Non-Abelian Anyons and Topological Quantum Computation

73 Pages·2008·1.16 MB·English

Non-Abelian Anyons and Topological Quantum Computation
ChetanNayak1,2,StevenH.Simon3,AdyStern4,MichaelFreedman1,SankarDasSarma5,
1Microsoft Station Q, University of California, Santa Barbara, CA 93108
2Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547
3Alcatel-Lucent, Bell Labs, 600 Mountain Avenue, Murray Hill, New Jersey 07974
4Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel
5DepartmentofPhysics,UniversityofMaryland,CollegePark,MD20742
Topologicalquantumcomputationhasrecentlyemergedasoneofthemostexcitingapproachestoconstructinga
fault-tolerantquantumcomputer.Theproposalreliesontheexistenceoftopologicalstatesofmatterwhosequasi-
particleexcitationsareneitherbosonsnorfermions,butareparticlesknownasNon-Abeliananyons,meaningthat
theyobeynon-Abelianbraidingstatistics. Quantuminformationisstoredinstateswithmultiplequasiparticles,
whichhaveatopologicaldegeneracy.Theunitarygateoperationswhicharenecessaryforquantumcomputation
arecarriedoutbybraidingquasiparticles,andthenmeasuringthemulti-quasiparticlestates. Thefault-tolerance
ofatopologicalquantumcomputerarisesfromthenon-localencodingofthestatesofthequasiparticles,which
makesthemimmunetoerrorscausedbylocalperturbations. Todate,theonlysuchtopologicalstatesthought
tohavebeenfoundinnaturearefractionalquantumHallstates,mostprominentlytheν = 5/2state,although
severalotherprospectivecandidates havebeenproposedinsystemsasdisparateasultra-coldatomsinoptical
latticesandthinfilmsuperconductors. Inthisreviewarticle,wedescribecurrentresearchinthisfield,focusing
onthegeneraltheoreticalconceptsofnon-Abelianstatisticsasitrelatestotopologicalquantumcomputation,on
understandingnon-AbelianquantumHallstates,onproposedexperimentstodetectnon-Abeliananyons,andon
proposedarchitecturesforatopologicalquantumcomputer.Weaddressboththemathematicalunderpinningsof
topologicalquantumcomputationandthephysicsofthesubjectusingtheν=5/2fractionalquantumHallstate
asthearchetypeofanon-Abeliantopologicalstateenablingfault-tolerantquantumcomputation.
Contents A. ν=5/2QubitsandGates 52
B. FibonacciAnyons:aSimpleExamplewhichisUniversalfor
I. Introduction 1 QuantumComputation 53
C. UniversalTopologicalQuantumComputation 57
II. BasicConcepts 3 D. Errors 59
A. Non-AbelianAnyons 3
1. Non-AbelianBraidingStatistics 3 V. FutureChallengesforTheoryandExperiment 60
2. EmergentAnyons 6
B. TopologicalQuantumComputation 7 Acknowledgments 64
1. BasicsofQuantumComputation 7
2. Fault-TolerancefromNon-AbelianAnyons 9 A. ConformalFieldTheory(CFT)forPedestrians 64
C. Non-AbelianQuantumHallStates 11
1. RapidReviewofQuantumHallPhysics 11 References 65
2. PossibleNon-AbelianStates 13
3. InterferenceExperiments 16
4. AFractionalQuantumHallQuantumComputer 18
I. INTRODUCTION
5. PhysicalSystemsandMaterialsConsiderations 19
D. OtherProposedNon-AbelianSystems 20
In recent years, physicists’ understanding of the quantum
III. TopologicalPhasesofMatterandNon-AbelianAnyons 22 properties of matter has undergone a major revolution pre-
A. TopologicalPhasesofMatter 23
cipitatedbysurprisingexperimentaldiscoveriesandprofound
1. Chern-SimonsTheory 23
theoreticalrevelations. Landmarksincludethediscoveriesof
2. TQFTsandQuasiparticleProperties 26
B. Superconductorswithp+ippairingsymmetry 29 the fractional quantum Hall effect and high-temperature su-
1. VorticesandFermionZeroModes 29 perconductivity and the advent of topological quantum field
2. TopologicalPropertiesofp+ipSuperconductors 31 theories. At the same time, new potential applications for
C. Chern-SimonsEffectiveFieldTheories,theJonesPolynomial,
quantummatterburstonthescene,punctuatedbythediscov-
andNon-AbelianTopologicalPhases 33
eriesofShor’sfactorizationalgorithmandquantumerrorcor-
1. Chern-SimonsTheoryandLinkInvariants 33
2. CombinatorialEvaluationofLinkInvariantsand rectionprotocols. Remarkably,therehasbeenaconvergence
QuasiparticleProperties 35 betweenthesedevelopments. Nowhereisthismoredramatic
D. Chern-SimonsTheory,ConformalFieldTheory,andFractional thanintopologicalquantumcomputation,whichseekstoex-
QuantumHallStates 37
ploittheemergentpropertiesofmany-particlesystemstoen-
1. TheRelationbetweenChern-SimonsTheoryandConformal
codeandmanipulatequantuminformationinamannerwhich
FieldTheory 37
2. QuantumHallWavefunctionsfromConformalFieldTheory 39 isresistanttoerror.
E. EdgeExcitations 43 It is rare for a new scientific paradigm, with its attendant
F. InterferometrywithAnyons 46 conceptsand mathematicalformalism, to developin parallel
G. LatticeModelswithP,T-InvariantTopologicalPhases 49
withpotentialapplications,withalloftheirdetailedtechnical
IV. QuantumComputingwithAnyons 52 issues. However,the physicsoftopologicalphasesofmatter
2
isnotonlyevolvingalongsidetopologicalquantumcomputa- icalsystem‘condenses’intoanon-Abeliantopologicalphase.
