticles, withN 3 for k 3, k = 4, 8, and N 4 for k = 8, can beapproximated to any desired accuracy by a braid.5,6

The main purpose of this paper is to give an efficientmethod for determining braids which can be used to carryout a universal set of a quantum gates (i.e. single-qubitrotations and controlled-NOT gates) on encoded qubitsfor the case k = 3, thought to be physically relevant forthe experimentally observed30 = 12/5 fractional quan-tum Hall effect27,31 ( = 12/5 corresponds to = 2/5 inthe second Landau level, and this is the particle-hole con-

jugate of = 3/5 corresponding to k = 3). We refer tothe process of finding such braids as topological quan-tum compiling since these braids can then be used totranslate a given quantum algorithm into the machinecode of a topological quantum computer. This is anal-

ogous to the action of an ordinary compiler which trans-lates instructions written in a high level programminglanguage into the machine code of a classical computer.

It should be noted that the proof of universality forSU(2)3 quasiparticles is a constructive one,

5,6 and there-fore, as a matter of principle, it provides a prescriptionfor compiling quantum gates into braids. However, inpractice, for two-qubit gates (such as controlled-NOT)

this prescription, if followed straightforwardly, is pro-hibitively difficult to carry out, primarily because it in-volves searching the space of braids with six or morestrands. We address this difficulty by dividing our two-qubit gate constructions into a series of smaller construc-tions, each of which only involves searching the space ofthree-stranded braids (three-braids). The required three-braids then canbe found efficiently and used to constructthe desired two-qubit gates. This divide and conquerapproach does not, in general, yield the most accuratebraid of a given length which approximates a desiredquantum gate. However, we believe that it does yieldthe most accurate (or at least among the most accurate)

braids which can be obtained for a given fixed amount ofclassical computing power.This paper is organized as follows. In Sec. II we re-

view the basic properties of the SU(2)k Hilbert space,and show that the case SU(2)3 is, for our purposes,equivalent to the case SO(3)3 the so-called Fibonaccianyon model. Section III then presents a quick reviewof the mathematical machinery needed to compute withFibonacci anyons. In Sec. IV we outline how, in prin-ciple, these particles can be used to encode qubits suit-able for quantum computation. Section V then describeshow to find braiding patterns for three Fibonacci anyonswhich can be used to carry out any allowed operationon the Hilbert space of these quasiparticles to any de-

sired accuracy, thus effectively implementing the proce-dure given in Ref. 5 for carrying out single-qubit rota-tions. In Sec. VI we discuss the more difficult case of two-qubit gates, and give two classes of explicit gate construc-tions one, first discussed by the authors in Ref. 32, inwhich a pair of quasiparticles from one qubit is woventhrough the quasiparticles in the second qubit, and an-other, presented here for the first time, in which only asingle quasiparticle is woven. Finally, in Sec. VII we ad-dress the question of to what extent the constructions wefind are special to the k = 3 case, and in Sec. VIII wesummarize our results.

II. FUSION RULES AND HILBERT SPACE

Consider a system with quasiparticle excitations de-scribed by the SU(2)k CSW theory. It is convenientto describe the properties of this system using the so-called quantum group language.28 The relevant quantumgroups are deformed versions of the representation the-ory ofSU(2), i.e. the theory of ordinary spin, and muchof the intuition for thinking about ordinary spin can be


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FIG. 1: Bratteli diagrams for SU(2)k for (a) k = 2 and (b)k = 3. Here N is the number of q-spin 1/2 quasiparticles andS is the total q-spin of those quasiparticles. The number ata given (N, S) vertex of each diagram indicates the numberof paths to that vertex starting from the (0, 0) point. Thisnumber gives the dimensionality of the Hilbert space of Nq-spin 1/2 quasiparticles with total q-spin S.

carried over to the quantum group case.In the quantum group description of an SU(2)k CSW

theory, each quasiparticle has a half-integer q-deformedspin (q-spin) quantum number. Just as for ordinary spin,there are rules for combining q-spin known as fusion rules.The fusion rules for the SU(2)k theory are similar to theusual triangle rule for adding ordinary spin, except thatthey are truncated so that there are no states with totalq-spin > k/2. Specifically, the fusion rules for the level ktheory are,33

s1 s2 = |s1 s2| |s1 s2| + 1 . . .. . . min(s1 + s2, k s1 s2). (1)

Note that in the quantum group description of non-abelian anyons, states are distinguished only by their to-tal q-spin quantum numbers. The q-deformed analogsof the Sz quantum numbers are physically irrelevant there is no degeneracy associated with them, and theyplay no role in any computation involving braiding.28

The situation is somewhat analogous to that of a col-lection of ordinary spin-1/2 particles in which the onlyallowed operations, including measurement, are rotation-ally invariant and hence independent of Sz, as is the casein exchange-based quantum computation.34

The fusion rules of the SU(2)k theory fix the structureof the Hilbert space of the system. For a collection ofquasiparticles with q-spin 1/2, a useful way to visualizethis Hilbert space is in terms of its so-called Bratteli dia-gram. This diagram shows the different fusion paths forN q-spin 1/2 quasiparticles in which these quasiparticlesare fused, one at a time, going from left to right in thediagram. Bratteli diagrams for the cases k = 2 and k = 3are shown in Fig. 1.

The dimensionality of the Hilbert space for N q-spin

FIG. 2: (color online). Graphical proof of the equivalenceof braiding q-spin-1/2 and q-spin-1 objects for SU(2)3. Part(a) shows a braiding pattern for a collection of objects, somehaving q-spin 1/2 and some having q-spin 1. Part (b) showsthe same braiding pattern but with the q-spin-1/2 objectsrepresented by q-spin 1 objects fused with q-spin-3/2 objects,which, for SU(2)3, has a unique fusion channel. Finally, part(c) shows the same braid with the q-spin-3/2 objects removed.Because these q-spin-3/2 ob jects are effectively abelian forSU(2)3, removing them from the braid will only result inan overall phase factor which will be irrelevant for quantumcomputing.

1/2 quasiparticles with total q-spin S can be determinedby counting the number of paths in the Bratteli diagramfrom the origin to the point (N, S). The results of thispath counting are also shown in Fig. 1, where one cansee the well-known 2N/21 Hilbert space degeneracy forthe k = 2 (Moore-Read) case,13,16 and the Fibonnacidegeneracy for the k = 3 case.27

In this paper we will focus on the k = 3 case, which isthe lowest k value for which SU(2)k nonabelian anyonsare universal for quantum computation.5,6 In fact, we willshow that two-qubit gates are particularly simple for thiscase. Before proceeding, it is convenient to introduce an

important property of the SU(2)3 theory, namely thatthe braiding properties of q-spin 1/2 quasiparticles arethe same as those with q-spin 1 (up to an overall abelianphase which is irrelevant for topological quantum compu-tation). This is a useful observation because the theory ofq-spin 1 quasiparticles in SU(2)3 is equivalent to SO(3)3,a theory also known as the Fibonacci anyon theory35,36

a particularly simple theory with only two possiblevalues of q-spin, 0 and 1, for which the fusion rules are

0 0 = 0, 0 1 = 1 0 = 1, 1 1 = 0 1. (2)Here we give a rough proof of this equivalence. This

proof is based on the fact that for k = 3 the fusion rules

involving q-spin 3/2 quasiparticles take the following sim-ple form

3

2 s = 3

2 s. (3)

The key observation is that since for k = 3 the highestpossible q-spin is 3/2, when fusing a q-spin-3/2 objectwith any other object (here we use the term object todescribe either a single quasiparticle or a group of quasi-particles viewed as a single composite entity), the Hilbert


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space dimensionality does not grow. This implies thatmoving a q-spin-3/2 object around other objects can, atmost, produce an overall abelian phase factor. While thisphase factor may be important physically, particularly indetermining the outcome of interference experiments in-volving nonabelian quasiparticles,21,22,23,24 it is irrelevantfor quantum computing, and thus does not matter whendetermining braids which correspond to a given compu-

tation. Because (3) implies that a q-spin-1/2 object canbe viewed as the result of fusing a q-spin-1 object with aq-spin-3/2 object, it follows that the braid matrices for q-spin-1/2 objects are the same as that for q-spin-1 objectsup to an overall phase (as can be explicitly checked).

