Topological quantum compilingweb2.physics.fsu.edu/~bonesteel/papers/prb07.pdf“partially” topological quantum computation using a mixture of topological and nontopological gates

Topological quantum compiling

L. Hormozi, G. Zikos, and N. E. BonesteelDepartment of Physics and National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310, USA

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

�Received 17 October 2006; published 11 April 2007�

A method for compiling quantum algorithms into specific braiding patterns for non-Abelian quasiparticlesdescribed by the so-called Fibonacci anyon model is developed. The method is based on the observation thata universal set of quantum gates acting on qubits encoded using triplets of these quasiparticles can be builtentirely out of three-stranded braids �three-braids�. These three-braids can then be efficiently compiled andimproved to any required accuracy using the Solovay-Kitaev algorithm.

DOI: 10.1103/PhysRevB.75.165310 PACS number�s�: 73.43.�f, 03.67.Lx, 03.67.Pp

I. INTRODUCTION

The requirements for realizing a fully functioning quan-tum computer are daunting. There must be a scalable systemof qubits which can be initialized and individually measured.It must be possible to enact a universal set of quantum gateson these qubits. And all this must be done with sufficientaccuracy so that quantum error correction can be used toprevent decoherence from spoiling any computation.

The problems of error and decoherence are particularlydifficult ones for any proposed quantum computer. While thestates of classical computers are typically stored in macro-scopic degrees of freedom which have a built-in redundancyand thus are resistant to errors, building similar redundancyinto quantum states is less natural. To protect quantum infor-mation it is necessary to encode it using quantum error-correcting code states.1,2 These states are highly entangled,and have the property that code states corresponding to dif-ferent logical qubit states can be distinguished from one an-other only by global �“topological”� measurements. Unlikestates whose macroscopic degrees of freedom are effectivelyclassical �think of the magnetic moment of a small part of ahard drive�, such highly entangled “topologically degener-ate” states do not typically emerge as the ground states ofphysical Hamiltonians. One route to fault-tolerant quantumcomputation is therefore to build the encoding and fault-tolerant gate protocols into the software of the quantumcomputer.3

A remarkable recent development in the theory of quan-tum computation which directly addresses these issues hasbeen the realization that certain exotic states of matter in twospace dimensions, so-called non-Abelian states, may providea natural medium for storing and manipulating quantuminformation.4–7 In these states, localized quasiparticle excita-tions have quantum numbers that are in some ways similar toordinary spin quantum numbers. However, unlike ordinaryspins, the quantum information associated with these quan-tum numbers is stored globally, throughout the entire system,and so is intrinsically protected against decoherence. Further-more, these quasiparticles satisfy so-called non-Abelian sta-tistics. This means that when two quasiparticles are adiabati-cally moved around one another, while being keptsufficiently far apart, the action on the Hilbert space is rep-

resented by a unitary matrix which depends only on the to-pology of the path used to carry out the exchange. Topologi-cal quantum computation can then be carried out bymoving quasiparticles around one another in two spacedimensions.4,5 The quasiparticle world-lines form topologi-cally nontrivial braids in three �=2+1� -dimensional space-time, and because these braids are topologically robust �i.e.,they cannot be unbraided without cutting one of the strands�the resulting computation is protected against error.

Non-Abelian states are expected to arise in a variety ofquantum many-body systems, including spin systems,8–10 ro-tating Bose gases,11 and Josephson junction arrays.12 Ofthose states which have actually been experimentally ob-served, the most likely to possess non-Abelian quasiparticleexcitations are certain fractional quantum Hall states. Mooreand Read13 were the first to propose that quasiparticle exci-tations which obey non-Abelian statistics might exist in thefractional quantum Hall effect. Their proposal was based onthe observation that the conformal blocks associated withcorrelation functions in the conformal field theory describingthe two-dimensional Ising model could be interpreted asquantum Hall wave functions. These wave functions describeboth the ground state of a half-filled Landau level of spin-polarized electrons, as well as states with some number offractionally charged quasihole excitations �charge e /4�. Theparticular ground state this construction produces, the so-called Pfaffian or Moore-Read state, is considered the mostlikely candidate for the observed fractional quantum Hallstate at Landau level filling fraction �=5/2 ��=1/2 in thesecond Landau level�.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. Themonodromy—or braiding properties—of these conformalblocks are then assumed to describe the unitary transforma-tions acting on the Hilbert space produced by adiabaticallybraiding quasiholes around one another.13 Explicit wavefunctions for these states were worked out in Ref. 16, and thenon-Abelian braiding properties have been verified numeri-cally in Ref. 17. In an alternate approach, the Moore-Readstate can be viewed as a composite fermion superconductorin a so-called weak pairing px+ ipy phase.

18 In this descrip-

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tion, the finite-dimensional Hilbert space arises from zero-energy solutions of the Bogoliubov–de Gennes equationsin the presence of vortices,18 and the vortices themselves arenon-Abelian quasiholes whose braiding properties have beenshown to agree with the conformal field theory result.19,20

Recently, a number of experiments have been proposedto directly probe the non-Abelian nature of theseexcitations.21–24

Unfortunately, the braiding properties of quasihole excita-tions in the Moore-Read state are not sufficiently rich tocarry out purely topological quantum computation, although“partially” topological quantum computation using a mixtureof topological and nontopological gates has been shown tobe possible.25,26 However, Read and Rezayi27 have shownthat the Moore-Read state is just one of a sequence of stateslabeled by an index k corresponding to electrons at fillingfractions �=k / �2+k�, with k=1 corresponding to the �=1/3 Laughlin state and k=2 to the Moore-Read state. Thewave functions for these states can be written as correlationfunctions in the Zk parafermion conformal field theory,

27 andthe braiding properties of the quasihole excitations wereworked out in detail in Ref. 28. There it was shown that thequasiholes are described by the SU�2�k Chern-Simons-Witten �CSW� theories, up to overall Abelian phase factorswhich are irrelevant for quantum computation. More re-cently, explicit quasihole wave functions have been workedout for the k=3 Read-Reazyi state,29 with results consistentwith the predicted SU�2�3 braiding properties. The elemen-tary braiding matrices for the SU�2�k CSW theory for k=3and k�5 have been shown to be sufficiently rich to carry outuniversal quantum computation, in the sense that any desiredunitary operation on the Hilbert space of N quasiparticles,with N�3 for k�3,k�4,8 and N�4 for k=8, can be ap-proximated 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-qubit ro-tations and controlled-NOT gates� on encoded qubits for thecase k=3, thought to be physically relevant for the experi-mentally observed30 �=12/5 fractional quantum Halleffect27,31 ��=12/5 corresponds to �=2/5 in the second Lan-dau level, and this is the particle-hole conjugate of �=3/5corresponding to k=3�. We refer to the process of findingsuch braids as “topological quantum compiling” since thesebraids can then be used to translate a given quantum algo-rithm into the machine code of a topological quantum com-puter. This is analogous to the action of an ordinary compilerwhich translates instructions written in a high-level program-ming language into the machine code of a classical com-puter.

It should be noted that the proof of universality forSU�2�3 quasiparticles is a constructive one,5,6 and therefore,as a matter of principle, it provides a prescription for com-piling quantum gates into braids. However, in practice, fortwo-qubit gates �such as controlled-NOT gates� this prescrip-tion, if followed straightforwardly, is prohibitively difficultto carry out, primarily because it involves searching thespace of braids with six or more strands. We address thisdifficulty by dividing our two-qubit gate constructions into a

series of smaller constructions, each of which involvessearching only the space of three-stranded braids �three-braids�. The required three-braids then can be found effi-ciently and used to construct the desired two-qubit gates.This divide and conquer approach does not, in general, yieldthe most accurate braid of a given length which approxi-mates a desired quantum gate. However, we believe that itdoes yield the most accurate �or at least among the mostaccurate� braids which can be obtained for a given fixedamount of classical computing power.

This paper is organized as follows. In Sec. II we reviewthe basic properties of the SU�2�k Hilbert space, and showthat the case SU�2�3 is, for our purposes, equivalent to thecase SO�3�3—the so-called Fibonacci anyon model. SectionIII then presents a quick review of the mathematical machin-ery needed to compute with Fibonacci anyons. In Sec. IV weoutline how, in principle, these particles can be used to en-code qubits suitable for quantum computation. Section Vthen describes how to find braiding patterns for three Fi-bonacci anyons which can be used to carry out any allowedoperation on the Hilbert space of these quasiparticles to anydesired accuracy, thus effectively implementing the proce-dure given in Ref. 5 for carrying out single-qubit rotations.In Sec. VI we discuss the more difficult case of two-qubitgates, and give two classes of explicit gate constructions—one, first discussed by the authors in Ref. 32, in which a pairof quasiparticles from one qubit is “woven” through the qua-siparticles in the second qubit, and another, presented herefor the first time, in which only a single quasiparticle is wo-ven. Finally, in Sec. VII we address the question of to whatextent the constructions we find are special to the k=3 case,and in Sec. VIII we summarize our results.