tion but is even informed by it. Therefore, this review must In Section III, we describe the universal low-energy, long-
necessarily be rather sweepingin scope, simply to introduce distancephysicsofsuchphases.Wealsodiscusshowtheycan
theconceptsofnon-Abeliananyonsandtopologicalquantum be experimentallydetected in the quantum Hall regime, and
computation,theirinter-connections,andhowtheymaybere- whentheymightoccurinotherphysicalsystems. Ourfocus
alized in physical systems, particularly in several fractional isonasequenceofuniversalityclassesofnon-Abeliantopo-
quantum Hall states. (For a popular account, see Collins, logicalphases,associated withSU(2) Chern-Simonstheory
k
2006;foraslightlymoretechnicalone,seeDasSarmaetal., whichwedescribeinsectionIII.A.Thefirstinterestingmem-
2006a.)Thisexpositionwilltakeusonatourextendingfrom ber of this sequence, k = 2, is realized in chiral p-wave
knottheoryandtopologicalquantumfieldtheorytoconformal superconductorsand in the leading theoreticalmodelfor the
fieldtheoryandthequantumHalleffecttoquantumcomputa- ν = 5/2 fractionalquantumHall state. Section III.B shows
tionandallthewaytothephysicsofgalliumarsenidedevices. how this universality class can be understood with conven-
Thebodyofthispaperiscomposedofthreeparts,Sections tionalBCStheory.InsectionIII.C,wedescribehowthetopo-
II, III, and IV. Section II is rather general, avoids technical logicalpropertiesoftheentiresequenceofuniversalityclasses
details, and aims to introduceconceptsat a qualitative level. (of which k = 2 is a special case) can be understood using
Section II should be of interest, andshould be accessible, to Witten’scelebratedconnectionbetweenChern-Simonstheory
allreaders. InSectionIIIwe describethetheoryoftopolog- andtheJonespolynomialofknottheory. InsectionIII.D,we
icalphasesinmoredetail. InSectionIV, wedescribehowa describe an alternate formalism for understanding the topo-
topologicalphasecanbeusedasaplatformforfault-tolerant logical properties of Chern-Simons theory, namely through
quantumcomputation. Thesecondandthirdpartsareproba- conformal field theory. The discussion revolves around the
blyofmoreinteresttotheorists,experiencedresearchers,and applicationofthisformalismtofractionalquantumHallstates
thosewhohopetoconductresearchinthisfield. and explains how non-Abelian quantum Hall wavefunctions
canbe constructedwith conformalfieldtheory. AppendixA
SectionII.A.1beginsbydiscussingtheconceptofbraiding
givesahighly-condensedintroductiontoconformalfieldthe-
statistics in 2+1-dimensions. We define the idea of a non-
ory. InSectionIII.E,wediscussthegaplessedgeexcitations
Abelian anyon, a particle exhibiting non-Abelian braiding
whichnecessarilyaccompanychiral(i.e. parity,P andtime-
statistics. Section II.A.2 discusses how non-Abelian anyons
reversal T-violating) topological phases. These excitations
can arise in a many-particlesystem. We then review the ba-
are useful for interferometry experiments, as we discuss in
sicideasofquantumcomputation,andtheproblemsoferrors
SectionIII.F.Finally,inSectionIII.G,wediscusstopological
anddecoherenceinsectionII.B.1. Thosefamiliarwithquan-
phaseswhichdonotviolateparityandtime-reversalsymme-
tumcomputationmaybeabletoskipmuchofthissection.We
tries. These phases emergein modelsof electrons, spins, or
explaininsectionII.B.2hownon-Abelianstatisticsnaturally
bosons on lattices which could describe transition metal ox-
leadstotheideaoftopologicalquantumcomputation,andex-
ides,Josephsonjunctionarrays,orultra-coldatomsinoptical
plainwhyitisagoodapproachtoerror-freequantumcompu-
lattices.
tation. In section II.C, we briefly describe the non-Abelian
quantum Hall systems which are the most likely arena for
observing non-Abelian anyons (and, hence, for producing a
InSectionIV,wediscusshowquasiparticlesintopological
topological quantum computer). Section II.C.1 gives a very
phases can be used for quantum computation. We first dis-
basic review of quantum Hall physics. Experts in quantum
cuss the case of SU(2) , which is the leading candidate for
Hallphysicsmaybeabletoskipmuchofthissection.Section 2
theν = 5/2fractionalquantumHallstate. We showinSec-
II.C.2introducesnon-AbelianquantumHallstates. Thissec-
tionIV.Ahowqubitsandgatescanbemanipulatedinagated
tionalsoexplainstheimportance(andsummarizestheresults)
GaAsdevicesupportingthisquantumHallstate. We discuss
ofnumericalworkinthisfieldfordeterminingwhichquantum
whyquasiparticlebraidingaloneisnotsufficientforuniversal
Hallstatesare(ormightbe)non-Abelian. SectionII.C.3de-
quantumcomputationandhowthislimitationoftheν = 5/2
scribessomeoftheproposedinterferenceexperimentswhich
state can be circumvented. Section IV.B discusses in detail
may be able to distinguish Abelian from non-Abelian quan-
how topological computations can be performed in the sim-
tum Hall states. Section II.C.4 shows how qubits and ele-
plestnon-Abeliantheorythatiscapableofuniversaltopolog-
mentarygatescanberealizedinaquantumHalldevice. Sec-
ical quantum computation, the so-called “Fibonacci-Anyon”
tion II.C.5 discusses some of the engineering issues associ-
theory. In IV.C, we show that the SU(2) theories support
ated with the physical systems where quantum Hall physics k
universaltopologicalquantumcomputationforallintegersk
is observed. In section II.D we discuss some of the other,
exceptk =1,2,4.InIV.D,wediscussthephysicalprocesses
non-quantum-Hall systems where it has been proposed that
whichwillcauseerrorsinatopologicalquantumcomputer.
non-Abeliananyons(andhencetopologicalquantumcompu-
tation)mightoccur.
SectionsIIIand IV are still written to be accessible to the Finally, we briefly conclude in section V. We discuss
broadest possible audiences, but they should be expected to questionsfortheimmediatefuture,primarilycenteredonthe
be somewhathardergoingthanSection II. SectionIII intro- ν = 5/2 and ν = 12/5 fractionalquantum Hall states. We
ducesthetheoryoftopologicalphasesin detail. Topological alsodiscussabroadersetofquestionrelatingtonon-Abelian
quantumcomputationcanonlybecomearealityifsomephys- topologicalphasesandfault-tolerantquantumcomputation.