In fact, based on this argument we can make a strongerstatement. Imagine a collection of SU(2)3 objects whicheach have either q-spin 1 or q-spin 1/2. It is then possi-ble to carry out topological quantum computation, evenif we do not know which objects have q-spin 1 and whichhave q-spin 1/2. The proof is illustrated in Fig. 2. Figure2(a) shows a braiding pattern for a collection of objects,some of which have q-spin 1/2 and some of which have q-

spin 1. Fig. 2(b) then shows the same braiding pattern,but now all objects with q-spin 1/2 are represented byobjects with q-spin 1 fused to objects with q-spin 3/2.Because, as noted above, the q-spin 3/2 objects havetrivial (abelian) braiding properties, the unitary trans-formation produced by this braid is the same, up to anoverall abelian phase, as that produced by braiding noth-ing but q-spin 1 objects, as shown in Fig. 2(c). It followsthat provided one can measure whether the total q-spinof some object belongs to the class 1 {1, 1/2} or theclass 0 {0, 3/2} something which should, in princi-ple, be possible by performing interference experimentsas described in Refs. 37 and 38 then quantum compu-tation is possible, even if we do not know which objects

have q-spin 1/2 and which have q-spin 1.

III. FIBONACCI ANYON BASICS

Having reduced the problem of compiling braids forSU(2)3 to compiling braids for SO(3)3, i.e. Fibonaccianyons, it is useful for what follows to give more detailsabout the mathematical structure associated with thesequasiparticles. For an excellent review of this topic seeRef. 35, and for the mathematics of nonabelian particlesin general see Ref. 39.

Note that for the rest of this paper, except for Sec.VII, it should be understood that each quasiparticle is aq-spin 1 Fibonacci anyon. It should also be understoodthat from the point of view of their nonabelian prop-erties quasihole excitations are also q-spin 1 Fibonaccianyons, even though they have opposite electric chargeand give opposite abelian phase factors when braided.Because it is the nonabelian properties which are relevantfor topological quantum computation, for our purposesquasiparticles and quasiholes can be viewed as identicalnonabelian particles. Unless it is important to distin-

FIG. 3: (color online). Basis states for the Hilbert space of(a) two and (b) three Fibonacci anyons. SU(2)3 Bratteli dia-grams showing fusion paths corresponding to the basis statesfor the Hilbert space of two and three q-spin 1/2 quasiparticlesare shown. The q-spin axes on these diagrams are labeled both

by the SU(2)3 q-spin quantum numbers 0, 1/2, 1 and 3/2 and,to the left of these in bold, the corresponding Fibonacci q-spinquantum numbers 0 {0, 3/2} and 1 {1/2, 1}. Beneatheach Bratteli diagram the same state is represented using anotation in which dots correspond to Fibonacci anyons, andgroups of Fibonacci anyons enclosed in ovals labeled by q-spinquantum numbers are in the corresponding q-spin eigenstates.

guish between the two (as when we discuss creating andfusing quasiparticles and quasiholes in Sec. IV) we willsimply use the terms quasiparticle or Fibonacci anyon torefer to either excitation.

Figure 3 establishes some of the notation for repre-senting Fibonacci anyons which will be used in the restof the paper. This figure shows SU(2)3 Bratteli dia-grams in which the q-spin axis is labeled both by theSU(2)3 q-spin quantum numbers and, in boldface, thecorresponding Fibonacci q-spin quantum numbers, i.e. 0for {0, 3/2} and 1 for {1/2, 1}. In Fig. 3(a) Bratteli di-agrams showing fusion paths corresponding to two basisstates spanning the two-dimensional Hilbert space of twoFibonacci anyons are shown. Beneath each Bratteli di-agram an alternate representation of the correspondingstate is also shown. In this representation dots corre-spond to Fibonacci anyons and ovals enclose collectionsof Fibonacci anyons which are in q-spin eigenstates when-

ever the oval is labeled by a total q-spin quantum number.(Note: If the oval is not labeled, it should be understoodthat the enclosed quasiparticles may not be in a q-spineigenstate).

In the text we will use the notation to represent aFibonacci anyon, and the ovals will be represented byparentheses. In this notation, the two states shown inFig. 3(a) are denoted (, )0 and (, )1.

Fig. 3(b) shows Bratteli diagram, again with bothSU(2)3 and Fibonacci quantum numbers, with fusion


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paths which this time correspond to three basis statesof the three-dimensional Hilbert space of three Fibonaccianyons. Beneath these diagrams the oval represen-tations of these three states are also shown, which inthe text will be represented ((, )0, )1, ((, )1, )1 and((, )1, )0.

In addition to fusion rules, all theories of nonabeliananyons possess additional mathematical structure which

allows one to calculate the result of any braiding oper-ation. This structure is characterized by the F (fusion)and R (rotation) matrices.35,39,40

To define the F matrix, note that the Hilbert spaceof three Fibonacci anyons is spanned both by the threestates labeled ((, )a, )c, and the three states labeled(, (, )b)c. The F matrix is the unitary transformationwhich maps one of these bases to the other,

(, (, )a)c =

b

Fcab ((, )b , )c , (4)

and has the form

F =

1

, (5)where = (

5 1)/2 is the inverse of the golden mean.

In this matrix the upper left 22 block, F1ab, acts onthe two-dimensional total q-spin 1 sector of the three-quasiparticle Hilbert space and the lower right matrixelement, F011 = 1, acts on the unique total q-spin 0 state.Note that this F matrix can be applied to any threeobjects which each have q-spin 1, where each object canconsist of more than one Fibonacci anyon. Furthermore,if one considers three objects for which one or more of

the objects has q-spin 0, then the state of these objects isuniquely determined by the total q-spin of all three, andin this case the F matrix is trivially the identity. Thus,for the case of Fibonacci anyons, the matrix (5) is allthat is needed to make arbitrary basis changes for anynumber of Fibonacci anyons.

The R matrix gives the phase factor produced whentwo Fibonacci anyons are moved around one another witha certain sense. One can think of these phase factors asthe q-deformed versions of the 1 or +1 phase factorsone obtains when interchanging two ordinary spin-1/2quasiparticles when they are in a singlet or triplet state,respectively. This phase factor depends on the overallq-spin of the two quasiparticles involved in the exchange,

so for Fibonacci anyons there are two such phase factorswhich are summarized in the R matrix,

R =

ei4/5 0

0 ei3/5

. (6)

Here the upper left and lower right matrix elements are,respectively, the phase factor that two Fibonacci anyonsacquire if they are interchanged in a clockwise sense whenthey have total q-spin 0 or q-spin 1. Again, this matrix

FIG. 4: (color online). (a) Four-quasparticle and (b) three-quasiparticle qubit encodings for Fibonacci anyons. Part (a)shows two states which span the Hilbert space of four quasi-particles with total q-spin 0 which can be used as the logical|0L and |1L states of a qubit. Part (b) shows two statesspanning the Hilbert space of three quasiparticles with to-tal q-spin 1 which can also be used as logical qubit states|0L and |1L. This three-quasiparticle qubit can be obtainedby removing the rightmost quasiparticle from the two statesshown in (a). The third state shown in Part (b), labeled |NCfor noncomputational, is the unique state of three quasipar-ticles which has total q-spin 0.

also applies if we exchange two objects that both havetotal q-spin 1, even if these objects consist of more thanone Fibonacci anyon. And if one or both ob jects has q-spin 0, the result of this interchange is the identity. Againwe emphasize that in the k = 3 Read-Rezayi state, therewill be additional abelian phases present, which may havephysical consequences for some experiments, but whichwill be irrelevant for topological quantum computation.

Typically the sequence of F and R matrices used tocompute the unitary operation produced by a given braid

is not unique. To guarantee that the result of any suchcomputation is independent of this sequence, the F andR matrices must satisfy certain consistency conditions.These consistency conditions, the so-called pentagon andhexagon equations,35,39,40 are highly restrictive, and, infact, for the case of Fibonacci anyons essentially fix theF and R matrices to have the forms given above (up toa choice of chirality, and Abelian phase factors which areagain irrelevant to our purposes here).35

Finally, we point out an obvious, but important, con-sequence of the structure of the F and R matrices. Wheninterchanging any two quasiparticles which are part of alarger set of quasiparticles with a well-defined total q-spin

quantum number, this total q-spin quantum number willnot change.

IV. QUBIT ENCODING AND GENERAL

COMPUTATION SCHEME

Before proceeding, it will be useful to have a spe-cific scheme in mind for how one might actually carryout topological quantum computation with Fibonacci


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anyons. Here we follow the scheme outlined in Ref. 7,which, for completeness, we briefly review below.