II. FUSION RULES AND HILBERT SPACE

Consider a system with quasiparticle excitations describedby the SU�2�k CSW theory. It is convenient to describe theproperties of this system using the so-called quantum grouplanguage.28 The relevant quantum groups are “deformed”versions of the representation theory of SU�2�, i.e., thetheory of ordinary spin, and much of the intuition for think-ing about ordinary spin can be carried over to the quantumgroup case.

In the quantum group description of an SU�2�k CSWtheory, each quasiparticle has a half-integer q-deformed spin�q-spin� quantum number. Just as for ordinary spin, there arerules for combining q-spin known as fusion rules. The fusionrules for the SU�2�k theory are similar to the usual trianglerule for adding ordinary spin, except that they are truncatedso that there are no states with total q-spin �k /2. Specifi-cally, the fusion rules for the level k theory are33

s1 � s2 = �s1 − s2� � �s1 − s2� + 1 � ¯

� min�s1 + s2,k − s1 − s2� . �1�

Note that, in the quantum group description of non-Abeliananyons, states are distinguished only by their total q-spinquantum numbers. The q-deformed analogs of the Sz quan-tum numbers are physically irrelevant—there is no degen-

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eracy associated with them, and they play no role in anycomputation involving braiding.28 The situation is somewhatanalogous to that of a collection of ordinary spin-1/2 par-ticles in which the only allowed operations, including mea-surement, are rotationally invariant and hence independent ofSz, as is the case in exchange-based quantum computation.

34

The fusion rules of the SU�2�k theory fix the structure ofthe Hilbert space of the system. For a collection of quasipar-ticles with q-spin-1/2, a useful way to visualize this Hilbertspace is in terms of its so-called Bratteli diagram. This dia-gram shows the different fusion paths for N q-spin-1/2 qua-siparticles in which these quasiparticles are fused, one at atime, going from left to right in the diagram. Bratteli dia-grams for the cases k=2 and 3 are shown in Fig. 1.

The dimensionality of the Hilbert space for N q-spin-1/2quasiparticles with total q-spin S can be determined bycounting the number of paths in the Bratteli diagram fromthe origin to the point �N ,S�. The results of this path count-ing are also shown in Fig. 1, where one can see the well-known 2N/2−1 Hilbert space degeneracy for the k=2 �Moore-Read� case,13,16 and the Fibonacci degeneracy for the k=3case.27

In this paper we will focus on the k=3 case, which is thelowest k value for which SU�2�k non-Abelian anyons areuniversal for quantum computation.5,6 In fact, we will showthat two-qubit gates are particularly simple for this case. Be-fore proceeding, it is convenient to introduce an importantproperty of the SU�2�3 theory, namely, that the braidingproperties of q-spin-1/2 quasiparticles are the same as thosewith q-spin 1 �up to an overall Abelian phase which is irrel-evant for topological quantum computation�. This is a usefulobservation because the theory of q-spin-1 quasiparticles inSU�2�3 is equivalent to SO�3�3, a theory also known as theFibonacci anyon theory35,36—a particularly simple theorywith only two possible values of q-spin, 0 and 1, for whichthe 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. Thisproof is based on the fact that for k=3 the fusion rules in-

volving q-spin-3/2 quasiparticles take the following simpleform:

3

2� s =

3

2− s . �3�

The key observation is that, since for k=3 the highest pos-sible q-spin is 3/2, when fusing a q-spin-3/2 object with anyother object �here we use the term object to describe either asingle quasiparticle or a group of quasiparticles viewed as asingle composite entity�, the Hilbert space dimensionalitydoes not grow. This implies that moving a q-spin-3/2 objectaround other objects can, at most, produce an overall Abelianphase factor. While this phase factor may be important physi-cally, particularly in determining the outcome of interferenceexperiments involving non-Abelian quasiparticles,21–24 it isirrelevant for quantum computing, and thus does not matterwhen determining braids which correspond to a given com-putation. Because �3� implies that a q-spin 1/2 object can beviewed as the result of fusing a q-spin-1 object with aq-spin-3/2 object, it follows that the braid matrices forq-spin-1/2 objects are the same as those 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 which eachhave either q-spin 1 or q-spin 1/2. It is then possible to carryout topological quantum computation, even if we do notknow which objects have q-spin 1 and which have q-spin1/2. The proof is illustrated in Fig. 2. Figure 2�a� shows abraiding pattern for a collection of objects, some of whichhave q-spin 1/2 and some of which have q-spin 1. Figure2�b� then shows the same braiding pattern, but now all ob-jects with q-spin 1/2 are represented by objects with q-spin 1fused to objects with q-spin 3/2. Because, as noted above, theq-spin-3/2 objects have trivial �Abelian� braiding properties,the unitary transformation produced by this braid is thesame, up to an overall Abelian phase, as that produced bybraiding nothing but q-spin-1 objects, as shown in Fig. 2�c�.It follows that, provided one can measure whether the totalq-spin of some object belongs to the class 1��1,1 /2� or theclass 0��0,3 /2�—something which should, in principle, be

FIG. 1. Bratteli diagrams for SU�2�k fork��a� 2 and �b� 3. Here N is the number ofq-spin-1/2 quasiparticles and S is the total q-spinof those quasiparticles. The number at a given�N ,S� vertex of each diagram indicates the num-ber of paths to that vertex starting from the �0,0�point. This number gives the dimensionality ofthe Hilbert space of N q-spin-1/2 quasiparticleswith total q-spin S.

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possible by performing interference experiments as describedin Refs. 37 and 38—then quantum computation is possible,even if we do not know which objects have q-spin 1/2 andwhich 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 these qua-siparticles. For an excellent review of this topic see Ref. 35,and for the mathematics of non-Abelian particles in generalsee Ref. 39.

Note that for the rest of this paper, except for Sec. VII, itshould be understood that each quasiparticle is a q-spin-1Fibonacci anyon. It should also be understood that, from thepoint of view of their non-Abelian properties quasihole ex-citations are also q-spin-1 Fibonacci anyons, even thoughthey have opposite electric charge and give opposite Abelianphase factors when braided. Because it is the non-Abelianproperties that are relevant for topological quantum compu-tation, for our purposes quasiparticles and quasiholes can beviewed as identical non-Abelian particles. Unless it is impor-tant to distinguish between the two �as when we discuss cre-ating and fusing quasiparticles and quasiholes in Sec. IV� wewill simply use the terms quasiparticle or Fibonacci anyon torefer to either excitation.

Figure 3 establishes some of the notation for representingFibonacci anyons which will be used in the rest of the paper.This figure shows SU�2�3 Bratteli diagrams in which theq-spin axis is labeled by both the SU�2�3 q-spin quantumnumbers and, in boldface, the corresponding Fibonacciq-spin quantum numbers, i.e., 0 for �0,3 /2� and 1 for�1/2 ,1�. In Fig. 3�a� Bratteli diagrams showing fusion pathscorresponding to two basis states spanning the two-dimensional Hilbert space of two Fibonacci anyons areshown. Beneath each Bratteli diagram an alternate represen-tation of the corresponding state is also shown. In this rep-resentation dots correspond to Fibonacci anyons and ovalsenclose collections of Fibonacci anyons which are in q-spineigenstates whenever the oval is labeled by a total q-spin

quantum number. �Note: If the oval is not labeled, it shouldbe understood that the enclosed quasiparticles may not be ina q-spin eigenstate.�

In the text we will use the notation • to represent a Fi-bonacci anyon, and the ovals will be represented by paren-theses. In this notation, the two states shown in Fig. 3�a� aredenoted �• , • �0, and �• , • �1.

Figure 3�b� shows a Bratteli diagram, again with bothSU�2�3 and Fibonacci quantum numbers, with fusion pathsthat this time correspond to three basis states of the three-dimensional Hilbert space of three Fibonacci anyons. Be-neath these diagrams the oval representations of these threestates are also shown, which in the text will be represented(�• , • �0 , • )1, (�• , • �1 , • )1, and (�• , • �1 , • )0.

In addition to fusion rules, all theories of non-Abeliananyons possess additional mathematical structure which al-lows one to calculate the result of any braiding operation.This structure is characterized by the F �fusion� and R �rota-tion� matrices.35,39,40

To define the F matrix, note that the Hilbert space of threeFibonacci anyons is spanned by both the three states labeled(�• , • �a , • )c, and the three states labeled (• , �• , • �b)c. The Fmatrix is the unitary transformation which maps one of thesebases to the other,

„•,�•, • �a…c = �b

Fabc„�•, • �b, • …c, �4�

and has the form

F = � ��

� − �1

, �5�

where �= �5−1� /2 is the inverse of the golden mean. In thismatrix the upper left 2�2 block Fab

1 acts on the two-dimensional total q-spin-1 sector of the three-quasiparticleHilbert space, and the lower right matrix element F11

0 =1 actson the unique total q-spin-0 state. Note that this F matrix canbe applied to any three objects which each have q-spin 1,where each object can consist of more than one Fibonaccianyon. Furthermore, if one considers three objects for whichone or more of the objects has q-spin 0, then the state of

FIG. 2. �Color online� Graphical proof of the equivalence of braiding q-spin-1/2 and q-spin-1 objects for SU�2�3. �a� shows a braidingpattern for a collection of objects, some having q-spin 1/2 and some having q-spin 1. �b� shows the same braiding pattern but with theq-spin-1/2 objects represented by q-spin-1 objects fused with q-spin-3/2 objects, which, for SU�2�3, has a unique fusion channel. Finally, �c�shows the same braid with the q-spin-3/2 objects removed. Because these q-spin-3/2 objects are effectively Abelian for SU�2�3, removingthem from the braid will only result in an overall phase factor which will be irrelevant for quantum computing.