3
II. BASICCONCEPTS Consequently,thenotionofawindingofoneparticlearound
another in two dimensions is well-defined. Then, when two
A. Non-AbelianAnyons particlesareinterchangedtwice ina clockwisemanner,their
trajectoryinvolvesanon-trivialwinding,andthesystemdoes
1. Non-AbelianBraidingStatistics notnecessarilycomebacktothesamestate. Thistopological
difference between two and three dimensions, first realized
Quantum statistics is one of the basic pillars of the quan- byLeinaasandMyrheim,1977andbyWilczek,1982a,leads
tum mechanical view of the world. It is the property which toaprofounddifferenceinthepossiblequantummechanical
distinguishes fermions from bosons: the wave function that properties,atleastasa matterof principle,forquantumsys-
describesa system of manyidenticalparticlesshould satisfy temswhenparticlesareconfinedto2+1D(seealsoGoldin
thepropersymmetryunderthe interchangeofanytwo parti- etal.,1981andWu,1984). (Asanaside,wementionthatin
cles. In3spatialdimensionandonetimedimension(3+1D) 1+1D, quantumstatistics isnotwell-definedsinceparticle
thereareonlytwopossiblesymmetries—thewavefunction interchangeisimpossiblewithoutoneparticlegoingthrough
ofbosonsissymmetricunderexchangewhilethatoffermions another,andbosonswithhard-corerepulsionareequivalentto
isanti-symmetric. Onecannotoveremphasize,ofcourse,the fermions.)
importance of the symmetry of the wavefunction, which is Supposethatwehavetwoidenticalparticlesintwodimen-
therootofthePauliprinciple,superfluidity,themetallicstate, sions. Then when one particle is exchanged in a counter-
Bose-Einsteincondensation,andalonglistofotherphenom- clockwise manner with the other, the wavefunction can
ena. changebyanarbitraryphase,
The limitation to one of two possible types of quantum
ψ(r ,r ) eiθψ(r ,r ) (1)
symmetry originates from the observation that a process in 1 2 1 2
→
which two particles are adiabatically interchanged twice is
The phase need not be merely a sign because a second
equivalent to a process in which one of the particles is adi- ±
counter-clockwiseexchangeneed notlead backto the initial
abaticallytakenaroundtheother. Since,inthreedimensions,
statebutcanresultinanon-trivialphase:
wrappingoneparticleallthewayaroundanotheristopolog-
ically equivalent to a process in which none of the particles ψ(r1,r2) e2iθψ(r1,r2) (2)
→
move at all, the wave function should be left unchanged by
Thespecialcasesθ =0,πcorrespondtobosonsandfermions,
two such interchanges of particles. The only two possibili-
respectively. Particleswithothervaluesofthe‘statisticalan-
ties are for the wavefunction to change by a sign under a
± gle’θ arecalledanyons(Wilczek,1990). Wewilloftenrefer
singleinterchange,correspondingto thecasesofbosonsand
tosuchparticlesasanyonswithstatisticsθ.
fermions,respectively.
LetusnowconsiderthegeneralcaseofN particles,where
We can recast this in path integrallanguage. Suppose we
a more complex structure arises. The topological classes of
consider all possible trajectories in 3+1 dimensions which
trajectories which take these particles from initial positions
take N particles from initial positions R , R , ..., R at
1 2 N R ,R ,...,R attimet tofinalpositionsR ,R ,...,R
time t to final positions R , R , ..., R at time t . If the 1 2 N i 1 2 N
i 1 2 N f attimet areinone-to-onecorrespondencewiththeelements
particles are distinguishable, then there are no topologically f
of the braid group . An element of the braid group can
non-trivial trajectories, i.e. all trajectories can be continu- BN
bevisualizedbythinkingoftrajectoriesofparticlesasworld-
ously deformed into the trajectory in which the particles do
lines (or strands) in 2+1 dimensional space-time originating
not move at all (straight lines in the time direction). If the
atinitialpositionsandterminatingatfinalpositions,asshown
particles are indistinguishable, then the different trajectories
inFigure1. Thetimedirectionwillberepresentedvertically
fallintotopologicalclassescorrespondingtotheelementsof
on the page, with the initial time at the bottom and the final
the permutation group S , with each element of the group
N time at the top. An elementof the N-particlebraidgroupis
specifyinghowtheinitialpositionsarepermutedtoobtainthe
anequivalenceclassof suchtrajectoriesupto smoothdefor-
finalpositions.Todefinethequantumevolutionofsuchasys-
mations. To represent an element of a class, we will draw
tem, we mustspecifyhowthepermutationgroupactsonthe
the trajectories on paper with the initial and final points or-
statesofthe system. Fermionsandbosonscorrespondto the
deredalonglinesattheinitialandfinaltimes. Whendrawing
only two one-dimensional irreducible representations of the
the trajectories, we must be careful to distinguish when one
permutationgroupofN identicalparticles.1
strandpassesoverorunderanother,correspondingtoaclock-
Two-dimensional systems are qualitatively different from
wise or counter-clockwise exchange. We also require that
three(andhigherdimensions)inthisrespect. Aparticleloop
anyintermediatetimeslicemustintersectN strands. Strands
that encircles another particle in two dimensions cannot be
cannot ‘double back’, which would amount to particle cre-
deformedtoapointwithoutcuttingthroughtheotherparticle.
ation/annihilationatintermediatestages. Wedonotallowthis
because we assume that the particle number is known. (We
will considerparticle creation/annihilationlater in this paper
whenwediscussfieldtheoriesofanyonsand,fromamathe-
1Higher dimensional representations of the permutation group, known as
maticalperspective,whenwediscusstheideaofa“category”
‘parastatistics’, canalways bedecomposedintofermionsorbosonswith
in section IV below.) Then, the multiplication of two ele-
anadditionalquantumnumberattachedtoeachparticle(Doplicheretal.,
1971,1974). ments of the braid group is simply the successive execution
4
which one particle winds around the other an integer num-
beroftimes. Thesetopologicalclassescorrespondtotheele-
mentsofthe‘pure’braidgroup,whichisthesubgroupofthe
σ σ
1 2 braidgroupcontainingonlyelementswhichbringeachparti-
cle back to its own initial position, notthe initial positionof
time oneoftheotherparticles. Therichnessofthe braidgroupis
thekeyfactenablingquantumcomputationthroughquasipar-
ticlebraiding.