The computer can be initialized by pullingquasiparticle-quasihole pairs out of the vacuum,(by vacuum we mean the ground state of the k = 3Read-Rezayi state or any other state which supportsFibonacci anyon excitations). Each such pair will consistof two q-spin 1 excitations in a state with total q-spin

0, i.e. the state (, )0. In principle, this pair can alsoexist in a state with total q-spin 1, provided there areother quasiparticles present to ensure the total q-spinof the system is 0, so one can imagine using this pairas a qubit. However, it is impossible to carry outarbitrary single-qubit operations by braiding only thetwo quasiparticles forming such a qubit this braidingnever changes the total q-spin of the pair, and so onlygenerates rotations about the z-axis in the qubit space.

For this reason it is convenient to encode qubits us-ing more than two Fibonacci anyons. Thus, to cre-ate a qubit, two quasiparticle-quasihole pairs can bepulled out of the vacuum. The resulting state is then

((, )0, (, )0)0 which again has total q-spin 0. TheHilbert space of four Fibonacci anyons with total q-spin0 is two dimensional, with basis states, which we cantake as logical qubit states, |0L = ((, )0, (, )0)0 and|1L = ((, )1, (, )1)0, (see Fig 4(a)). The state of sucha four-quasiparticle qubit is determined by the total q-spin of either the rightmost or leftmost pair of quasipar-ticles. Note that the fusion rules (2) imply that the totalq-spin of these two pairs must be the same because thetotal q-spin of all four quasiparticles is 0.

For this encoding, in addition to the two-dimensionalcomputational qubit space of four quasiparticles withtotal q-spin 0, there is a three-dimensional noncom-putational Hilbert space of states with total q-spin 1spanned by the states ((, )0, (, )1)1, ((, )1, (, )0)1and ((, )1, (, )1)1. When carrying out topologicalquantum computation it is crucial to avoid transitionsinto this noncomputational space.

Fortunately, single-qubit rotations can be carried outby braiding quasiparticles within a given qubit and, asdiscussed in Sec. III, such operations will not change thetotal q-spin of the four quasiparticles involved. Single-qubit operations can therefore be carried out without anyundesirable transitions out of the encoded computationalqubit space.

Two-qubit gates, however, will require braiding quasi-particles from different qubits around one another. This

will in general lead to transitions out of the encoded qubitspace. Nevertheless, given the so-called density resultof Ref. 6 it is known that, as a matter of principle, onecan always find two-qubit braiding patterns which willentangle the two qubits, and also stay within the com-putational space to whatever accuracy is required for agiven computation. The main purpose of this paper is toshow how such braiding patterns can be efficiently found.

Note that the action of braiding the two leftmostquasiparticles in a four-quasiparticle qubit (referring to

FIG. 5: (color online). Space-time paths corresponding to the

initialization, manipulation through braiding, and measure-ment of an encoded qubit. Two quasiparticle-quasihole pairsare pulled out of the vacuum, with each pair having total q-spin 0. The resulting state corresponds to a four-quasiparticlequbit in the state |0L (see Fig. 4(a)). After some braiding,the qubit is measured by trying to fuse the bottommost pair(in this case a quasiparticle-quasihole pair). If they fuse backinto the vacuum the result of the measurement is |0L, oth-erwise it is |1L. Because only the three lower quasiparticlesare braided, the encoded qubit can also be viewed as a three-quasiparticle qubit (see Fig. 4(b)) which is initialized in thestate |0L.

Fig. 4(a)) is equivalent to that of braiding the two right-most quasiparticles with the same sense. This is becauseas long as we are in the computational qubit space boththe leftmost and rightmost quasiparticle pairs must havethe same total q-spin, and so interchanging either pairwill result in the same phase factor from the R matrix.It is therefore not necessary to braid all four quasiparti-cles to carry out single-qubit rotations one need onlybraid three.

In fact, one may consider qubits encoded using onlythree quasiparticles with total q-spin 1, as originally pro-posed in Ref. 5. Such qubits can be initialized by first cre-ating a four-quasiparticle qubit in the state |0L, as out-lined above, and then simply removing one of the quasi-particles. In this three-quasiparticle encoding, shown inFig. 4(b), the logical qubit states can be taken to be|0L = ((, )0, )1 and |1L = ((, )1, )1. For thisencoding there is just a single noncomputational state|N C = ((, )1, )0, also shown in Fig. 4(b). As for thefour-quasiparticle qubit, when carrying out single-qubitrotations by braiding within a three-quasiparticle qubitthe total q-spin of the qubit, in this case 1, remains un-changed and there are no transitions from the compu-tational qubit space into the state |N C. However, justas for four-quasiparticle qubits, when carrying out two-qubit gates these transitions will in general occur and wemust work hard to avoid them. Henceforth we will refer

to these unwanted transitions as leakage errors.Note that, because each three-quasiparticle qubit has

total q-spin 1, when more than one of these qubits ispresent the state of the system is not entirely charac-terized by the internal q-spin quantum numbers whichdetermine the computational qubit states. It is also nec-essary to specify the state of what we will refer to as theexternal fusion space the Hilbert space associatedwith fusing the total q-spin 1 quantum numbers of eachqubit. When compiling braids for three-quasiparticle


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qubits it is crucial that the operations on the computa-tional qubit space not depend on the state of this externalfusion space if they did, these two spaces would be-come entangled with one another leading to errors. For-tunately, we will see that it is indeed possible to findbraids which do not lead to such errors.

For the rest of this paper (except Sec. VII) we willuse this three-quasiparticle qubit encoding. It should be

noted that any braid which carries out a desired oper-ation on the computational space for three-quasiparticlequbits will carry out the same operation on the computa-tional space of four-quasiparticle qubits, with one quasi-particle in each qubit acting as a spectator. The braidswe find here can therefore be used for either encoding.

We can now describe how topological quantum com-putation might actually proceed, again following Ref. 7.A quantum circuit consisting of a sequence of one- andtwo-qubit gates which carries out a particular quantumalgorithm would first be translated (or compiled) intoa braid by compiling each individual gate to whateveraccuracy is required. Qubits would then be initialized

by pulling quasiparticle-quasihole pairs out of the vac-uum. These localized excitations would then be adia-batically dragged around one another so that their world-lines trace out a braid in three-dimensional space-timewhich is topologically equivalent to the braid compiledfrom the quantum algorithm. Finally, individual qubitswould be measured by trying to fuse either the two right-most or two leftmost excitations within them (referringto Fig. 4(a)) for four-quasiparticle qubits, or just the twoleftmost excitations (referring to Fig. 4(b)) for three-quasiparticle qubits. If this pair of excitations consistsof a quasiparticle and a quasihole (and it will always bepossible to arrange this), then, if the total q-spin of thepair is 0, it will be possible for them to fuse back into

the vacuum. However, if the total q-spin is 1 this willnot be possible. The resulting difference in the chargedistribution of the final state would then be measured todetermine if the qubit was in the state |0L or |1L. Al-ternatively, as already mentioned in Sec. II, interferenceexperiments37,38 could be used to initialize and read outencoded qubits.

As a simple illustration, Fig. 5 shows a computationin which a four-quasiparticle qubit (which can also beviewed as a three-quasiparticle qubit if the top quasi-particle is ignored) is initialized by pulling quasiparticle-quasihole pairs out of the vacuum, a single-qubit opera-tion is carried out by braiding within the qubit, and the

final state of the qubit is measured by fusing a quasipar-ticle and quasihole together and observing the outcome.

V. COMPILING THREE-BRAIDS AND

SINGLE-QUBIT GATES

We now focus on the problem of finding braids for threeFibonacci anyons (three-braids) which approximate anyallowed unitary transformation on the Hilbert space of

FIG. 6: (color online). Elementary three-braids and the de-composition of a general three-braid into a series of elemen-tary braids. The unitary operation produced by this braidis computed by multiplying the corresponding sequence ofelementary braid matrices, 1 and 2 (see text) and their in-verses, as shown. Here the (unlabeled) ovals represent a par-ticular basis choice for the three-quasiparticle Hilbert space,consistent with that used in the text. In this and all sub-sequent figures which show braids, quasiparticles are alignedvertically, and we adopt the convention that reading frombottom to top in the figures corresponds to reading from leftto right in expressions such as ((, )a, )c in the text. Itshould be noted that these figures are only meant to representthe topology of a given braid. In any actual implementationof topological quantum computation, quasiparticles will cer-tainly not be arranged in a straight line, and they will haveto be kept sufficiently far apart while being braided to avoidlifting the topological degeneracy.

these quasiparticles. This is important not only becauseit allows one to find braids which carry out arbitrarysingle-qubit rotations,5 but also because, as will be shownin Sec. VI, it is possible to reduce the problem of con-structing braids which carry out two-qubit gates to thatof finding a series of three-braids approximating specificoperations.