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these objects is uniquely determined by the total q-spin of allthree, and in 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 any num-ber of Fibonacci anyons.

The R matrix gives the phase factor produced when twoFibonacci anyons are moved around one another with a cer-tain sense. One can think of these phase factors as theq-deformed versions of the −1 or +1 phase factors one ob-tains when interchanging two ordinary spin-1/2 quasiparti-cles when they are in a singlet or triplet state, respectively.This phase factor depends on the overall q-spin of the twoquasiparticles involved in the exchange, so for Fibonaccianyons there are two such phase factors which are summa-rized in the R matrix,

R = �e−i4�/5 00 ei3�/5

� . �6�Here the upper left and lower right matrix elements are, re-spectively, the phase factor that two Fibonacci anyons ac-quire if they are interchanged in a clockwise sense when theyhave total q-spin 0 or q-spin 1. Again, this matrix also ap-plies if we exchange two objects that both have total q-spin1, even if these objects consist of more than one Fibonaccianyon. And if one or both objects has q-spin 0 the result ofthis interchange is the identity. Again we emphasize that inthe k=3 Read-Rezayi state, there will be additional Abelianphases present, which may have physical consequences for

some experiments, but which will be irrelevant for topologi-cal quantum computation.

Typically the sequence of F and R matrices used to com-pute the unitary operation produced by a given braid isnot unique. To guarantee that the result of any such compu-tation is independent of this sequence, the F and R matricesmust satisfy certain consistency conditions. These consis-tency conditions, the so-called pentagon and hexagonequations,35,39,40 are highly restrictive, and, in fact, for thecase of Fibonacci anyons essentially fix the F and R matricesto have the forms given above �up to a choice of chirality,and Abelian phase factors which are again irrelevant to ourpurposes here�.35

Finally, we point out an obvious, but important, conse-quence of the structure of the F and R matrices. When inter-changing any two quasiparticles which are part of a larger setof quasiparticles with a well-defined total q-spin quantumnumber, this total q-spin quantum number will not change.

IV. QUBIT ENCODING AND GENERAL COMPUTATIONSCHEME

Before proceeding, it will be useful to have a specificscheme in mind for how one might actually carry out topo-logical quantum computation with Fibonacci anyons. Herewe follow the scheme outlined in Ref. 7, which, for com-pleteness, we briefly review below.

The computer can be initialized by pulling quasiparticle-quasihole pairs out of the “vacuum” �by vacuum we meanthe ground state of the k=3 Read-Rezayi state or any other

FIG. 3. �Color online� Basis states for the Hilbert space of �a� two and �b� three Fibonacci anyons. SU�2�3 Bratteli diagrams showingfusion paths corresponding to the basis states for the Hilbert space of two and three q-spin-1/2 quasiparticles are shown. The q-spin axes onthese diagrams are labeled by both the SU�2�3 q-spin quantum numbers 0, 1/2, 1 and 3/2 and, to the left of these in bold, the correspondingFibonacci q-spin quantum numbers 0��0,3 /2� and 1��1/2 ,1�. Beneath each Bratteli diagram the same state is represented using anotation in which dots correspond to Fibonacci anyons, and groups of Fibonacci anyons enclosed in ovals labeled by q-spin quantumnumbers are in the corresponding q-spin eigenstates.


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state that supports Fibonacci anyon excitations�. Each suchpair will consist of two q-spin-1 excitations in a state withtotal q-spin 0, i.e., the state �• , • �0. In principle, this pair canalso exist in a state with total q-spin 1, provided there areother quasiparticles present to ensure the total q-spin of thesystem is 0, so one can imagine using this pair as a qubit.However, it is impossible to carry out arbitrary single-qubitoperations by braiding only the two quasiparticles formingsuch a qubit—this braiding never changes the total q-spin ofthe pair, and so only generates rotations about the z axis inthe qubit space.

For this reason it is convenient to encode qubits usingmore than two Fibonacci anyons. Thus, to create a qubit, twoquasiparticle-quasihole pairs can be pulled out of thevacuum. The resulting state is then (�• , • �0 , �• , • �0)0 whichagain has total q-spin 0. The Hilbert space of four Fibonaccianyons with total q-spin 0 is two dimensional, with basisstates, which we can take as logical qubit states �0L= (�• , • �0 , �• , • �0)0 and �1L= (�• , • �1 , �• , • �1)0 �see Fig. 4�a��.The state of such a four-quasiparticle qubit is determined bythe total q-spin of either the rightmost or leftmost pair ofquasiparticles. Note that the fusion rules �2� imply that thetotal q-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 with totalq-spin 0, there is a three-dimensional noncomputational Hil-bert space of states with total q-spin 1 spanned by the states(�• , • �0 , �• , • �1)1, (�• , • �1 , �• , • �0)1, and (�• , • �1 , �• , • �1)1.When carrying out topological quantum computation it iscrucial to avoid transitions into this noncomputational space.

Fortunately, single-qubit rotations can be carried out bybraiding quasiparticles within a given qubit and, as discussedin Sec. III, such operations will not change the total q-spin ofthe four quasiparticles involved. Single-qubit operations cantherefore be carried out without any undesirable transitionsout of the encoded computational qubit space.

Two-qubit gates, however, will require braiding quasipar-ticles from different qubits around one another. This will ingeneral lead to transitions out of the encoded qubit space.

Nevertheless, given the so-called “density” result of Ref. 6 itis known that, as a matter of principle, one can always findtwo-qubit braiding patterns which will entangle the two qu-bits, and also stay within the computational space to what-ever accuracy is required for a given computation. The mainpurpose of this paper is to show how such braiding patternscan be efficiently found.

Note that the action of braiding the two leftmost quasipar-ticles in a four-quasiparticle qubit �referring to Fig. 4�a�� isequivalent to that of braiding the two rightmost quasiparti-cles with the same sense. This is because as long as we are inthe computational qubit space both the leftmost and right-most quasiparticle pairs must have the same total q-spin, andso interchanging either pair will result in the same phasefactor from the R matrix. It is therefore not necessary tobraid all four quasiparticles to carry out single-qubitrotations—one need only braid three.

In fact, one may consider qubits encoded using only threequasiparticles with total q-spin 1, as originally proposed inRef. 5. Such qubits can be initialized by first creating a four-quasiparticle qubit in the state �0L, as outlined above, andthen simply removing one of the quasiparticles. In this three-quasiparticle encoding, shown in Fig. 4�b�, the logical qubitstates can be taken to be �0L= (�• , • �0 , • )1 and �1L= (�• , • �1 , • )1. For this encoding there is just a single non-computational state �NC= (�• , • �1 , • )0, also shown in Fig.4�b�. As for the four-quasiparticle qubit, when carrying outsingle-qubit rotations by braiding within a three-quasiparticlequbit the total q-spin of the qubit, in this case 1, remainsunchanged and there are no transitions from the computa-tional qubit space into the state �NC. However, just as forfour-quasiparticle qubits, when carrying out two-qubit gatesthese transitions will in general occur and we must workhard to avoid them. Henceforth we will refer to these un-wanted transitions as leakage errors.

Note that, because each three-quasiparticle qubit has totalq-spin 1, when more than one of these qubits is present thestate of the system is not entirely characterized by the “in-ternal” q-spin quantum numbers which determine the com-putational qubit states. It is also necessary to specify the stateof what we will refer to as the “external fusion space”—theHilbert space associated with fusing the total q-spin-1 quan-tum numbers of each qubit. When compiling braids for three-quasiparticle qubits it is crucial that the operations on thecomputational qubit space not depend on the state of thisexternal fusion space—if they did, these two spaces wouldbecome entangled with one another leading to errors. Fortu-nately, we will see that it is indeed possible to find braidswhich do not lead to such errors.

For the rest of this paper �except Sec. VII� we will usethis three-quasiparticle qubit encoding. It should be notedthat any braid which carries out a desired operation on thecomputational space for three-quasiparticle qubits will carryout the same operation on the computational space of four-quasiparticle qubits, with one quasiparticle in each qubit act-ing as a spectator. The braids we find here can therefore beused for either encoding.

We can now describe how topological quantum computa-tion might actually proceed, again following Ref. 7. A quan-tum circuit consisting of a sequence of one- and two-qubit

FIG. 4. �Color online� �a� Four-quasiparticle and �b� three-quasiparticle qubit encodings for Fibonacci anyons. �a� shows twostates that span the Hilbert space of four quasiparticles with totalq-spin 0 which can be used as the logical �0L and �1L states of aqubit. �b� shows two states spanning the Hilbert space of threequasiparticles with total q-spin 1 which can also be used as logicalqubit states �0L and �1L. This three-quasiparticle qubit can be ob-tained by removing the rightmost quasiparticle from the two statesshown in �a�. The third state shown in �b�, labeled �NC for non-computational, is the unique state of three quasiparticles that hastotal q-spin 0.