Todefinethequantumevolutionofasystem,wemustnow
specifyhowthebraidgroupactsonthestates ofthe system.
=
Thesimplestpossibilitiesareone-dimensionalrepresentations
6
ofthebraidgroup. Inthesecases, thewavefunctionacquires
a phase θ when one particle is taken around another, analo-
gous to Eqs. 1, 2. The special cases θ = 0,π are bosons
and fermions, respectively, while particles with other values
of θ are anyons (Wilczek, 1990). These are straightforward
many-particlegeneralizationsofthetwo-particlecaseconsid-
ered above. An arbitrary element of the braid group is rep-
resented by the factor eimθ where m is the total number of
times that one particle winds around another in a counter-
= clockwisemanner(minusthenumberoftimesthataparticle
winds around another in a clockwise manner). These repre-
sentationsare Abelian since the orderof braidingoperations
inunimportant.However,theycanstillhaveaquiterichstruc-
ture since there can be n differentparticle species with pa-
s
rametersθ ,wherea,b=1,2,...,n ,specifyingthephases
FIG. 1 Top: The two elementary braid operations σ and σ on ab s
1 2 resultingfrombraidingaparticleoftypeaaroundaparticleof
three particles. Middle: Here we show σ σ = σ σ , hence the
2 1 6 1 2 typeb. Sincedistinguishableparticlescanbraidnon-trivially,
braid group is Non-Abelian. Bottom: The braid relation (Eq. 3)
i.e. θ can be non-zero for a = b as well as for a = b,
σiσi+1σi=σi+1σiσi+1. ab 6
anyonic‘statistics’is,perhaps,betterunderstoodasakindof
topologicalinteractionbetweenparticles.
ofthecorrespondingtrajectories, i.e. theverticalstackingof We now turn to non-Abelian braiding statistics, which
thetwodrawings. (Asmaybeseenfromthefigure,theorder areassociatedwithhigher-dimensionalrepresentationsofthe
in which they are multiplied is important because the group braid group. Higher-dimensional representations can occur
isnon-Abelian,meaningthatmultiplicationis notcommuta- whenthereisadegeneratesetofgstateswithparticlesatfixed
tive.) positionsR1,R2,...,Rn. Letusdefineanorthonormalbasis
Thebraidgroupcanberepresentedalgebraicallyintermsof ψα, α = 1,2,...,g ofthese degeneratestates. Thenan ele-
generatorsσi,with1 i N 1.Wechooseanarbitraryor- mentofthebraidgroup–sayσ1,whichexchangesparticles1
deringoftheparticles≤1,2≤,...,−N.2 σi isacounter-clockwise and2–isrepresentedbyag×g unitarymatrixρ(σ1)acting
exchangeoftheithand(i+1)thparticles. σi−1is,therefore,a onthesestates.
clockwiseexchangeoftheith and(i+1)th particles. Theσ s
i
ψ [ρ(σ )] ψ (4)
satisfythedefiningrelations(seeFig. 1), α → 1 αβ β
Ontheotherhand,exchangingparticles2and3leadsto:
σ σ = σ σ for i j 2
i j j i
| − |≥
σ σ σ = σ σ σ for1 i n 1 (3)
i i+1 i i+1 i i+1 ≤ ≤ − ψα →[ρ(σ2)]αβψβ (5)
The only difference from the permutation group S is that
N
σi2 6= 1, but this makes an enormous difference. While cBeost,hwρh(iσc1h)daenfidneρ(uσn2it)arayretrgan×sfogrmdiamtieonnswiointhalinutnhietasryubmspaatrcie-
the permutation group is finite, the number of elements in
ofdegenerategroundstates. Ifρ(σ )andρ(σ )donotcom-
the group S = N!, the braid group is infinite, even for 1 1
N
| | mute, [ρ(σ )] [ρ(σ )] = [ρ(σ )] [ρ(σ )] , the parti-
justtwoparticles. Furthermore,therearenon-trivialtopolog- 1 αβ 2 βγ 6 2 αβ 1 βγ
cles obey non-Abelianbraiding statistics. Unless they com-
icalclassesoftrajectoriesevenwhenthe particlesaredistin-
mute foranyinterchangeof particles, in whichcase the par-
guishable, e.g. in the two-particle case those trajectories in
ticles’ braiding statistics is Abelian, braiding quasiparticles
will cause non-trivial rotations within the degenerate many-
quasiparticleHilbertspace.Furthermore,itwillessentiallybe
trueatlowenergiesthattheonlywaytomakenon-trivialuni-
2Choosingadifferentorderingwouldamounttoarelabelingoftheelements
taryoperationsonthisdegeneratespaceisbybraidingquasi-
ofthebraidgroup,asgivenbyconjugationbythebraidwhichtransforms
oneorderingintotheother. particlesaroundeachother. Thisstatementisequivalenttoa
5
statementthatnolocalperturbationcanhavenonzeromatrix areso-namedforreasonswhichwillbecomeclearinsections
elementswithinthisdegeneratespace. III.DandIII.E),SU(2) ,andchiralp-superconductors.There
2
A systemwithanyonicparticlesmustgenerallyhavemul- areslightdifferencesbetweenthesethreetheories,relatingto
tipletypesofanyons. Forinstance,inasystemwithAbelian Abelianphases,buttheseareunimportantforthepresentdis-
anyons with statistics θ, a bound state of two such particles cussion.Thismodelhasthreedifferenttypesofanyons,which
hasstatistics4θ. Evenifnosuchstableboundstateexists,we canbe variouslycalled1,σ,ψ or0,1,1. (Unfortunately,the
2
may wish to bringtwo anyonsclose togetherwhile all other notation is a little confusing because the trivial particle is
particles are much further away. Then the two anyons can called ‘1’ in the first model but ‘0‘ in the second, however,
be approximated as a single particle whose quantum num- we will avoid confusionbyusing bold-faced1 to denotethe
bers are obtained by combining the quantum numbers, in- trivialparticle.)Thefusionrulesforsuchanyonsare
cluding the topological quantum numbers, of the two parti-
cles. As a result, a complete description of the system must σ σ = 1+ψ, σ ψ =σ, ψ ψ =1,
× × ×
also include these ‘higher’ particle species. For instance, if 1 x = x for x=1,σ,ψ (7)
×
there are θ = π/m anyons in system, then there are also
θ =4π/m,9π/m,...,(m 1)2π/m.Sincethestatisticspa- (Translating these rules into the notation of SU(2)2, we see
rameterisonlywell-defined−upto2π, θ = (m 1)2π/m = that these fusion rules are very similar to the decomposition