A. Elementary Braid Matrices

Using the F and R matrices, it is straightforward todetermine the elementary braiding matrices that act onthe three-dimensional Hilbert space of three Fibonaccianyons. If, as in Fig. 6, we take the basis states for thethree-quasiparticle Hilbert space to be the states labeled((, )a, )c then, in the ac = {01, 11, 10} basis, the ma-trix 1 corresponding to a clockwise interchange of thetwo bottommost quasiparticles in the figure (or leftmost

in the ((, )a, )c representation) is

1 =

e

i4/5 0

0 ei3/5

ei3/5

, (7)

where the upper left 22 block acts on the total q-spin1 sector (|0L and |1L) of the three quasiparticles, andthe lower right matrix element is a phase factor acquiredby the q-spin 0 state (|N C). This matrix is easily read


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off from the R matrix, since the total q-spin of the twoquasiparticles being exchanged is well defined in this ba-sis.

To find the matrix 2 corresponding to a clockwiseinterchange of the two topmost (or rightmost in the((, )a, )c representation) quasiparticles, we must firstuse the F matrix to change bases to one in which thetotal q-spin of these quasiparticles is well defined. In

this basis, the braiding matrix is simply 1, and so, afterchanging back to the original basis, we find

2 = F11F =

e

i/5

ei3/5 ei3/5

ei3/5

. (8)

The unitary transformation corresponding to a giventhree-braid can now be computed by representing it as asequence of elementary braid operations and multiplyingthe corresponding sequence of 1 and 2 matrices andtheir inverses, as shown in Fig. 6.

If we are only concerned with single-qubit rotations,

then we only care about the action of these matrices onthe encoded qubit space with total q-spin 1, and notthe total q-spin 0 sector corresponding to the noncom-putational state. However, in our two-qubit gate con-structions, various three-braids will be embedded into thebraiding patterns of six quasiparticles, and in this casethe action on the full three-dimensional Hilbert spacedoes matter.

To understand this action note that 1 can be written

1 =

ei/10

ei7/10 0

0 ei7/10

ei3/5

, (9)

where the upper 2 2 block acting on the total q-spin1 sector is an SU(2) matrix, (i.e., a 2 2 unitary ma-trix with determinant 1), multiplied by a phase factor ofeither +ei/10 or ei/10, and the lower right matrixelement, ei3/5, is the phase acquired by the total q-spin0 state. The phase factor pulled out of the upper 2 2block is only defined up to 1 because any SU(2) matrixmultiplied by 1 is also an SU(2) matrix.

From (8) it follows that 2 can be written in a sim-ilar fashion, with the same phase factors. Each clock-wise braiding operation then corresponds to applying anSU(2) operation multiplied by a phase factor of

ei/10

to the q-spin 1 sector, while at the same time multiplyingthe q-spin 0 sector by a phase factor of ei3/5. Likewise,each counterclockwise braiding operation corresponds toapplying an SU(2) operation multiplied by a phase fac-tor ofe+i/10 to the q-spin 1 sector and a phase factorof ei3/5 to the q-spin 0 sector.

We define the winding, W(B), of a given three-braid B,to be the total number of clockwise interchanges minusthe total number of counterclockwise interchanges. Itthen follows that the unitary operation corresponding to

an arbitrary braid B can always be expressed

U(B) =

eiW(B)/10 [SU(2)]

ei3W(B)/5

, (10)

where [SU(2)] indicates an SU(2) matrix. Thus, for agiven three-braid, the phase relation between the totalq-spin 1 and total q-spin 0 sectors of the corresponding

unitary operation is determined by the winding of thebraid. We will refer to (10) often in what follows. Ittells us precisely what unitary operations can be approx-imated by three-braids, and places useful restrictions ontheir winding.

B. Weaving and Brute Force Search

At this point it is convenient to restrict ourselves toa subclass of braids which we will refer to as weaves. Aweave is any braid which is topologically equivalent tothe space-time paths of some number of quasiparticles in

which only a single quasiparticle moves. It was shown inRef. 41 that this restricted class of braids is universal forquantum computation, provided the unitary representa-tion of the braid group is dense in the space of all unitarytransformations on the relevant Hilbert space, which isthe case for Fibonacci anyons.

Following Ref. 41 we will borrow some weaving termi-nology and refer to the mobile quasiparticle (or collectionof quasiparticles) as the weft quasiparticle(s) and thestatic quasiparticles as the warp quasiparticles.

One reason for focusing on weaves is that weaving willlikely be easier to accomplish technologically than gen-eral braiding. This is true even if the full computationinvolves not just weaving a single quasiparticle, as was

proposed in Ref. 41, but possibly weaving several quasi-particles at the same time in different regions of the com-puter carrying out quantum gates on different qubitsin parallel.

Considering weaves has the added (and more imme-diate) benefit of simplifying the problem of numericallysearching for three-braids which approximate desiredgates. For the full braid group, even on just three strands,there is a great deal of redundancy since braids which aretopologically equivalent will yield the same unitary oper-ation. Weaves, however, naturally provide a unique rep-resentation in which the warp strands are straight, andthe weft weaves around them. There is therefore no triv-

ial double counting of topologically equivalent weaveswhen one does a brute force numerical search of weavesup to some given length.

The unitary operations performed by weaving threequasiparticles in which the weft quasiparticle starts andends in the middle position, will always have the form

Uweave({ni}) = nm1 nm12 n31 n22 n11 . (11)Here the sequence of exponents n2, n3 nm1 all taketheir values from {2, 4}, and n1 and nm can take the


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values {0, 2, 4}. Because these exponents are all even,each factor in this sequence takes the weft quasiparticleall the way around one of the two warp quasiparticleseither once or twice with either a clockwise or counter-clockwise sense, returning it to the middle position. Weallow n1 and nm to be 0 to account for the possibilitythat the initial or final weaving operations could eachbe either n1 or

n2 with n = 2 or 4. Note that we

need only consider exponents ni up to 4 (i.e., movingthe weft quasiparticle at most two times around a warpquasiparticle) because of the fact that 10i = 1 for Fi-bonacci anyons, implying, e.g., 6i =

4i . We define the

length L of such weaves to be equal to the total numberof elementary crossings, thus L =

mi=1 |ni|.

We will also consider weaves in which the weft quasi-particle begins and/or ends at a position other than themiddle. These possibilities can easily be taken into ac-count by multiplying Uweave({ni}), as defined in (11), bythe appropriate factors of 1 or 2 on the right and/orleft. Thus, for example, the unitary operation producedby a weave in which the weft quasiparticle starts in thetop position and ends in the middle position can be writ-ten Uweave({ni})2, where, because of the extra factorof 2, the first braiding operations carried out by thisweave will be n2 where n is an odd power, n = 1, 3or 5. This will weave the weft quasiparticle from thetop position to the middle position after which Uweavewill simply continue weaving this quasiparticle eventu-ally ending with it in the middle position. (Note thatby multiplying Uweave on the right by 2, and not

12 ,

we are not requiring the initial elementary braid to beclockwise, since Uweave may have n1 = 0 and n2 = 2or 4 so that the initial 2 is immediately multiplied by2 to a negative power.) Similarly, the unitary operationproduced by a weave in which the weft particle starts in

the top position and ends in the bottom position can bewritten 1Uweave({ni})2, and so on.

To find a weave for which the corresponding unitaryoperation Uweave({ni}) approximates a particular desiredunitary operation, the most straightforward approach isto simply perform a brute force search over all weaves,i.e. all sequences {ni} as described above, up to a cer-tain length L, in order to find the Uweave({ni}) which isclosest to the target operation. Here we will take as ameasure of the distance between two operators U and Vthe operator norm distance (U, V) = ||U V|| where||O|| is the operator norm, defined to be the square rootof the highest eigenvalue of OO. Again, if we are inter-

ested in fixing the relative phase of the total q-spin 1 andtotal q-spin 0 sectors then we would restrict the windingof the weaves so that the phases in (10) match those ofthe desired target gate.

For example, imagine our goal is to find a weave whichapproximates the unitary operation,

iX =

0 ii 0

1

. (12)

10 20 30 40 50

L

0

2

4

6

8

ln

1

FIG. 7: ln 1

vs. braid length L for weaves approximating thegate iX. Here is the distance (defined in terms of operatornorm) between iX and the unitary transformation producedby a weave of length L which best approximates it. The lineis a guide to the eye.

If the resulting weave were to be used only for a single-qubit operation, then we would only require that theweave approximate the upper left 2 2 block ofiX up toan overall phase and we would not care about the phasefactor appearing in the lower right matrix element. Therewould then be no constraint on the winding of the braid.However, for this example we will assume that this weavewill be used in a two-qubit gate construction, for whichthe overall phase and/or the phase difference between thetotal q-spin 1 and total q-spin 0 sectors will matter.