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gates which carries out a particular quantum algorithmwould first be translated �or “compiled”� into a braid bycompiling each individual gate to whatever accuracy is re-quired. Qubits would then be initialized by pullingquasiparticle-quasihole pairs out of the vacuum. These local-ized excitations would then be adiabatically dragged aroundone another so that their world-lines trace out a braid inthree-dimensional space-time which is topologically equiva-lent to the braid compiled from the quantum algorithm. Fi-nally, individual qubits would be measured by trying to fuseeither the two rightmost or two leftmost excitations withinthem �referring to Fig. 4�a�� for four-quasiparticle qubits, orjust the two leftmost excitations �referring to Fig. 4�b�� forthree-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 the pair is0, it will be possible for them to fuse back into the vacuum.However, if the total q-spin is 1 this will not be possible. Theresulting difference in the charge distribution of the finalstate would then be measured to determine if the qubit was inthe state �0L or �1L. Alternatively, as already mentioned inSec. II, interference experiments37,38 could be used to initial-ize and read out encoded qubits.

As a simple illustration, Fig. 5 shows a computation inwhich a four-quasiparticle qubit �which can also be viewedas a three-quasiparticle qubit if the top quasiparticle is ig-nored� is initialized by pulling quasiparticle-quasihole pairsout of the vacuum, a single-qubit operation is carried out bybraiding within the qubit, and the final state of the qubit ismeasured by fusing a quasiparticle and quasihole togetherand observing the outcome.

V. COMPILING THREE-BRAIDS AND SINGLE-QUBITGATES

We now focus on the problem of finding braids for threeFibonacci anyons �three-braids� which approximate any al-lowed unitary transformation on the Hilbert space of these

quasiparticles. This is important not only because it allowsone to find braids which carry out arbitrary single-qubitrotations,5 but also because, as will be shown in Sec. VI, it ispossible to reduce the problem of constructing braids whichcarry out two-qubit gates to that of finding a series of three-braids approximating specific operations.

A. Elementary braid matrices

Using the F and R matrices, it is straightforward to deter-mine the elementary braiding matrices that act on the three-dimensional Hilbert space of three Fibonacci anyons. If, as inFig. 6, we take the basis states for the three-quasiparticleHilbert space to be the states labeled (�• , • �a , • )c then, in theac= �01,11,10� basis, the matrix 1 corresponding to aclockwise interchange of the two bottommost quasiparticlesin 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 2�2 block acts on the total q-spin-1sector ��0L and �1L� of the three quasiparticles, and thelower right matrix element is a phase factor acquired by theq-spin 0 state ��NC�. This matrix is easily read off from theR matrix, since the total q-spin of the two quasiparticlesbeing exchanged is well defined in this basis.

To find the matrix 2 corresponding to a clockwise inter-change of the two topmost �or rightmost in the (�• , • �a , • )crepresentation� quasiparticles, we must first use the F matrixto change bases to one in which the total q-spin of thesequasiparticles is well defined. In this basis, the braiding ma-

FIG. 5. �Color online� Space-time paths corresponding to theinitialization, manipulation through braiding, and measurement ofan encoded qubit. Two quasiparticle-quasihole pairs are pulled outof the vacuum, with each pair having total q-spin 0. The resultingstate corresponds to a four-quasiparticle qubit in the state �0L �seeFig. 4�a��. After some braiding, the qubit is measured by trying tofuse the bottommost pair �in this case a quasiparticle-quasiholepair�. If they fuse back into the vacuum the result of the measure-ment is �0L; otherwise it is �1L. Because only the three lowerquasiparticles are braided, the encoded qubit can also be viewed asa three-quasiparticle qubit �see Fig. 4�b�� which is initialized in thestate �0L.

FIG. 6. �Color online� Elementary three-braids and the decom-position of a general three-braid into a series of elementary braids.The unitary operation produced by this braid is computed by mul-tiplying the corresponding sequence of elementary braid matrices,1 and 2 �see text� and their inverses, as shown. Here the �unla-beled� ovals represent a particular basis choice for the three-quasiparticle Hilbert space, consistent with that used in the text. Inthis and all subsequent figures which show braids, quasiparticles arealigned vertically, and we adopt the convention that reading frombottom to top in the figures corresponds to reading from left to rightin expressions such as (�• , • �a , • )c in the text. It should be noted thatthese figures are only meant to represent the topology of a givenbraid. In any actual implementation of topological quantum compu-tation, quasiparticles will certainly not be arranged in a straight line,and they will have to be kept sufficiently far apart while beingbraided to avoid lifting the topological degeneracy.


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trix is simply 1, and so, after changing back to the originalbasis, we find

2 = F−11F = �− �e

−i�/5 �e−i3�/5�e−i3�/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 multiplying thecorresponding sequence of 1 and 2 matrices and their in-verses, as shown in Fig. 6.

If we are only concerned with single-qubit rotations, thenwe only care about the action of these matrices on the en-coded qubit space with total q-spin 1, and not the totalq-spin-0 sector corresponding to the noncomputational state.However, in our two-qubit gate constructions, various three-braids will be embedded into the braiding patterns of sixquasiparticles, and in this case the action on the full three-dimensional Hilbert space does matter.

To understand this action note that 1 can be written

1 = �±e−i�/10�±e−i7�/10 0

0 ± ei7�/10�

ei3�/5

, �9�

where the upper 2�2 block acting on the total q-spin 1sector is an SU�2� matrix, �i.e., a 2�2 unitary matrix withdeterminant 1�, multiplied by a phase factor of either +e−i�/10or −e−i�/10, and the lower right matrix element ei3�/5 is thephase acquired by the total q-spin-0 state. The phase factorpulled out of the upper 2�2 block is only defined up to ±1because any SU�2� matrix multiplied by −1 is also an SU�2�matrix.

From �8� it follows that 2 can be written in a similarfashion, with the same phase factors. Each clockwise braid-ing operation then corresponds to applying an SU�2� opera-tion multiplied by a phase factor of ±e−i�/10 to the q-spin-1sector, while at the same time multiplying the q-spin-0 sectorby a phase factor of ei3�/5. Likewise, each counterclockwisebraiding operation corresponds to applying an SU�2� opera-tion multiplied by a phase factor of ±e+i�/10 to the q-spin-1sector and a phase factor of e−i3�/5 to the q-spin-0 sector.

We define the winding W�B� of a given three-braid B tobe the total number of clockwise interchanges minus the totalnumber of counterclockwise interchanges. It then followsthat the unitary operation corresponding to an arbitrary braidB can always be expressed

U�B� = �±e−iW�B��/10�SU�2��ei3W�B��/5

� , �10�where �SU�2�� indicates an SU�2� matrix. Thus, for a giventhree-braid, the phase relation between the total q-spin-1 andtotal q-spin-0 sectors of the corresponding unitary operationis determined by the winding of the braid. We will refer to�10� often in what follows. It tells us precisely what unitaryoperations can be approximated by three-braids, and placesuseful restrictions on their winding.

B. Weaving and brute force search

At this point it is convenient to restrict ourselves to asubclass of braids which we will refer to as weaves. A weaveis any braid that is topologically equivalent to the space-timepaths of some number of quasiparticles in which only asingle quasiparticle moves. It was shown in Ref. 41 that thisrestricted class of braids is universal for quantum computa-tion, provided the unitary representation of the braid group isdense in the space of all unitary transformations on the rel-evant Hilbert space, which is the case for Fibonacci anyons.

Following Ref. 41 we will borrow some weaving termi-nology and refer to the mobile quasiparticle �or collection ofquasiparticles� as the “weft” quasiparticle�s� and the staticquasiparticles as the “warp” quasiparticles.

One reason for focusing on weaves is that weaving willlikely be easier to accomplish technologically than generalbraiding. This is true even if the full computation involvesnot just weaving a single quasiparticle, as was proposed inRef. 41, but possibly weaving several quasiparticles at thesame time in different regions of the computer—carrying outquantum gates on different qubits in parallel.

Considering weaves has the added �and more immediate�benefit of simplifying the problem of numerically searchingfor three-braids which approximate desired gates. For the fullbraid group, even on just three strands, there is a great dealof redundancy since braids which are topologically equiva-lent will yield the same unitary operation. Weaves, however,naturally provide a unique representation in which the warpstrands are straight, and the weft weaves around them. Thereis therefore no trivial double counting of topologicallyequivalent weaves when one does a brute force numericalsearch of weaves up to some given length.

The unitary operations performed by weaving three qua-siparticles in which the weft quasiparticle starts and ends inthe middle position will always have the form

Uweave��ni�� = 1nm2

nm−1¯ 1

n32n21

n1. �11�

Here the sequence of exponents n2 ,n3 , . . . ,nm−1 all take theirvalues from �±2, ±4�, and n1 and nm can take the values�0, ±2, ±4�. Because these exponents are all even, each fac-tor in this sequence takes the weft quasiparticle all the wayaround one of the two warp quasiparticles either once ortwice with either a clockwise or counterclockwise sense, re-turning it to the middle position. We allow n1 and nm to be 0to account for the possibility that the initial or final weavingoperations could each be either 1

n or 2n with n= ±2 or ±4.