π/m for m evenand π π/m for m odd. T−he formation rulesfortensorproductsofirreducibleSU(2)representations,
−ofadifferenttypeofanyo−nbybringingtogethertwoanyons butdifferintheimportantrespectthat1isthemaximumspin
iscalledfusion. Whenastatisticsπ/mparticleisfusedwith sothat 21×21 =0+1,asintheSU(2)case,but 21×1= 12 and
a statistics π/m particle, the resulthas statistics θ = 0. It 1 1 = 0.) Notethattherearetwodifferentfusionchannels
isconvenie−nttocallthisthe‘trivial’particle. Asfarastopo- fo×rtwoσs. Asaresult,iftherearefourσswhichfusetogether
logicalpropertiesareconcerned,suchabosonisjustasgood togive1, thereisa two-dimensionalspaceofsuchstates. If
as the absence of any particle, so the ‘trivial’particle is also we divided the four σs into two pairs, by groupingparticles
sometimessimplycalledthe‘vacuum’. Wewilloftendenote 1,2 and 3,4, then a basis for the two-dimensional space is
thetrivialparticleby1. givenbythestateinwhich1,3fuseto1or1,3fusetoψ(2,4
With Abelian anyonswhich are made by forming succes- mustfusetothesameparticletypeas1,3doinorderthatall
sivelylargercompositesofπ/m particles, the fusionrule is: fourparticlesfuseto1). WecancallthesestatesΨ1 andΨψ;
n2π k2π = (n+k)2π. (Wewillusea btodenoteafused they are a basis for the four-quasiparticleHilbert space with
m × m m × totaltopologicalcharge1. (Similarly,iftheyallfusedtogive
with b.) However, for non-Abelian anyons, the situation is
ψ,therewouldbeanothertwo-dimensionaldegeneratespace;
morecomplicated. Aswithordinaryquantumnumbers,there
onebasisisgivenbythestateinwhichthefirstpairfusesto1
mightnotbeauniquewayofcombiningtopologicalquantum
whilethesecondfusestoψandthestateinwhichtheopposite
numbers(e.g. two spin-1/2particlescouldcombineto form
occurs.)
either a spin-0 or a spin-1 particle). The different possibili-
Of course, our division of the four σs into two pairs was
ties are called the different fusion channels. This is usually
arbitrary. We could have divided them differently, say, into
denotedby
the pairs 1,3 and 2,4. We would thereby obtain two dif-
φa×φb = Nacbφc (6) f1eroernttobaψs,isresstapteecst,ivΨ˜el1y.anTdhΨi˜sψis, ijnuswt haicdhiffbeorethntpbaairssisfuinsethtoe
c
X same two-dimensionalspace. The matrix parametrizingthis
which represents the fact that when a particle of species a
basis change (see also Appendix A) is called the F-matrix:
fuses with one of species b, the result can be a particle of
Ψ˜ = F Ψ , where a,b = 1,ψ. There should really be
species c if Nc = 0. For Abelian anyons, the fusion mul- a ab b
ab 6 6 indices on F if we include indices to specify the 4 parti-
tiplicities Nc = 1 foronly onevalue ofc andNc′ = 0 for
ab ab cletypes: Fijk ,butwehavedroppedtheseotherindices
ia.lel.c′N6=k′c′. F=orδparticles.oFfotyrpneokn-wAibthelsiatantiastniycsonθsk,=theπrke2i/smat, since i =hjl=ikab= l = σ in our case. The F-matrices
kk′ k+k′,k′′
least one a, b such that there are multiple fusion channels c aresometimescalled6j symbolssincetheyareanalogousto
withNc =0. Intheexampleswhichwewillbeconsidering thecorrespondingquantitiesforSU(2)representations.Recall
in this apbap6 er,Nc = 0 or 1, butthere are theoriesforwhich thatinSU(2),therearemultiplestatesinwhichspinsj1,j2,j3
ab
Nc >1forsomea,b,c.Inthiscase,aandbcanfusetoform coupletoformatotalspinJ. Forinstance,j1 andj2 canadd
ab
cinNc > 1differentdistinctways. Wewillusea¯todenote toformj12,whichcanthenaddwithj3 togiveJ. Theeigen-
theantaipbarticleofparticlespeciesa. Whenaanda¯fuse,they statesof(j12)2 forma basisofthedifferentstateswithfixed
can always fuse to 1 in precisely one way, i.e. Na1a¯ = 1; in j1,j2,j3,andJ. Alternatively,j2 andj3 canaddtoformj23,
thenon-Abeliancase,theymayormaynotbeabletofuseto whichcanthenaddwithj togiveJ. Theeigenstatesof(j )2
1 23
otherparticletypesaswell. formadifferentbasis. The6j symbolgivesthebasischange
Thedifferentfusionchannelsareonewayofaccountingfor betweenthetwo. TheF-matrixofa system of anyonsplays
thedifferentdegeneratemulti-particlestates. Letusseehow thesamerolewhenparticlesoftopologicalchargesi,j,kfuse
thisworksinonesimplemodelofnon-Abeliananyonswhich tototaltopologicalchargel. Ifiandj fusetoa, whichthen
wediscussinmoredetailinsectionIII. Aswediscussinsec- fuseswithktogivetopologicalchargel,thedifferentallowed
tion III, this model is associated with ‘Ising anyons’ (which a define a basis. If j and k fuse to b and then fuse with i to
6
give topologicalcharge l, this defines another basis, and the nottelluswhenandwheretheymightoccurinnature. Elec-
F-matrixistheunitarytransformationbetweenthetwobases. trons, protons, atoms, and photons, are all either fermions
States with more than 4 quasiparticlescan be understoodby or bosons even when they are confined to move in a two-
successivelyfusingadditionalparticles,inamannerdescribed dimensionalplane.However,ifasystemofmanyelectrons(or
inSectionIII.A. TheF-matrixcanbeappliedtoanysetof4 bosons,atoms,etc.) confinedtoatwo-dimensionalplanehas
consecutivelyfusedparticles. excitations which are localized disturbances of its quantum-
The different states in this degenerate multi-anyon state mechanicalgroundstate,knownasquasiparticles,thenthese
spacetransformintoeachotherunderbraiding.However,two quasiparticles can be anyons. When a system has anyonic
particlescannotchangetheirfusionchannelsimplybybraid- quasiparticleexcitationsaboveitsgroundstate,itisinatopo-
ing with each other since their total topological charge can logicalphaseofmatter. (Amoreprecisedefinitionofatopo-
bemeasuredalongafardistantloopenclosingthetwoparti- logicalphaseofmatterwillbegiveninSectionIII.)