In this case, by comparing iX to (10), we see that thewinding W of any weave approximating iX must satisfyei3W/5 = 1 or W = 0 (modulo 10). Results of a brute

force search over weaves satisfying this winding require-ment which approximate iX are shown in Fig. 7. In thisfigure, ln 1 is plotted vs. braid length L, where is theminimum distance between Uweave and iX for weaves oflength L. It is expected that, for any such brute forcesearch for weaves approximating a generic target opera-tion, the length should scale with distance according toL log 1 , because the number of braids grows exponen-tially with L. The results shown in Fig. 7 are consistentwith such logarithmic scaling.

All the brute force searches used to find braids inthis paper are straightforward sequential searches, meantmainly to demonstrate proof of principle. No doubt moresophisticated brute force search methods (e.g. bidirec-tional search) could be used to perform deeper searchesresulting in longer and more accurate braids. Neverthe-less, the exponential growth in the number of braids withL implies that finding optimal braids by any brute forcesearch method will rapidly become infeasible as L in-creases. Fortunately one can still systematically improvea given braid to any desired accuracy by applying theSolovay-Kitaev algorithm,42,43 which we now briefly re-view.


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C. Implementation of the Solovay-Kitaev

Algorithm for Braids

The general result of the Solovay-Kitaev theorem tellsus that we can efficiently improve the accuracy of anygiven braid without the need to perform exhaustive bruteforce searches of ever improving accuracy.42,43 The essen-tial ingredient in this procedure is an -net a discrete

set of operators which in the present case correspond tofinite braids up to some given length, with the propertythat for any desired unitary operator there exists an el-ement of the -net which is within some given distance0 of that operator. Provided 0 is sufficiently small, theSolovay-Kitaev algorithm gives us a clever way to pick afinite number of braid segments out of the -net and sewthem together so that the resulting gate will be an ap-proximation to the desired gate with improved accuracy.

The implementation of the Solovay-Kitaev algorithmwe use here follows closely that described in detail inRefs. 44 and 45. The first step of this algorithm is tofind a braid which approximates the desired gate, U, by

performing a brute force search over the -net. Let U0denote the result of this search. Since we know that(U0, U) 0 it follows that C = U U10 is an operatorwhich is within a distance 0 of the identity.

The next step is to decompose C as a group commu-tator. This means that we find two unitary operatorsA and B for which C = ABA1B1. The unitary op-erators A and B are chosen so that their action on thecomputational qubit space corresponds to small rotationsthrough the same angle but about perpendicular axes.For this choice, ifA and B are then approximated by op-erators A0 and B0 in the -net, it can readily be shownthat the operator C0 = A0B0A

10 B

10 , will approximate

C to a distance of order 3/20 . It follows that the op-erator U1 = A0B0A

10 B

10 U0 is an approximation to U

within a distance 1 c3/20 , where c is a constant whichdetermines the size of the -net needed to guarantee animprovement in accuracy.

What we have just described corresponds to one iter-ation of the Solovay-Kitaev algorithm. Subsequent iter-ations are carried out recursively. Thus, at the secondlevel of approximation each search within the -net isreplaced by the procedure described above, and so on,so that at the nth level all approximations are made atthe (n 1)st level. The result of this recursive processis a braid whose accuracy grows superexponentially inn, with the distance to the desired gate being of ordern (c20)(3/2)n at the nth level of recursion, whilethe braid length grows only exponentially in n, withL 5nL0, where L0 is a typical braid length in the ini-tial -net. Thus, as the distance of the approximate gatefrom the desired target gate, , goes to zero, the braidlength grows only polylogarithmically, with L log 1where = l n 5/ ln(3/2) 3.97. While this scaling is,of course, worse than the logarithmic scaling for bruteforce searching, it is still only a polylogarithmic increase

FIG. 8: (color online). One iteration of the Solovay-Kitaevalgorithm applied to finding a braid which approximates theoperation U = iX. The braid U0 is the result of a bruteforce search over weaves up to length 44 which best approx-imates the desired gate U = iX, with an operator norm

distance between U and U0 of 8.5 104

. The braidsA0 and B0 are the results of similar brute force searchesto approximate unitary operations A and B whose groupcommutator satisfies ABA1B1 = UU1

0. The new braid

U1 = A0B0A1

0B10

U0 is then five times longer than U0, andthe accuracy has improved so that the distance to the targetgate is now 1 4.2 10

5. Given the group commutatorstructure of the A0B0A

1

0B10

factor, the winding of the U1braid is the same as the U0 braid. Note that when joiningbraids to form U1 it is possible that elementary braid op-erations from one braid will multiply their own inverses inanother braid, allowing the total braid to be shortened. Herewe have left these redundant braids in U1, as the carefulreader should be able to find.

in braid length which is sufficient for quantum compu-tation. Similar arguments44,45 can be used to show thatthe classical computer time t required to carry out theSolovay-Kitaev algorithm also only scales polylogarith-mically in the desired accuracy, with t log 1 where = ln 3/ ln(3/2) 2.71.

It is worth noting that there is a particularly nice fea-ture of this implementation of the Solovay-Kitaev algo-rithm when applied to compiling three-braids. Recallthat when carrying out two-qubit gates it will be crucialto maintain the phase difference between the total q-spin1 and total q-spin 0 sectors of the three-quasiparticleHilbert space associated with a given three-braid, and,according to (10), this can be done by fixing the windingof the braid (modulo 10). Because of the group commuta-tor structure of the Solovay-Kitaev algorithm, the wind-ing of the nth-level approximation Un will be the same asthat of the initial approximation U0. This is because allsubsequent improvements involve multiplying this braidby group commutators of the form AnBnA1n B

1n which

automatically have zero winding. The phase relationshipbetween the total q-spin 1 and total q-spin 0 sectors is


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FIG. 9: (color online). Two encoded qubits and a generic

braid. Because quasiparticles are braided outside of theirstarting qubits these braids will generally lead to leakage outof the computational qubit space, i.e. the q-spin of each groupof three quasiparticles forming these qubits will in general nolonger be 1.

therefore preserved at every level of the construction.Fig. 8 shows the application of one iteration of the

Solovay-Kitaev algorithm applied to finding a braidwhich generates a unitary operation approximating iX.The braid labeled U0 is the result of a brute force searchwith L = 44 corresponding to the best approximationshown in Fig. 7. (Note that although this braid is drawnas a sequence of elementary braid operations, it is topo-logically equivalent to a weave. In fact precisely thisbraid, drawn explicitly as a weave, is shown in Fig. 13.)The braids labeled A0 and B0 generate unitary opera-tions which approximate operators A and B whose groupcommutator gives U U10 where U = iX. Finally, thebraid labeled U1 is the new, more accurate, approximateweave.

VI. TWO-QUBIT GATES

We have seen that single-qubit gates are easy in thesense that as long as we braid within an encoded qubitthere will be no leakage errors (the overall q-spin of thegroup of three quasiparticles will remain 1). Further-more, the space of unitary operators acting on the three-quasiparticle Hilbert space (essentially SU(2)) is smallenough to find excellent approximate braids by perform-ing brute force searches and subsequent improvement us-ing the Solovay-Kitaev algorithm. We now turn to thesignificantly harder problem of finding braids which ap-proximate entangling two-qubit gates.

A. Divide and Conquer Approach

Figure 9 depicts six quasiparticles encoding two qubitsand a general braiding pattern. To entangle these qubits,quasiparticles from one qubit must be braided aroundquasiparticles from the other qubit, and this will in-evitably lead to leakage out of the encoded qubit space,(i.e. the overall q-spin of the three quasiparticles con-stituting a qubit may no longer be 1). Furthermore,the space of all operators acting on the Hilbert space ofsix quasiparticles is much bigger than for three, making

FIG. 10: (color online). A two-qubit gate construction in

which a pair of quasiparticles from the top (control) qubit iswoven through the bottom (target) qubit. The mobile pairof quasiparticles is referred to as the control pair and has atotal q-spin of 0 if the control qubit is in the state |0L, and1 if the control qubit is in the state |1L. Since weaving anobject with total q-spin 0 yields the identity operation, thisconstruction is guaranteed to result in a transformation of thetarget qubit state only if the control qubit is in the state |1L.Note that in this and subsequent figures world-lines of mobilequasiparticles will always be dark blue.

brute force searching extremely difficult. Here the uni-tary operations acting on this space are in SU(5)SU(8),(up to winding dependent phase factors as in (10)), whichhas 87 free parameters as opposed to 3 for the three quasi-particle case of SU(2).