Note that we need only consider exponents ni up to ±4 �i.e.,moving the weft quasiparticle at most two times around awarp quasiparticle� because of the fact that i

10=1 for Fi-bonacci anyons, implying, e.g., i

6=i−4. We define the

length L of such weaves to be equal to the total number ofelementary crossings; thus L=�i=1

m �ni�.We will also consider weaves in which the weft quasipar-

ticle begins and/or ends at a position other than the middle.These possibilities can easily be taken into account by mul-tiplying Uweave��ni��, as defined in �11�, by the appropriatefactors of 1 or 2 on the right and/or left. Thus, for ex-ample, the unitary operation produced by a weave in which


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the weft quasiparticle starts in the top position and ends inthe middle position can be written Uweave��ni��2, where, be-cause of the extra factor of 2, the first braiding operationscarried out by this weave will be 2

n where n is an odd power,n= ±1, ±3 or 5. This will weave the weft quasiparticle fromthe top position to the middle position after which Uweave willsimply continue weaving this quasiparticle eventually endingwith it in the middle position. �Note that by multiplyingUweave on the right by 2, and not 2

−1, we are not requiringthe initial elementary braid to be clockwise, since Uweave mayhave n1=0 and n2=−2 or −4 so that the initial 2 is imme-diately multiplied by 2 to a negative power.� Similarly, theunitary operation produced by a weave in which the weftparticle starts in the top position and ends in the bottomposition can be written 1Uweave��ni��2, and so on.

To find a weave for which the corresponding unitary op-eration Uweave��ni�� approximates a particular desired unitaryoperation, the most straightforward approach is to simplyperform a brute force search over all weaves, i.e., all se-quences �ni� as described above, up to a certain length L, inorder to find the Uweave��ni�� which is closest to the targetoperation. Here we will take as a measure of the distancebetween two operators U and V the operator norm distance

�U ,V�= �U−V� where �O� is the operator norm, defined tobe the square root of the highest eigenvalue of O†O. Again, ifwe are interested in fixing the relative phase of the totalq-spin-1 and total q-spin-0 sectors then we would restrict thewinding of the weaves so that the phases in �10� match thoseof the desired target gate.

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

iX = �0 ii 01

. �12�

If the resulting weave were to be used only for a single-qubitoperation, then we would only require that the weave ap-proximate the upper left 2�2 block of iX up to an overallphase and we would not care about the phase factor appear-ing in the lower right matrix element. There would then beno constraint on the winding of the braid. However, for thisexample we will assume that this weave will be used in atwo-qubit gate construction, for which the overall phaseand/or the phase difference between the total q-spin-1 andtotal 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 satisfyei3�W/5=1 or W=0 �modulo 10�. Results of a brute forcesearch over weaves satisfying this winding requirementwhich approximate iX are shown in Fig. 7. In this figure,ln�1/� is plotted vs braid length L, where is the minimumdistance between Uweave and iX for weaves of length L. It isexpected that, for any such brute force search for weavesapproximating a generic target operation, the length shouldscale with distance according to L� log�1/�, because thenumber of braids grows exponentially with L. The resultsshown in Fig. 7 are consistent with such logarithmic scaling.

All the brute force searches used to find braids in thispaper are straightforward sequential searches, meant mainlyto demonstrate proof of principle. No doubt more sophisti-cated brute force search methods �e.g., bidirectional search�could be used to perform deeper searches resulting in longerand more accurate braids. Nevertheless, the exponentialgrowth in the number of braids with L implies that findingoptimal braids by any brute force search method will rapidlybecome infeasible as L increases. Fortunately one can stillsystematically improve a given braid to any desired accuracyby applying the Solovay-Kitaev algorithm,42,43 which wenow briefly review.

C. Implementation of the Solovay-Kitaev algorithm for braids

The general result of the Solovay-Kitaev theorem tells usthat we can efficiently improve the accuracy of any givenbraid without the need to perform exhaustive brute forcesearches of ever improving accuracy.42,43 The essential ingre-dient in this procedure is an -net—a discrete set of operatorswhich in the present case correspond to finite braids up tosome given length, with the property that for any desiredunitary operator there exists an element of the -net that iswithin some given distance 0 of that operator. Provided 0 issufficiently small, the Solovay-Kitaev algorithm gives us aclever way to pick a finite number of braid segments out ofthe -net and sew them together so that the resulting gate willbe an approximation to the desired gate with improved accu-racy.

The implementation of the Solovay-Kitaev algorithm weuse here follows closely that described in detail in Refs. 44and 45. The first step of this algorithm is to find a braidwhich approximates the desired gate, U, by performing abrute force search over the -net. Let U0 denote the result ofthis search. Since we know that �U0 ,U��0 it follows thatC=UU0

−1 is an operator that is within a distance 0 of theidentity.

The next step is to decompose C as a group commutator.This means that we find two unitary operators A and B for

FIG. 7. ln�1/� vs braid length L for weaves approximating thegate iX. Here is the distance �defined in terms of operator norm�between iX and the unitary transformation produced by a weave oflength L which best approximates it. The line is a guide to the eye.


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which C=ABA−1B−1. The unitary operators A and B are cho-sen so that their action on the computational qubit spacecorresponds to small rotations through the same angle butabout perpendicular axes. For this choice, if A and B are thenapproximated by operators A0 and B0 in the -net, it canreadily be shown that the operator C0=A0B0A0

−1B0−1, will ap-

proximate C to a distance of order 03/2. It follows that the

operator U1=A0B0A0−1B0

−1U0 is an approximation to U withina distance 1�c0

3/2, where c is a constant which determinesthe size of the -net needed to guarantee an improvement inaccuracy.

What we have just described corresponds to one iterationof the Solovay-Kitaev algorithm. Subsequent iterations arecarried out recursively. Thus, at the second level of approxi-mation each search within the -net is replaced by the pro-cedure described above, and so on, so that at the nth level allapproximations are made at the �n−1�st level. The result ofthis recursive process is a braid whose accuracy grows su-perexponentially in n, with the distance to the desired gatebeing of order n��c20��3/2�

nat the nth level of recursion,

while the braid length grows only exponentially in n, withL�5nL0, where L0 is a typical braid length in the initial

-net. Thus, as the distance of the approximate gate from thedesired target gate, , goes to zero, the braid length growsonly polylogarithmically, with L� log��1/� where �=ln 5/ ln�3/2��3.97. While this scaling is, of course, worsethan the logarithmic scaling for brute force searching, it isstill only a polylogarithmic increase in braid length which issufficient for quantum computation. Similararguments44,45can be used to show that the classical com-puter time t required to carry out the Solovay-Kitaev algo-rithm also only scales polylogarithmically in the desired ac-curacy, with t� log�1/� where =ln 3/ ln�3/2��2.71.

It is worth noting that there is a particularly nice featureof this implementation of the Solovay-Kitaev algorithmwhen applied to compiling three-braids. Recall that whencarrying out two-qubit gates it will be crucial to maintain thephase difference between the total q-spin-1 and total q-spin-0sectors of the three-quasiparticle Hilbert space associatedwith a given three-braid, and, according to �10�, this can bedone by fixing the winding of the braid �modulo 10�. Be-cause of the group commutator structure of the Solovay-Kitaev algorithm, the winding of the nth-level approximationUn will be the same as that of the initial approximation U0.This is because all subsequent improvements involve multi-plying this braid by group commutators of the formAnBnAn

−1Bn−1 which automatically have zero winding. The

phase relationship between the total q-spin-1 and totalq-spin-0 sectors is therefore preserved at every level of theconstruction.

Figure 8 shows the application of one iteration of theSolovay-Kitaev algorithm applied to finding a braid whichgenerates a unitary operation approximating iX. The braidlabeled U0 is the result of a brute force search with L=44corresponding to the best approximation shown in Fig. 7.�Note that although this braid is drawn as a sequence ofelementary braid operations, it is topologically equivalent toa weave. In fact precisely this braid, drawn explicitly as aweave, is shown in Fig. 13.� The braids labeled A0 and B0

generate unitary operations which approximate operators Aand B whose group commutator gives UU0

−1 where U= iX.Finally, the braid labeled U1 is the new, more accurate, ap-proximate weave.

VI. TWO-QUBIT GATES

We have seen that single-qubit gates are “easy” in thesense that as long as we braid within an encoded qubit therewill be no leakage errors �the overall q-spin of the group ofthree quasiparticles will remain 1�. Furthermore, the space ofunitary operators acting on the three-quasiparticle Hilbertspace �essentially SU�2�� is small enough to find excellentapproximate braids by performing brute force searches andsubsequent improvement using the Solovay-Kitaev algo-rithm. We now turn to the significantly harder problem offinding braids which approximate entangling two-qubitgates.