cles. Theymustbraidwithathirdparticleinordertochange Let us see how anyons might arise as an emergent prop-
their fusion channel. Consequently, when two particles fuse ertyofamany-particlesystem. Forthesakeofconcreteness,
in a particular channel (rather than a linear superposition of consider the ground state of a 2+1 dimensional system of
channels), the effect of taking one particle around the other ofelectrons,whosecoordinatesare(r ,...,r ). Weassume
1 n
is just multiplication by a phase. This phase resulting from that the ground state is separated from the excited states by
a counter-clockwise exchange of particles of types a and b anenergygap(i.e,itisincompressible),asisthesituationin
which fuse to a particle of type c is called Rab. In the Ising fractional quantum Hall states in 2D electron systems. The
c
anyoncase,aswewillderiveinsectionIIIandAppendixA.1, lowest energy electrically-charged excitations are known as
Rσσ = e πi/8, Rσσ = e3πi/8, Rψψ = 1, Rσψ = i. For quasiparticles or quasiholes, depending on the sign of their
1 − ψ 1 − σ
an exampleofhowthisworks, supposethatwe createa pair electric charge. (The term “quasiparticle” is also sometimes
ofσ quasiparticlesoutofthe vacuum. Theywillnecessarily usedinagenericsensetomeanbothquasiparticleandquasi-
fuseto1. Ifwetakeonearoundanother,thestatewillchange hole as in the previous paragraph). These quasiparticles are
byaphasee πi/8. Ifwetakeathirdσ quasiparticleandtake localdisturbancestothewavefunctionoftheelectronscorre-
−
itaroundone,butnotboth,ofthefirsttwo,thenthefirsttwo spondingtoaquantizedamountoftotalcharge.
willnowfusetoψ,aswewillshowinSec.III.Ifwenowtake We now introduce into the system’s Hamiltonian a scalar
oneofthefirsttwoaroundtheother,thestatewillchangeby potential composed of many local “traps”, each sufficient to
aphasee3πi/8. capture exactly one quasiparticle. These traps may be cre-
Inorderto fullyspecifythe braidingstatistics of a system ated by impurities, by very small gates, or by the potential
of anyons, it is necessary to specify (1) the particle species, createdbytipsof scanningmicroscopes. The quasiparticle’s
(2) the fusion rules Nc , (3) the F-matrices, and (4) the R- charge screens the potential introduced by the trap and the
ab
matrices.InsectionIV,wewillintroducetheothersetsofpa- “quasiparticle-tip” combination cannot be observed by local
rameters,namelythe topologicalspinsΘa andtheS-matrix, measurementsfromfar away. Letus denote the positionsof
which, together with the parameters 1-4 above fully charac- these traps to be (R ,...,R ), and assume that these posi-
1 k
terizethetopologicalpropertiesofasystemofanyons. Some tions are well spaced from each other compared to the mi-
readersmaybefamiliar withthe incarnationofthese mathe- croscopic length scales. A state with quasiparticles at these
maticalstructuresinconformalfieldtheory(CFT),wherethey positionscanbeviewedasanexcitedstateoftheHamiltonian
occurforreasonswhichweexplaininsectionIII.D;webriefly of the system without the trap potential or, alternatively, as
reviewthesepropertiesintheCFTcontextinAppendixA. the groundstate in the presence of the trap potential. When
Quasiparticles obeying non-Abelian braiding statistics or, werefertothegroundstate(s)ofthesystem,wewilloftenbe
simplynon-Abeliananyons,werefirstconsideredinthecon- referringtomulti-quasiparticlestatesinthelattercontext.The
text of conformal field theory by Moore and Seiberg, 1988, quasiparticles’coordinates(R ,...,R )areparametersboth
1 k
1989 and in the context of Chern-Simons theory by Witten, intheHamiltonianandintheresultinggroundstatewavefunc-
1989. They were discussed in the context of discrete gauge tionfortheelectrons.
theories and linked to the representation theory of quantum
Weareconcernedherewiththeeffectoftakingthesequasi-
groupsbyBais, 1980;Bais etal.,1992,1993a,b. Theywere particles around each other. We imagine making the quasi-
discussed in a more general context by Fredenhagen et al., particles coordinates R = (R ,...,R ) adiabatically time-
1 k
1989 and Fro¨hlich and Gabbiani, 1990. The properties of
dependent.Inparticular,weconsideratrajectoryinwhichthe
non-Abelianquasiparticlesmakethemappealingforuseina
final configuration of quasiparticles is just a permutation of
quantumcomputer.Butbeforediscussingthis,wewillbriefly
the initial configuration (i.e. at the end, the positions of the
reviewhowtheycouldoccurinnatureandthenthebasicideas
quasiparticles are identical to the intial positions, but some
behindquantumcomputation.
quasiparticlesmay have interchangedpositionswith others.)