Still, as a matter of principle, it is possible to per-form a brute force search of sufficient depth so thatit corresponds to a fine enough -net to carry out theSolovay-Kitaev algorithm in this larger space.42 This isessentially the program outlined in Ref. 5 as an exis-tence proof that universal quantum computation is pos-sible; however, it is not at all clear that, even if onecould do this, it would be the most efficient procedurefor compiling braids. For the same amount of classicalcomputing power required to directly compile braids inSU(5)

SU(8), we believe one can find much more effi-

cient (in the sense of having a more accurate computa-tion with a shorter braid) braids by breaking the probleminto smaller problems, each consisting of finding a spe-cific three-braid embedded in the full six-braid space. Asweve shown above, these three-braids can then be veryefficiently compiled.

Here we present two classes of two-qubit gate construc-tions based on this divide and conquer approach. Thefirst of these were originally introduced by the authors inRef. 32 and are characterized by the weaving of a pair ofquasiparticles from one qubit through the quasiparticlesforming the second qubit. The second class, presentedhere for the first time, can be carried out by weaving

only a single quasiparticle from one qubit around oneother quasiparticle from the same qubit, and two quasi-particles from the second qubit.

B. Two-Quasiparticle Weave Construction

We now review the two-qubit gate constructions firstdiscussed in Ref. 32. The basic idea behind these con-structions is illustrated in Fig. 10. This figure shows two


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FIG. 11: (color online). An effective braiding weave, anda two-qubit gate constructed using this weave. The effec-tive braiding weave is a woven three-braid which produces aunitary operation which is a distance 2.3 103 fromthat produced by simply interchanging the two target parti-cles (21). When the control pair is woven through the targetqubit using this weave the resulting two-qubit gate approxi-mates a controlled-(22) gate to a distance 1.9 10

3 or 1.6 103 when the total q-spin of the two qubits is 0 or1, respectively.

qubits and a braiding pattern in which a pair of quasi-particles from the top qubit (the control qubit) is wo-

ven through the quasiparticles forming the bottom qubit(the target qubit). Throughout this braiding the pair istreated as a single immutable object which, at the end ofthe braid, is returned to its original position.

If, as in Fig. 10, we choose the pair of weft quasipar-ticles to be the two quasiparticles whose total q-spin de-termines the logical state of the qubit, then we refer tothis pair as the control pair. We can then immediatelysee why this construction naturally suggests itself. If thecontrol qubit is in the state |0L the control pair willhave total q-spin 0, and weaving this pair through thetarget qubit will have no effect. We are thus guaranteedthat if the control qubit is in the state |0L the identityoperation is performed on the target qubit.The only non-trivial effect of this weaving pattern oc-curs when the control qubit is in the state |1L. In thiscase, the control pair has total q-spin 1 and so behaves asa single Fibonacci anyon. The problem of constructing atwo-qubit controlled gate then corresponds to finding aweaving pattern in which a single Fibonacci anyon weavesthrough the three quasiparticles of the target qubit, in-ducing a transition on this qubit without inducing leak-age error out of the computational qubit space, or at leastkeeping such leakage as small as required for a particularcomputation. This reduces the problem of finding a two-qubit gate to that of finding a weaving pattern in whichone Fibonacci anyon weaves around three others aproblem involving only four Fibonacci anyons. However,following our divide and conquer philosophy, we willfurther narrow our focus to weaving a single Fibonaccianyon through only two others at a time.

We define an effective braiding weave, to be a woventhree-braid in which the weft quasiparticle starts at thetop position, and returns to the top position at the end ofthe weave, with the requirement that the unitary trans-formation it generates be approximately equal to thatproduced by m clockwise interchanges of the two warp

FIG. 12: (color online). An injection weave, and step onein our injection based gate construction. The box labeled Irepresents an ideal (infinite) injection weave which is approx-imated by the weave shown to a distance 1.5 103. Instep one of our gate construction, this injection weave is usedto weave the control pair into the target qubit. If the con-trol qubit is in the state |1L then a = 1 and the result is toproduce a target qubit with the same quantum numbers asthe original, but with its middle quasiparticle replaced by thecontrol pair.

quasiparticles. To find such weaves we perform a bruteforce search, as outlined in Sec. V, over sequences {ni}which approximately satisfy

2 Uweave({ni}) 2 m1 . (13)

If both sides of this equation are expressed using (10) itbecomes evident that the winding of any effective braid-ing weave must satisfy W = m (modulo 10). Since theweft particle starts and ends in the top position, W mustbe even, thus effective braiding weaves only exist for evenm.

An example of an m = 2 effective braiding weave found

through a brute force search is shown in Fig. 11. Thecorresponding unitary operation approximates that of in-terchanging the two warp quasiparticles twice to a dis-tance 103. (This is a typical distance for a wo-ven three-braid of length L 46 which approximatesa desired operation precise distances of approximateweaves are given in the figure captions.) As for all ap-proximate weaves considered here, the Solovay-Kitaev al-gorithm outlined in Sec. V.C can be used to improve theaccuracy of this weave so that can be made as small asrequired with only a polylogarithmic increase in length.

The construction of a two-qubit gate using this effec-tive braiding weave is also shown in Fig. 11. In thisconstruction the control pair is woven through the toptwo quasiparticles of the target qubit using this weave.As described above, if the control qubit is in the state|0L, the control pair has q-spin 0 and the target qubit isunchanged. But, if the control qubit is in the state |1L,the control pair has q-spin 1 and the action on the targetqubit is approximately equivalent to that of interchang-ing the top two quasiparticles twice, with the approxima-tion becoming more accurate as the length of the effectivebraiding weave is increased, either by deeper brute forcesearching or by applying the Solovay-Kitaev algorithm.


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FIG. 13: (color online). A weave which approximates iX (seeEq. 12), and step two in our injection based construction. Thebox labeled iX represents an ideal (infinite) iX weave which isapproximated by the weave shown to a distance = 8.5104

(this is the same weave which appears at the top of Fig. 8).In step two of our gate construction the control pair is wovenwithin the injected target qubit, following this weave, in orderto carry out an approximate iX gate when a = 1, as shown.

Because this effective braiding all occurs within an en-

coded qubit, leakage errors can be reduced to zero inthe limit 0. The resulting two-qubit gate is then acontrolled-22 gate which corresponds to controlled rota-tion of the target qubit through an angle of 6/5.

Unfortunately, due to the even m constraint, it is im-possible to find an effective braiding gate which corre-sponds to a controlled rotation of the target qubit.Such a gate would be equivalent to a controlled-NOT gateup to single-qubit rotations.43 Nonetheless, it is knownthat any entangling two-qubit gate, when combined withthe ability to carry out arbitrary single-qubit rotations,forms a universal set of quantum gates.46 Thus, the effi-cient compilation of single-qubit operations described inSec. V and the effective braiding construction just givenprovide direct procedures for compiling any quantum al-gorithm into a braid to any desired accuracy.

Although it can be used to form a universal set ofgates, this effective braiding construction is still ratherrestrictive. It is clearly desirable to be able to directlycompile a controlled-NOT gate into a braid. We nowgive a construction which can be used to efficiently com-pile any arbitrary controlled rotation of the target qubit including a controlled-NOT gate. This constructionis based on a class of woven three-braids which we callinjection weaves.

In an injection weave the weft quasiparticle againstarts at the top position but in this case ends at a differ-

ent position. At the same time we require that the uni-tary operation generated by this weave approximate theidentity. Thus the effect of an injection weave is to per-mute the quasiparticles involved without changing any ofthe underlying q-spin quantum numbers of the system.

Comparing the identity matrix to (10) we see that anythree-braid approximating the identity must have wind-ing W = 0 (modulo 10). The fact that this windingmust be even implies that the final position of the weftparticle must be at the bottom of the weave. Thus injec-

FIG. 14: (color online). An inverse injection weave and stepthree in our injection based construction. The box labeled I1

represents an ideal (infinite) inverse injection weave which isapproximated by the the inverse of the injection weave shownin Fig. 12, again to a distance 1.5 103. This weaveis used to extract the control pair out of the injected targetqubit and return it to the control qubit, as shown.

tion weaves correspond to sequences {ni} which approx-imately satisfy the equation,

1 Uweave({ni}) 2

1 00 1

1

. (14)

An injection weave obtained through brute force searchis shown in Fig. 12. The unitary operation producedby this weave approximates the identity operation to adistance 103.