A. Divide and conquer approach

Figure 9 depicts six quasiparticles encoding two qubitsand a general braiding pattern. To entangle these qubits, qua-siparticles from one qubit must be braided around quasipar-ticles from the other qubit, and this will inevitably lead toleakage out of the encoded qubit space, �i.e., the overallq-spin of the three quasiparticles constituting a qubit may no

FIG. 8. �Color online� One iteration of the Solovay-Kitaev al-gorithm applied to finding a braid which approximates the operationU= iX. The braid U0 is the result of a brute force search overweaves up to length 44 that best approximate the desired gate U= iX, with an operator norm distance between U and U0 of �8.5�10−4. The braids A0 and B0 are the results of similar brute forcesearches to approximate unitary operations A and B whose groupcommutator satisfies ABA−1B−1 � UU0

−1. The new braid U1=A0B0A0

−1B0−1U0 is then five times longer than U0, and the accuracy

has improved so that the distance to the target gate is now 1�4.2�10−5. Given the group commutator structure of theA0B0A0

−1B0−1 factor, the winding of the U1 braid is the same as the

U0 braid. Note that, when joining braids to form U1, it is possiblethat elementary braid operations from one braid will multiply theirown inverses in another braid, allowing the total braid to be short-ened. Here we have left these “redundant” braids in U1, as thecareful reader should be able to find.


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longer be 1�. Furthermore, the space of all operators actingon the Hilbert space of six quasiparticles is much bigger thanfor three, making brute force searching extremely difficult.Here the unitary operations acting on this space are inSU�5� � SU�8� �up to winding-dependent phase factors as in�10��, which has 87 free parameters as opposed to three forthe three quasiparticle case of SU�2�.

Still, as a matter of principle, it is possible to perform abrute force search of sufficient depth so that it corresponds toa fine enough -net to carry out the Solovay-Kitaev algo-rithm in this larger space.42 This is essentially the programoutlined in Ref. 5 as an “existence proof” that universalquantum computation is possible; however, it is not at allclear that, even if one could do this, it would be the mostefficient procedure for compiling braids. For the sameamount of classical computing power required to directlycompile braids in SU�5� � SU�8�, we believe one can findmuch more efficient braids �in the sense of having a moreaccurate computation with a shorter braid� by breaking theproblem into smaller problems, each consisting of finding aspecific three-braid embedded in the full six-braid space. Aswe have 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. The first ofthese were originally introduced by the authors in Ref. 32and are characterized by the weaving of a pair of quasiparti-cles from one qubit through the quasiparticles, forming thesecond qubit. The second class, presented here for the firsttime, can be carried out by weaving only a single quasipar-ticle from one qubit around one other quasiparticle from thesame qubit, and two quasiparticles from the second qubit.

B. Two-quasiparticle weave construction

We now review the two-qubit gate constructions first dis-cussed in Ref. 32. The basic idea behind these constructionsis illustrated in Fig. 10. This figure shows two qubits and abraiding pattern in which a pair of quasiparticles from thetop qubit �the control qubit� is woven through the quasipar-ticles forming the bottom qubit �the target qubit�. Throughoutthis braiding the pair is treated as a single immutable objectwhich, at the end of the braid, is returned to its originalposition.

If, as in Fig. 10, we choose the pair of weft quasiparticlesto be the two quasiparticles whose total q-spin determinesthe logical state of the qubit, then we refer to this pair as thecontrol pair. We can then immediately see why this construc-tion naturally suggests itself. If the control qubit is in thestate �0L the control pair will have total q-spin 0, and weav-ing this pair through the target qubit will have no effect. Weare thus guaranteed that if the control qubit is in the state �0Lthe identity operation is performed on the target qubit.

The only nontrivial effect of this weaving pattern occurswhen the control qubit is in the state �1L. In this case, thecontrol pair has total q-spin 1 and so behaves as a singleFibonacci anyon. The problem of constructing a two-qubitcontrolled gate then corresponds to finding a weaving patternin which a single Fibonacci anyon weaves through the threequasiparticles of the target qubit, inducing a transition on thisqubit without inducing leakage error out of the computa-tional qubit space, or at least keeping such leakage as smallas required for a particular computation. This reduces theproblem of finding a two-qubit gate to that of finding a weav-ing pattern in which one Fibonacci anyon weaves aroundthree others—a problem involving only four Fibonaccianyons. However, following our divide and conquer philoso-phy, we will further narrow our focus to weaving a singleFibonacci anyon 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 the topposition, and returns to the top position at the end of theweave, with the requirement that the unitary transformationit generates be approximately equal to that produced by mclockwise interchanges of the two warp quasiparticles. Tofind such weaves we perform a brute force search, as out-lined in Sec. V, over sequences �ni� which approximatelysatisfy

2Uweave��ni��2 � 1m. �13�

If both sides of this equation are expressed using �10� itbecomes evident that the winding of any effective braiding

FIG. 9. �Color online� Two encoded qubits and a generic braid.Because quasiparticles are braided outside of their starting qubitsthese braids will generally lead to leakage out of the computationalqubit space, i.e., the q-spin of each group of three quasiparticlesforming these qubits will in general no longer be 1.

FIG. 10. �Color online� A two-qubit gate construction in which apair of quasiparticles from the top �control� qubit is woven throughthe bottom �target� qubit. The mobile pair of quasiparticles is re-ferred to as the control pair and has a total q-spin of 0 if the controlqubit is in the state �0L, and 1 if the control qubit is in the state �1L.Since weaving an object with total q-spin 0 yields the identity op-eration, this construction is guaranteed to result in a transformationof the target qubit state only if the control qubit is in the state �1L.Note that in this and subsequent figures world-lines of mobile qua-siparticles will always be dark blue.


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weave must satisfy W=m �modulo 10�. Since the weft par-ticle starts and ends in the top position, W must be even; thuseffective braiding weaves exist only for even m.

An example of an m=2 effective braiding weave foundthrough a brute force search is shown in Fig. 11. The corre-sponding unitary operation approximates that of interchang-ing the two warp quasiparticles twice to a distance �10−3.�This is a typical distance for a woven three-braid of lengthL�46 which approximates a desired operation—precise dis-tances of approximate weaves are given in the figure cap-tions.� As for all approximate weaves considered here, theSolovay-Kitaev algorithm outlined in Sec. V C can be usedto improve the accuracy of this weave so that can be madeas small as required with only a polylogarithmic increase inlength.

The construction of a two-qubit gate using this effectivebraiding weave is also shown in Fig. 11. In this constructionthe control pair is woven through the top two quasiparticlesof the target qubit using this weave. As described above, ifthe control qubit is in the state �0L, the control pair hasq-spin 0 and the target qubit is unchanged. But, if the controlqubit is in the state �1L, the control pair has q-spin 1 and theaction on the target qubit is approximately equivalent to thatof interchanging the top two quasiparticles twice, with theapproximation becoming more accurate as the length of theeffective braiding weave is increased, either by deeper bruteforce searching or by applying the Solovay-Kitaev algo-rithm. Because this effective braiding all occurs within anencoded qubit, leakage errors can be reduced to zero in thelimit →0. The resulting two-qubit gate is then a controlled-2

2 gate which corresponds to controlled rotation of the targetqubit through an angle of 6� /5.

Unfortunately, due to the even m constraint, it is impos-sible to find an effective braiding gate which corresponds toa controlled � rotation of the target qubit. Such a gate wouldbe equivalent to a controlled-NOT gate up to single-qubitrotations.43 Nonetheless, it is known that any entangling two-qubit gate, when combined with the ability to carry out arbi-trary single-qubit rotations, forms a universal set of quantumgates.46 Thus, the efficient compilation of single-qubit opera-tions described in Sec. V and the effective braiding construc-

tion just given provide direct procedures for compiling anyquantum algorithm into a braid to any desired accuracy.

Although it can be used to form a universal set of gates,this effective braiding construction is still rather restrictive. Itis clearly desirable to be able to directly compile acontrolled-NOT gate into a braid. We now give a constructionwhich can be used to efficiently compile any arbitrary con-trolled rotation of the target qubit—including a controlled-NOT gate. This construction is based on a class of woventhree-braids which we call “injection weaves.”

In an injection weave the weft quasiparticle again starts atthe top position but in this case ends at a different position.At the same time we require that the unitary operation gen-erated by this weave approximate the identity. Thus the ef-fect of an injection weave is to permute the quasiparticlesinvolved without changing any of the underlying q-spinquantum numbers of the system.

Comparing the identity matrix to �10� we see that anythree-braid approximating the identity must have windingW=0 �modulo 10�. The fact that this winding must be evenimplies that the final position of the weft particle must be atthe bottom of the weave. Thus injection weaves correspondto sequences �ni� which approximately satisfy the equation

1Uweave��ni��2 � �1 00 11

. �14�

An injection weave obtained through brute force search isshown in Fig. 12. The unitary operation produced by thisweave approximates the identity operation to a distance

�10−3.