Ifthegroundstatewavefunctionissingle-valuedwithrespect
to (R ,..,R ), and if there is only one ground state for any
1 k
2. EmergentAnyons givensetofR ’s,thenthefinalgroundstatetowhichthesys-
i
temreturnstoafterthewindingisidenticaltotheinitialone,
The preceding considerations show that exotic braiding uptoaphase.Partofthisphaseissimplythedynamicalphase
statistics is a theoretical possibility in 2+ 1-D, but they do which depends on the energy of the quasiparticle state and
7
thelengthoftimefortheprocess. Intheadiabaticlimit,itis Where R(s), s [0,2π] is the closed trajectory of the par-
∈
dtE(R~(t)). There is also a a geometric phase which does ticles and the path-orderingsymbol is defined by the sec-
P
notdependonhowlongtheprocesstakes. ThisBerryphase ond equality. Again, the matrix Mab may be the productof
Ris(Berry,1984), topologicaland non-topologicalparts. In a system in which
quasiparticles obey non-Abelian braiding statistics, the non-
topological part will be Abelian, that is, proportional to the
α=i dR ψ(R) ψ(R) (8)
·h |∇R~| i unitmatrix.Onlythetopologicalpartwillbenon-Abelian.
I
Therequirementsforquasiparticlesto follownon-Abelian
where ψ(R) isthegroundstatewiththequasiparticlesatpo-
| i statistics arethen,first, thatthe N-quasiparticlegroundstate
sitionsR,andwheretheintegralistakenalongthetrajectory
is degenerate. In general, the degeneracy will not be exact,
R(t). Itismanifestlydependentonlyonthetrajectorytaken
but it should vanish exponentially as the quasiparticle sepa-
by the particles and not on how long it takes to move along
rations are increased. Second, that adiabatic interchange of
thistrajectory.
quasiparticlesappliesaunitarytransformationontheground
The phase α has a piece that dependson the geometry of
state,whosenon-Abelianpartisdeterminedonlybythetopol-
thepathtraversed(typicallyproportionaltotheareaenclosed
ogyofthebraid,whileitsnon-topologicalpartisAbelian. If
by all of the loops), and a piece θ that depends only on the
the particles are not infinitely far apart, and the degeneracy
topology of the loops created. If θ = 0, then the quasipar-
6 is only approximate, then the adiabatic interchange must be
ticles excitations of the system are anyons. In particular, if
donefasterthantheinverseoftheenergysplitting(Thouless
weconsiderthecasewhereonlytwoquasiparticlesareinter-
andGefen,1991)betweenstatesinthenearly-degeneratesub-
changedclockwise(withoutwrappingaroundanyotherquasi-
space (but, of course, still much slower than the energy gap
particles),θisthestatisticalangleofthequasiparticles.
betweenthissubspaceandtheexcitedstates). Third,theonly
Therewere two keyconditionsto ourabovediscussionof
waytomakeunitaryoperationsonthedegenerategroundstate
theBerryphase. Thesinglevaluednessofthewavefunction
space, so longas the particlesare keptfar apart, is bybraid-
isatechnicalissue. Thenon-degeneracyofthegroundstate,
ing.Thesimplest(albeituninteresting)exampleofdegenerate
however,isanimportantphysicalcondition. Infact,mostof
groundstatesmayariseifeachofthequasiparticlescarrieda
thispaperdealswiththesituationinwhichthisconditiondoes
spin1/2withavanishingg–factor. Ifthatwerethecase, the
nothold. Wewillgenerallybeconsideringsystemsinwhich,
system would satisfy the first requirement. Spin orbit cou-
oncethepositions(R ,..,R )ofthequasiparticlesarefixed,
1 k pling may conceivablylead to the second requirementbeing
there remain multiple degenerate ground states (i.e. ground
satisfied. Satisfying the third one, however, is much harder,
statesinthepresenceofapotentialwhichcapturesquasipar-
andrequiresthesubtlestructurethatwedescribebelow.
ticles atpositions(R ,..,R )), whichare distinguishedbya
1 k ThedegeneracyofN-quasiparticlegroundstatesiscondi-
set of internal quantum numbers. For reasons that will be-
tionedonthequasiparticlesbeingwellseparatedfromonean-
come clear later, we will refer to these quantum numbersas
other. When quasiparticles are allowed to approach one an-
“topological”.
other too closely, the degeneracy is lifted. In other words,
Whenthegroundstateisdegenerate,theeffectofaclosed
when non-Abelian anyonic quasiparticles are close together,
trajectory of the R ’s is not necessarily just a phase factor.
i theirdifferentfusionchannelsaresplitinenergy.Thisdepen-
Thesystemstartsandendsingroundstates,buttheinitialand
denceisanalogoustothewaytheenergyofasystemofspins
final ground states may be differentmembers of this degen-
dependsontheirinternalquantumnumberswhenthespinsare
erate space. The constraint imposed by adiabaticity in this
close together and their coupling becomes significant. The
caseisthattheadiabaticevolutionofthestateofthesystemis
splitting between different fusion channels is a means for a
confinedtothesubspaceofgroundstates. Thus,itmaybeex-
measurement of the internal quantum state, a measurement
pressedasaunitarytransformationwithinthissubspace. The
thatisofimportanceinthecontextofquantumcomputation.
inner productin (8) must be generalized to a matrix of such
innerproducts:
m = ψ (R) ~ ψ (R) (9) B. TopologicalQuantumComputation
ab a R b
h |∇ | i
where ψa(R) , a = 1,2,...,g are theg degenerateground 1. BasicsofQuantumComputation
| i
states. SincethesematricesatdifferentpointsR~ donotcom-
mute,wemustpath-ordertheintegralinordertocomputethe As the components of computers become smaller and
transformationruleforthestate,ψa Mabψbwhere smaller, we are approaching the limit in which quantum ef-
→
fectsbecomeimportant.Onemightaskwhetherthisisaprob-
lem or anopportunity. The foundersof the field of quantum
M = exp i dR m
ab P · computation (Manin, 1980, Feynman, 1982, 1986, Deutsch,
(cid:18) I (cid:19)
1985,andmostdramatically,Shor,1994)answeredinfavorof
= ∞ in 2πds s1ds ... sn−1ds R˙(s )m (R(s ))... thelatter. Theyshowedthatacomputerwhichoperatescoher-
1 2 n 1 · aa1 1
nX=0 Z0 Z0 Z0 h ently on quantum states has potentially much greater power
thanaclassicalcomputer(NielsenandChuang,2000).