Our two-qubit gate construction based on injectionweaving is carried out in three steps. In the first step,also shown in Fig. 12, the control pair is woven into thetarget qubit using the injection weave. If the control pair

has total q-spin 1 (the only nontrivial case) the effect ofthis weave is merely to replace the middle quasiparticleof the target qubit with the control pair. Because theunitary operation approximated by the injection weaveis the identity, in the 0 limit this injection is ac-complished without changing any of the q-spin quantumnumbers. The injected target qubit is therefore (approxi-mately) in the same quantum state as the original targetqubit.

In the second step of our construction, illustrated inFig. 13, we carry out an operation on the injected tar-get qubit by simply weaving the control pair within thetarget. Because for a = 1 all of this weaving takes placewithin the injected target qubit, there will be no leakageerror (again, strictly speaking, only in the limit of an ex-act injection weave). The only constraint on this weaveis that the control pair must both start and end in themiddle position, and so it must have even winding.

If our goal is to produce a gate which is equivalent toa controlled-NOT gate up to single-qubit rotations thenwe must apply a rotation to the target qubit. Unfortu-nately, this cannot be accomplished by any finite weavewith even winding, so we must again consider approx-imate weaves. Figure 13 shows the control pair being


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FIG. 15: (color online). Injection-weave based compilation of a controlled-NOT gate into a braid. A controlled-NOT gate canbe expressed as a controlled-(iX) gate and a single-qubit operation R(/2 z) = exp(iz/4) acting on the control qubit.The single-qubit rotation can be compiled following the procedure outlined in Sec. V, and the controlled-(iX) gate can bedecomposed into ideal injection (I), iX, and inverse injection (I1) operations which can be similarly compiled. The fullapproximate controlled-(iX) braid obtained by replacing I, iX and I1 with the weaves shown in the previous three figures isshown at bottom. The resulting gate approximates a controlled-(iX) to a distance 1.8 103 and 1.2 103 when thetotal q-spin of the two qubits is 0 or 1, respectively.

woven through the injected target qubit using a weavefound by a brute force search which approximates a par-ticular rotation the operator iX defined in (12) to a distance 103 (this is, in fact, the same weaveshown at the top of Fig. 8).

The third step in our construction is the extraction ofthe control pair from the target qubit. This is accom-plished, as shown in Fig. 14, by applying the inverse ofthe injection weave to the control pair. The effect of thisextraction is to restore the control qubit to its originalstate, and replace the control pair inside the target qubitwith the quasiparticle which originally occupied that po-sition.

The full construction is summarized in Fig. 15, whichprovides a recipe for compiling a controlled-NOT gateinto a two-quasiparticle weave. A quantum circuitshowing that a controlled-NOT gate is equivalent toa controlled-(iX) gate and a single-qubit operation isshown in the top part of the figure. The single-qubitoperation can be compiled to whatever accuracy is re-quired following Sec. V, and the controlled-(iX) gate canbe decomposed into injection, iX, and inverse injectionoperations, as is also shown in the top part of the fig-ure. These operations can then all be similarly compiledfollowing Sec. V.

The full braid shown at the bottom of Fig. 15 cor-responds to using the approximate woven three-braidsshown in Figs. 12-14 to carry out a controlled-(iX) gate.In this braid, if the control qubit is in the state |0L thecontrol pair has total q-spin 0 and the resulting unitarytransformation is exactly the identity. However, if thecontrol qubit is in the state |1L the control pair has to-tal q-spin 1 and behaves like a single Fibonacci anyon.This pair is then woven into the target qubit using an in-

jection weave, woven within the target in order to carryout the iX operation, and finally woven out of the tar-get and back into the control qubit using the inverse ofthe injection weave. The resulting gate is therefore a

controlled-(iX) gate.By replacing the iX weave with an even winding weave

which carries out an arbitrary operation U this construc-tion will give a controlled-U gate. The only restriction onU is that its overall phase must be consistent with (10)with even winding W. However, this phase can be eas-ily set to any desired value by applying the appropriatesingle-qubit rotation to the control qubit, as in Fig. 15.

Finally, note that at no point in either the effectivebraiding or injection weave constructions described abovedid we make reference to the total q-spin of the twoqubits involved. It follows that, in the limit of exact ef-fective braiding or injection weaves, the action of the cor-responding two-qubit gates on the computational qubitspace does not depend on the state of the external fusion

space associated with the q-spin 1 quantum numbers ofeach qubit (see Sec. IV). These gates will therefore notentangle the computational qubit space with this exter-nal fusion space.

C. One-Quasiparticle Weave Constructions

We now show that two-qubit gates can be carried outwith only a single mobile quasiparticle. This possibil-ity follows from the general result of Ref. 41 that forany system of nonabelian quasiparticles in which generalbraids are universal for quantum computation (such asFibonacci anyons), single quasiparticle weaves are uni-versal as well. However, the proof of principle weavesconstructed in that work were extremely inefficient involving a huge number of excess operations. Here weshow how to efficiently construct a single-quasiparticleweave corresponding to a controlled-NOT gate (up tosingle-qubit rotations).

Our construction is based on a class of weaves whichare similar to injection weaves in that they can be used toswap two q-spin 1 objects where one object is a pair of


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Fibonacci anyons with total q-spin 1 and the other objectis a single Fibonacci anyon while acting effectively asthe identity operation so that none of the other q-spinquantum numbers of the system are disturbed. How-ever, unlike injection weaves, this new class of weavesaccomplish this swap without moving the pair as a singleobject, and in fact can be carried out by moving just onequasiparticle.

The class of weaves we seek are those which approxi-mate the transformation

U((, )a, )c = ei(, (, )a)c, (15)where is an overall (irrelevant) phase which does notdepend on a or c. The relevant case for showing thesimilarity with injection is when a = 1, for which theinitial and final states in (15) consist of two q-spin 1 ob-

jects a single Fibonacci anyon and a pair of Fibonaccianyons with total q-spin 1. If both these objects arerepresented as single Fibonacci anyons then (15) can bewritten U(, )c = ei(, )c. In this representation Utherefore acts effectively as the identity operation (times

an irrelevant phase), similar to injection.Using the F matrix (5) to expand the right hand sideof (15) in the ((, ), ) basis yields

U((, )a, )c = ei

b

Fcab((, )b, )c. (16)

Comparing this with the action of a unitary operation Uwith matrix representation

U =

U

100 U

101

U110 U111

U011

, (17)

on the state ((, )a, )c,U((, )a, )c =

b

Ucab((, )b, )c, (18)

we see that the matrix representation of the U we seekis precisely the F matrix (up to a phase): U = eiF.While the F matrix describes a passive operation, i.e.a change of basis, the operator U can be viewed as anactive F operation which acts directly on the states ofthe Hilbert space. Note that, since F = F1, we alsohave

U(, (, )a)c = ei((, )a, )c. (19)

We will refer to weaves which approximate the opera-tion (15) (and thus also (19)) as F weaves. As we haveseen, the unitary operation U produced by an F weaveneed only approximate the F matrix (5) up to an overallirrelevant phase. To be consistent with (10) this phasemust be 1, as can be seen by writing the matrix F as

F =

i

i i

i i

1

, (20)

FIG. 16: (color online). An F weave, and step one of our Fweave based two-qubit gate construction. The box labeled Frepresents an ideal (infinite) F weave which is approximatedby the weave shown to a distance 3.1103. Applying theF weave to the initial two-qubit state, as shown, produces anintermediate state with q-spins labeled a and b which dependsimply on a and b the initial states of the two qubits (seeTable I).

where a factor ofi has been pulled out of the upper left22 block, leaving an SU(2) matrix (det = 2 + = 1).Comparing (20) with (10), it is also evident that any Fweave must have winding W = 5 (modulo 10), which isnecessarily odd.

The fact that F weaves must have an odd number ofwindings implies that if the weft quasiparticle starts atthe top position of the weave it must end at the middleposition. For this choice the F weave must then approx-imately satisfy the equation

Uweave({ni}) 2 F. (21)The result of a brute force search for an F weave whichapproximates the operation F to a distance 103is shown in Fig. 16.

The first step in our single-quasparticle weave con-struction is the application of an F weave to two qubits,also shown in Fig. 16. Note that in this figure for con-venience we have made a change of basis on the bot-tom qubit, so that the pair which determines its state(the control pair) consists of the top two quasiparticleswithin it rather than the bottom two. There is no lossof generality in doing so since this just corresponds to asingle-qubit rotation on the bottom qubit.