Our two-qubit gate construction based on injection weav-ing is carried out in three steps. In the first step, also shownin Fig. 12, the control pair is woven into the target qubitusing the injection weave. If the control pair has total q-spin1 �the only nontrivial case� the effect of this weave is merelyto replace the middle quasiparticle of the target qubit with

FIG. 11. �Color online� An effective braiding weave, and a two-qubit gate constructed using this weave. The effective braidingweave is a woven three-braid which produces a unitary operationwhich is a distance �2.3�10−3 from that produced by simplyinterchanging the two target particles �1

2�. When the control pair iswoven through the target qubit using this weave the resulting two-qubit gate approximates a controlled-�2

2� gate to a distance

�1.9�10−3 or 1.6�10−3 when the total q-spin of the two qubits is0 or 1, respectively.

FIG. 12. �Color online� An injection weave, and step 1 in ourinjection-based gate construction. The box labeled I represents anideal �infinite� injection weave which is approximated by the weaveshown to a distance �1.5�10−3. In step 1 of our gate construc-tion, this injection weave is used to weave the control pair into thetarget qubit. If the control qubit is in the state �1L then a=1 and theresult is to produce a target qubit with the same quantum numbersas the original, but with its middle quasiparticle replaced by thecontrol pair.


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the control pair. Because the unitary operation approximatedby the injection weave is the identity, in the →0 limit thisinjection is accomplished without changing any of the q-spinquantum numbers. The injected target qubit is therefore �ap-proximately� in the same quantum state as the original targetqubit.

In the second step of our construction, illustrated in Fig.13, we carry out an operation on the injected target qubit bysimply weaving the control pair within the target. Becausefor a=1 all of this weaving takes place within the injectedtarget qubit, there will be no leakage error �again, strictlyspeaking, only in the limit of an exact injection weave�. Theonly constraint on this weave is that the control pair mustboth start and end in the middle position, and so it must haveeven winding.

If our goal is to produce a gate which is equivalent to acontrolled-NOT gate up to single-qubit rotations then we mustapply a � rotation to the target qubit. Unfortunately, thiscannot be accomplished by any finite weave with even wind-ing, so we must again consider approximate weaves. Figure

13 shows the control pair being woven through the injectedtarget qubit using a weave found by a brute force searchwhich approximates a particular � rotation—the operator iXdefined in �12�—to a distance �10−3 �this is, in fact, thesame weave shown at the top of Fig. 8�.

The third step in our construction is the extraction of thecontrol pair from the target qubit. This is accomplished, asshown in Fig. 14, by applying the inverse of the injectionweave to the control pair. The effect of this extraction is torestore the control qubit to its original state, and replace thecontrol pair inside the target qubit with the quasiparticlewhich originally occupied that position.

The full construction is summarized in Fig. 15, whichprovides a recipe for compiling a controlled-NOT gate into atwo-quasiparticle weave. A quantum circuit showing that acontrolled-NOT gate is equivalent to a controlled-�iX� gateand a single-qubit operation is shown in the top part of thefigure. The single-qubit operation can be compiled to what-ever accuracy is required following Sec. V, and thecontrolled-�iX� gate can be decomposed into injection, iX,and inverse injection operations, as is also shown in the top

FIG. 13. �Color online� A weave that approximates iX �see Eq.�12��, and step 2 in our injection-based construction. The box la-beled iX represents an ideal �infinite� iX weave which is approxi-mated by the weave shown to a distance =8.5�10−4 �this is thesame weave that appears at the top of Fig. 8�. In step 2 of our gateconstruction the control pair is woven within the injected targetqubit, following this weave, in order to carry out an approximate iXgate when a=1, as shown.

FIG. 14. �Color online� An inverse injection weave and step 3 inour injection-based construction. The box labeled I−1 represents anideal �infinite� inverse injection weave which is approximated bythe inverse of the injection weave shown in Fig. 12, again to adistance �1.5�10−3. This weave is used to extract the controlpair out of the injected target qubit and return it to the control qubit,as shown.

FIG. 15. �Color online� Injection-weave based compilation of a controlled-NOT gate into a braid. A controlled-NOT gate can be expressedas a controlled-�iX� gate and a single-qubit operation R�−� /2ẑ�=exp�i�z /4� acting on the control qubit. The single-qubit rotation can becompiled following the procedure outlined in Sec. V, and the controlled-�iX� gate can be decomposed into ideal injection �I�, iX, and inverseinjection �I−1� operations which can be similarly compiled. The full approximate controlled-�iX� braid obtained by replacing I, iX, and I−1with the weaves shown in the previous three figures is shown at bottom. The resulting gate approximates a controlled-�iX� gate to a distance

�1.8�10−3 and 1.2�10−3 when the total q-spin of the two qubits is 0 or 1, respectively.


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part of the figure. These operations can then all be similarlycompiled following Sec. V.

The full braid shown at the bottom of Fig. 15 correspondsto using the approximate woven three-braids shown in Figs.12–14 to carry out a controlled-�iX� gate. In this braid, if thecontrol qubit is in the state �0L the control pair has totalq-spin 0 and the resulting unitary transformation is exactlythe identity. However, if the control qubit is in the state �1Lthe control pair has total q-spin 1 and behaves like a singleFibonacci anyon. This pair is then woven into the target qubitusing an injection weave, woven within the target in order tocarry out the iX operation, and finally woven out of the targetand back into the control qubit using the inverse of the in-jection weave. The resulting gate is therefore a controlled-�iX� gate.

By replacing the iX weave with an even winding weavewhich carries out an arbitrary operation U this constructionwill give a controlled-U gate. The only restriction on U isthat its overall phase must be consistent with �10� with evenwinding W. However, this phase can be easily set to anydesired value by applying the appropriate single-qubit rota-tion to the control qubit, as in Fig. 15.

Finally, note that at no point in either the effective braid-ing or injection weave constructions described above did wemake reference to the total q-spin of the two qubits involved.It follows that, in the limit of exact effective braiding orinjection weaves, the action of the corresponding two-qubitgates on the computational qubit space does not depend onthe state of the external fusion space associated with theq-spin-1 quantum numbers of each qubit �see Sec. IV�. Thesegates will therefore not entangle the computational qubitspace with this external fusion space.

C. One-quasiparticle weave constructions

We now show that two-qubit gates can be carried out withonly a single mobile quasiparticle. This possibility followsfrom the general result of Ref. 41 that for any system ofnon-Abelian quasiparticles in which general braids are uni-versal for quantum computation �such as Fibonacci anyons�,single quasiparticle weaves are universal as well. However,the “proof of principle” weaves constructed in that workwere extremely inefficient—involving a huge number of ex-cess operations. Here we show how to efficiently construct asingle-quasiparticle weave corresponding to a controlled-NOT gate �up to single-qubit rotations�.

Our construction is based on a class of weaves that aresimilar to injection weaves in that they can be used to swaptwo q-spin-1 objects—where one object is a pair of Fi-bonacci anyons with total q-spin 1 and the other object is asingle Fibonacci anyon—while acting effectively as the iden-tity operation so that none of the other q-spin quantum num-bers of the system are disturbed. However, unlike injectionweaves, this new class of weaves accomplishes this swapwithout moving the pair as a single object, and in fact can becarried out by moving just one quasiparticle.

The class of weaves we seek are those that approximatethe transformation

U„�•, • �a, • …c = ei�„•,�•, • �a…c, �15�

where � is an overall �irrelevant� phase that does not dependon a or c. The relevant case for showing the similarity with

injection is when a=1, for which the initial and final states in�15� consist of two q-spin-1 objects—a single Fibonaccianyon and a pair of Fibonacci anyons with total q-spin 1. Ifboth these objects are represented as single Fibonacci anyonsthen �15� can be written U�• , • �c=ei��• , • �c. In this represen-tation U therefore acts effectively as the identity operation�times an irrelevant phase�, similar to injection.

Using the F matrix �5� to expand the right-hand side of�15� in the (�• , • � , • ) basis yields

U„�•, • �a, • …c = ei��b

Fabc„�•, • �b, • …c. �16�

Comparing this with the action of a unitary operation U withmatrix representation

U = �U001 U01

1

U101 U11

1

U110 , �17�

on the state (�• , • �a , • )c,

U„�•, • �a, • …c = �b

Uabc„�•, • �b, • …c, �18�

we see that the matrix representation of the U we seek isprecisely the F matrix �up to a phase�: U=ei�F. While the Fmatrix describes a “passive” operation, i.e., a change of ba-sis, the operator U can be viewed as an “active” F operationwhich acts directly on the states of the Hilbert space. Notethat, since F=F−1, we also have

U„•,�•, • �a…c = ei�„�•, • �a, • …c. �19�

We will refer to weaves that approximate the operation�15� �and thus also �19�� as F weaves. As we have seen, theunitary operation U produced by an F weave need only ap-proximate the F matrix �5� up to an overall irrelevant phase.To be consistent with �10� this phase must be −1, as can beseen by writing the matrix −F as

FIG. 16. �Color online� An F weave, and step 1 of ourF-weave-based two-qubit gate construction. The box labeled F rep-resents an ideal �infinite� F weave which is approximated by theweave shown to a distance �3.1�10−3. Applying the F weave tothe initial two-qubit state, as shown, produces an intermediate statewith q-spins labeled a and b� which depend simply on a and b—theinitial states of the two qubits �see Table I�.