With this basis choice the initial state of the two qubitsis determined by the q-spins of their respective controlpairs which are indicated in Fig. 16 as a (top qubit) andb (bottom qubit). After carrying out the F weave, tak-ing the middle quasiparticle of the top qubit as the weftquasiparticle and weaving it around both the bottomquasiparticle of the top qubit and the top quasiparticleof the bottom qubit, the resulting state (again, strictlyspeaking, only in the limit of an exact F weave) is shownat the end of the two-qubit weave in Fig. 16. From (19) itfollows that the newly positioned weft quasiparticle andthe quasiparticle beneath will have total q-spin a. Whenthe quasiparticle beneath these two is also included, thethree quasiparticles form what we will refer to as the in-termediate state, (, (, )a)b, where the total q-spin of


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FIG. 17: (color online). A phase weave with = (seetext) which gives a phase shift to the intermediate statewhen b = 1, and step two of our F weave based construc-tion. The box labeled P represents an ideal (infinite) = phase weave which is approximated by the weave shown to adistance 1.9 103. Applying this phase weave to theintermediate state created by the F weave, as shown, resultsin a b dependent phase shift (see Table I with = ).

all three quasiparticles, b, has a well-defined value pro-vided a and b are well defined, as we now show.

First consider the case a = 1. As described above, theeffect of the F weave is then similar to that of the injec-tion weave from the previous construction it replacesthe topmost quasiparticle in the bottom qubit with a pairof quasiparticles with q-spin 1, and the bottommost pairof quasiparticles in the top qubit (which also has totalq-spin 1) with a single quasiparticle, without changingany of the other q-spin quantum numbers of the system.In the limit of an ideal F weave, this means that the bquantum number does not change after this swap andso b = b. The case a = 0 is simpler, since in this casethe intermediate state is (, (, )0)b for which the fusionrules (2) imply b = 1, regardless of the value of b. The

resulting dependence of b

on a and b is summarized inTable I.Having used the F weave to create the intermediate

state (, (, )a)b , the next step in our construction is theapplication of a weave which performs an operation onthis state which does not change a and b but which doesyield an a and b dependent phase factor. After carryingout such a weave, which we will refer to as a phase weave,we can then apply the inverse of the F weave to restorethe two qubits to their initial states a and b.

For any phase weave we will require that the weft

a b b Phase Factor

0 0 b = 1 1 ei

0 1 1 ei

1 0 b = b 0 1

1 1 1 ei

TABLE I: Values of b for different values of a and b afterapplying the F weave as shown in Fig. 16, and the phaseapplied to the resulting state by a phase weave with zerowinding. The value of b is determined by the fact that b = 1when a = 0 and b = b when a = 1, as shown in the text.

FIG. 18: (color online). An inverse F weave and step threein our F weave construction. The box labeled F1 is anideal (infinite) inverse F weave which is approximated by theinverse of the F weave shown in Fig. 16, again to a distance 3.1 103. By applying the inverse F weave to the stateobtained after applying the phase weave, as shown, the twoqubits are returned to their initial states, but now with an aand b dependent phase factor (see Table I).

quasiparticle both start and end in the top position sothat when we join it to the F weave and its inverse therewill be a single weft quasiparticle throughout the entiregate construction. The phase weave must therefore haveeven winding, and with no loss of generality we can con-sider the case for which the winding satisfies W = 0(modulo 10). The unitary operation produced by such aphase weave must then approximately satisfy the equa-tion

2Uweave({ni})2 F

e

i 0

0 ei

1

F1, (22)

where the F matrices are needed to change the Hilbert

space basis from that in which the operation produced bythe phase braid must be diagonal, (the (, (, )) basis),to that in which the 1 and 2 matrices are defined, (the((, ), ) basis).

We will see that a phase weave with = produces atwo-qubit gate which is equivalent to a controlled-NOTgate up to single-qubit rotations. The result of a bruteforce search for such a phase weave which approximatesthe desired operation to a distance 103 is shown inFig. 17. This figure also shows the action of the phaseweave on the intermediate state produced in Fig. 16. Inthis weave, the weft quasiparticle is now woven throughthe two quasiparticles beneath it, and returns to its orig-inal position. Because the phase weave produces a di-agonal operation in the basis shown for the intermediatestate, it does not change the values of a and b. Its onlyeffect is to give a phase factor of ei to the state witha = 0 (which necessarily has b = 1) and ei to thestate with a = 1 and b = 1. The state with a = 1 andb = 0 is unchanged. These phase factors are also shownin Table I.

The final step in this construction is to perform the in-verse of the F weave to return the two qubits to their orig-inal states. This is shown in Fig. 18. In the limit of exact


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FIG. 19: (color online). F weave based compilation of a controlled-NOT gate into a braid. A controlled-NOT gate is equivalentto a controlled-(Z) gate with the single-qubit operation R(/2 y) = exp(iy/4) and its inverse applied to the target qubitbefore and after the controlled-(Z). Again, the single-qubit operations can be trivially compiled, and the controlled-(Z)gate decomposed into ideal F, phase (P), and inverse F (F1) weaves which can be similarly compiled. The full approximatecontrolled-(Z) weave obtained by replacing F, P and F1 with the approximate weaves shown in the previous three figuresis shown at bottom. The resulting gate approximates a controlled-(Z) to a distance 4.9 103 and 3.2 103 whenthe total q-spin of the two qubits is 0 or 1, respectively.

F and phase weaves, the resulting operation on the com-putational qubit space in the basis ab = {00, 01, 10, 11}is then,

U =

ei 0 0 0

0 ei 0 0

0 0 1 0

0 0 0 ei

. (23)

If we take the top qubit to be the control qubit, and thebottom qubit to be the target qubit, then this gate corre-sponds, up to an irrelevant overall phase, to a controlled-(ei3/2eiz/2) operation. For the case = this is acontrolled-(Z) gate (where Z = z), i.e. a controlled-Phase gate, which, up to single-qubit rotations, is equiv-

alent to a controlled-NOT gate.The full F weave based gate construction is summa-rized in Fig. 19. A quantum circuit showing a controlled-NOT gate in terms of a controlled-(Z) gate and twosingle-qubit operations is shown in the top part of thefigure. As in our injection based construction, the single-qubit operations can be compiled to whatever accuracyis required following the procedure outlined in Sec. V.The controlled-(Z) gate can then be decomposed intoideal F, phase, and inverse F weaves as is also shownin the top part of the figure. Woven three-braids whichapproximate these operations can then be compiled towhatever accuracy is required, again following Sec. V.The full controlled-(

Z) weave corresponding to using

the approximate F and phase weaves shown in Figs. 16-18 is shown in the bottom part of the figure.

Finally, in this construction, as for the constructionsdescribed in Sec. VI.B, we at no point made referenceto the total q-spin of the two qubits involved. Thus,in the limit of exact F and phase weaves, the action ofthe two-qubit gates constructed here will not entanglethe computational qubit space with the external fusionspace associated with the q-spin 1 quantum numbers ofeach qubit.

VII. WHATS SPECIAL ABOUT k = 3?

All of the gate constructions discussed in this paperexploit the fact that the braiding and fusion propertiesof a pair of Fibonacci anyons are either trivial if their to-tal q-spin is 0, or equivalent to those of a single Fibonaccianyon if their total q-spin is 1. The fact that these arethe only two possibilities is a special property of the Fi-bonacci anyon model, and hence also the SU(2)3 model,given their effective equivalence. It is then natural toask to what extent our constructions can be generalizedto SU(2)k CSW theories for different values of the levelparameter k.

Of course we know from the results of Freedman etal.6 that the SU(2)k representations of the braid group

are dense for k = 3 and k > 4. Thus, for example,braids which approximate controlled-NOT gates on en-coded qubits exist and can, in principle, be found for allthese k values. However, we will show below that thingsare somewhat simpler for the case k = 3. Specificallywe will show that for k = 3, and only k = 3, it is possi-ble to carry out two-qubit entangling gates by braidingonly four quasiparticles, as, for example, in our effectivebraiding and F weave constructions.

Consider a pair ofSU(2)k four-quasiparticle qubits asshown in Fig. 20. Here each quasiparticle is assumedto have q-spin 1/2 and the total q-spin of each qubit isrequired to be 0. The state of a given qubit is then deter-mined by the q-spin of either the topmost or bottommostpair of quasiparticles within it, where, from the SU(2)kfusion rules (1), the q-spin of each pair must be the samefor the total q-spin of the qubit to be 0. Thus, in Fig. 20,the state of the top qubit is determined by the q-spin la-beled a and the state of the bottom qubit is determinedby the q-spin labeled b, where, again from the fusion rules(1), a and b can be either 0 or 1.

If we are only allowed to braid the middle four quasi-particles, as shown in Fig. 20, then the total q-spin ofthe two topmost quasiparticles of the top qubit and the


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FIG. 20: (color online). Two four-quasiparticle qubit

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