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− F = �±i� ±i� ± i�

±i� � i� �− 1

, �20�where a factor of ±i has been pulled out of the upper left 2�2 block, leaving an SU�2� matrix �det=�2+�=1�. Compar-ing �20� with �10�, it is also evident that any F weave musthave winding W=5 �modulo 10�, which is necessarily odd.

The fact that F weaves must have an odd number of wind-ings implies that if the weft quasiparticle starts at the topposition of the weave it must end at the middle position. Forthis choice the F weave must then approximately satisfy theequation

Uweave��ni��2 � − F . �21�

The result of a brute force search for an F weave whichapproximates the operation −F to a distance �10−3 isshown in Fig. 16.

The first step in our single-quasparticle weave construc-tion is the application of an F weave to two qubits, alsoshown in Fig. 16. Note that in this figure for convenience wehave made a change of basis on the bottom qubit, so that thepair which determines its state �the control pair� consists ofthe top two quasiparticles within it rather than the bottomtwo. There is no loss of generality in doing so since this justcorresponds to a single-qubit rotation on the bottom qubit.

With this basis choice the initial state of the two qubits isdetermined by the q-spins of their respective control pairswhich are indicated in Fig. 16 as a �top qubit� and b �bottomqubit�. After carrying out the F weave, taking the middlequasiparticle of the top qubit as the weft quasiparticle, andweaving it around both the bottom quasiparticle of the topqubit and the top quasiparticle of the bottom qubit, the re-sulting state �again, strictly speaking, only in the limit of anexact F weave� is shown at the end of the two-qubit weave inFig. 16. From �19� it follows that the newly positioned weftquasiparticle and the quasiparticle beneath will have totalq-spin a. When the quasiparticle beneath these two is alsoincluded, the three quasiparticles form what we will refer toas the intermediate state (• , �• , • �a)b�, where the total q-spinof 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, the ef-fect of the F weave is then similar to that of the injectionweave from the previous construction—it replaces the top-

most quasiparticle in the bottom qubit with a pair of quasi-particles with q-spin 1, and the bottom-most pair of quasi-particles in the top qubit �which also has total q-spin 1� witha single quasiparticle, without changing any of the otherq-spin quantum numbers of the system. In the limit of anideal F weave, this means that the b quantum number doesnot change after this swap and so b�=b. The case a=0 issimpler, since in this case the intermediate state is(• , �• , • �0)b� for which the fusion rules �2� imply b�=1, re-gardless of the value of b. The resulting dependence of b� ona and b is summarized in Table I.

Having used the F weave to create the intermediate state(• , �• , • �a)b�, the next step in our construction is the applica-tion of a weave which performs an operation on this statewhich does not change a and b� but which does yield an a-and b�-dependent phase factor. After carrying out such aweave, which we will refer to as a phase weave, we can thenapply the inverse of the F weave to restore the two qubits totheir initial states a and b.

For any phase weave we will require that the weft quasi-particle both start and end in the top position so that whenwe join it to the F weave and its inverse there will be a singleweft quasiparticle throughout the entire gate construction.

TABLE I. Values of b� for different values of a and b afterapplying the F weave as shown in Fig. 16, and the phase applied tothe resulting state by a phase weave with zero winding. The valueof b� is determined by the fact that b�=1 when a=0 and b�=b whena=1, as shown in the text.

a b b� Phase factor

0 0 b�=1 1 ei�

0 1 1 ei�

1 0 b�=b 0 1

1 1 1 e−i�

FIG. 17. �Color online� A phase weave with �=� �see text�which gives a � phase shift to the intermediate state when b�=1,and step 2 of our F-weave-based construction. The box labeled Prepresents an ideal �infinite� �=� phase weave which is approxi-mated by the weave shown to a distance �1.9�10−3. Applyingthis phase weave to the intermediate state created by the F weave,as shown, results in a b�-dependent � phase shift �see Table I with�=��.

FIG. 18. �Color online� An inverse F weave and step 3 in our Fweave construction. The box labeled F−1 is an ideal �infinite� in-verse F weave which is approximated by the inverse of the F weaveshown in Fig. 16, again to a distance �3.1�10−3. By applyingthe inverse F weave to the state obtained after applying the phaseweave, as shown, the two qubits are returned to their initial states,but now with an a- and b-dependent phase factor �see Table I�.


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The phase weave must therefore have even winding, andwith no loss of generality we can consider the case for whichthe winding satisfies W=0 �modulo 10�. The unitary opera-tion produced by such a phase weave must then approxi-mately satisfy the equation

2Uweave��ni��2 � F�ei� 0

0 e−i�

1

F−1, �22�

where the F matrices are needed to change the Hilbert spacebasis from that in which the operation produced by the phasebraid must be diagonal �the (• , �• , • �) basis�, to that in whichthe 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-NOT gateup to single-qubit rotations. The result of a brute force searchfor such a phase weave which approximates the desired op-eration to a distance �10−3 is shown in Fig. 17. This figurealso shows the action of the phase weave on the intermediatestate produced in Fig. 16. In this weave, the weft quasiparti-cle is now woven through the two quasiparticles beneath it,and returns to its original position. Because the phase weaveproduces a diagonal operation in the basis shown for theintermediate state, it does not change the values of a and b�.Its only effect is to give a phase factor of ei� to the state witha=0 �which necessarily has b�=1� and e−i� to the state witha=1 and b�=1. The state with a=1 and b�=0 is unchanged.These phase factors are also shown in Table I.

The final step in this construction is to perform the inverseof the F weave to return the two qubits to their originalstates. This is shown in Fig. 18. In the limit of exact F andphase weaves, the resulting operation on the computationalqubit 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 e−i�

. �23�

If we take the top qubit to be the control qubit and the bot-tom qubit to be the target qubit, then this gate corresponds,up to an irrelevant overall phase, to a controlled-�e−i3�/2ei�z/2� operation. For the case �=� this is acontrolled-�−Z� gate �where Z=z�, i.e., a controlled-phase

FIG. 19. �Color online� F-weave-based compilation of a controlled-NOT gate into a braid. A controlled-NOT gate is equivalent to acontrolled-�−Z� gate with the single-qubit operation R�� /2ŷ�=exp�−i�y /4� and its inverse applied to the target qubit before and after thecontrolled-�−Z�. Again, the single-qubit operations can be trivially compiled, and the controlled-�−Z� gate decomposed into ideal F, phase�P�, and inverse F �F−1� weaves which can be similarly compiled. The full approximate controlled-�−Z� weave obtained by replacing F, P,and F−1 with the approximate weaves shown in the previous three figures is shown at the bottom. The resulting gate approximates acontrolled-�−Z� to a distance �4.9�10−3 and 3.2�10−3 when the total q-spin of the two qubits is 0 or 1, respectively.

FIG. 20. �Color online� Two four-quasiparticle qubits and abraiding pattern in which only two quasiparticles from each qubitare braided. Here the quasiparticles are SU�2�k excitations withq-spin 1/2. The state of the top qubit is determined by the totalq-spin of the quasiparticle pairs labeled a and the state of the bot-tom qubit is determined by the total q-spin of the quasiparticle pairslabeled b. The overall q-spin of the four braided quasiparticles is d�a dashed oval is used because when a=b=1 these quasiparticleswill not be in a q-spin eigenstate�. For this braid to produce noleakage errors, the unitary operation it generates must be diagonalin a and b, though it can, of course, result in an a- and b-dependentphase factor. For k�3, d can take the values 0, 1, or 2, while fork=3 the only allowed values for d are 0 and 1. The existence of thed=2 state for k�3 makes it impossible to carry out an entanglingtwo-qubit gate by braiding only four quasiparticles �see text�.


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gate, which, up to single-qubit rotations, is equivalent to acontrolled-NOT gate.

The full F-weave-based gate construction is summarizedin Fig. 19. A quantum circuit showing a controlled-NOT gatein terms of a controlled-�−Z� gate and two single-qubit op-erations is shown in the top part of the figure. As in ourinjection based construction, the single-qubit operations canbe compiled to whatever accuracy is required following theprocedure outlined in Sec. V. The controlled-�−Z� gate canthen be decomposed into ideal F, phase, and inverse Fweaves as is also shown in the top part of the figure. Woventhree-braids which approximate these operations can then becompiled to whatever accuracy is required, again followingSec. V. The full controlled-�−Z� weave corresponding to us-ing 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 constructions de-scribed in Sec. VI B, we at no point made reference to thetotal q-spin of the two qubits involved. Thus, in the limit ofexact F and phase weaves, the action of the two-qubit gatesconstructed here will not entangle the computational qubitspace with the external fusion space associated with theq-spin 1 quantum numbers of each qubit.

VII. WHAT IS SPECIAL ABOUT k=3?

All of the gate constructions discussed in this paper ex-ploit the fact that the braiding and fusion properties of a pairof Fibonacci anyons are either trivial if their total q-spin is 0,or equivalent to those of a single Fibonacci anyon if theirtotal q-spin is 1. The fact that these are the only two possi-bilities is a special property of the Fibonacci anyon model,and hence also the SU�2�3 model, given their eff

Topological quantum compilingweb2.physics.fsu.edu/~bonesteel/papers/prb07.pdf“partially” topological quantum computation using a mixture of topological and nontopological gates

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