R˙(s ) m (R(s )) (10)
n · anb n The problem which Feynman had in mind for a quantum
i
8
computerwasthesimulationofaquantumsystem(Feynman, tem to traverse many trajectories in parallel, and determine
1982). He showed that certain many-body quantum Hamil- its state by their coherent sum. In some sense this coherent
tonianscouldbesimulatedexponentiallyfasteronaquantum sum amounts to a massive quantum parallelism. It should
computer than they could be on a classical computer. This not, however, be confusedwith classical parallelcomputing,
isanextremelyimportantpotentialapplicationofa quantum where many computers are run in parallel, and no coherent
computersinceitwouldenableustounderstandtheproperties sumtakesplace.
of complexmaterials, e.g. solve high-temperaturesupercon- Thebiggestobstacletobuildingapracticalquantumcom-
ductivity. Digital simulations of large scale quantum many- puterisposedbyerrors,whichwouldinvariablyhappendur-
bodyHamiltoniansareessentiallyhopelessonclassicalcom- ing any computation, quantumor classical. For any compu-
puters because of the exponentially-largesize of the Hilbert tationtobesuccessfulonemustdevisepracticalschemesfor
space.Aquantumcomputer,usingthephysicalresourceofan error correction which can be effectively implemented (and
exponentially-large Hilbert space, may also enable progress whichmustbesufficientlyfault-tolerant).Errorsaretypically
in the solution of lattice gaugetheoryand quantumchromo- correctedinclassicalcomputersthroughredundancies,i.e.,by
dynamics,thussheddinglightonstrongly-interactingnuclear keepingmultiplecopiesofinformationandcheckingagainst
forces. thesecopies.
In1994PeterShorfoundanapplicationofaquantumcom- With a quantum computer, however, the situation is more
puterwhichgeneratedwidespreadinterestnotjustinsidebut complex. If we measure a quantumstate duringan interme-
also outside of the physics community(Shor, 1994). He in- diatestageofacalculationtoseeifanerrorhasoccurred,we
vented an algorithm by which a quantum computer could collapse the wave function and thus destroy quantum super-
find the prime factors of an m digit number in a length of positions and ruin the calculation. Furthermore, errors need
time m2logmloglogm. This is much faster than the notbemerelyadiscreteflipof 0 to 1 ,butcanbecontinu-
fastes∼tknownalgorithmforaclassicalcomputer,whichtakes ous:thestatea0 +b1 maydr|ifti,due| tioanerror,tothestate
exp(m1/3) time. Since manyencryptionschemesdepend a0 +beiθ|1iwith| airbitraryθ.
∼onthedifficultyoffindingthesolutiontoproblemssimilarto → | i | i
Remarkably,inspiteofthesedifficulties,errorcorrectionis
findingtheprimefactorsofalargenumber,thereisanobvi-
possibleforquantumcomputers(CalderbankandShor,1996;
ousapplicationofaquantumcomputerwhichisofgreatbasic
Gottesman,1998;Preskill,2004;Shor,1995;Steane,1996a).
andappliedinterest.
Onecanrepresentinformationredundantlysothaterrorscan
Thecomputationmodelsetforthbythesepioneersofquan- beidentifiedwithoutmeasuringtheinformation.Forinstance,
tum computing (and refined in DiVincenzo, 2000), is based if we use three spins to represent each qubit, 0 000 ,
on three steps: initialization, unitaryevolutionand measure- 1 111 , and the spin-fliprate is low, then|wie→can|ideni-
ment. Weassumethatwehaveasystematourdisposalwith t|ifiy→erro|rs biy checking whether all three spins are the same
Hilbert space . We further assume that we can initialize (here, we represent an up spin by 0 and a down spin by 1).
H
the system in some known state ψ0 . We unitarily evolve Supposethatourspinsareininthestateα000 +β 111 . If
the system untilit is in some final|staite U(t)ψ0 . This evo- thefirstspinhasflippederroneously,theno|ursipinsa|reinithe
| i
lution will occur according to some Hamiltonian H(t) such state α100 +β 011 . We candetectthiserrorbychecking
thatdU/dt=iH(t)U(t)/~. Werequirethatwehaveenough whethe|rtheifirst|spiniis the same as the other two; this does
controloverthisHamiltoniansothatU(t)canbemadetobe notrequireustomeasurethestateofthequbit.(“Wemeasure
anyunitarytransformationthatwedesire. Finally,weneedto the errors, rather than the information.” (Preskill, 2004)) If
measure the state of the system at the end of this evolution. thefirstspinisdifferentfromtheothertwo,thenwejustneed
Such a process is called quantum computation (Nielsen and to flip it. We repeat this process with the second and third
Chuang, 2000). The Hamiltonian H(t) is the software pro- spins. Solongaswecanbesurethattwospinshavenoterro-
gramtoberun.Theinitialstateistheinputtothecalculation, neouslyflipped(i.e. solongasthebasicspin-fliprateislow),
andthefinalmeasurementistheoutput. thisprocedurewillcorrectspin-fliperrors. Amoreelaborate
The need for versatility, i.e., for one computer to effi- encodingisnecessaryinordertocorrectphaseerrors,butthe
cientlysolvemanydifferentproblems,requirestheconstruc- keyobservationisthataphaseerrorintheσ basisisabitflip
z
tionofthecomputeroutofsmallerpiecesthatcanbemanipu- errorintheσ basis.
x
latedandreconfiguredindividually. Typicallythefundamen- However,theerrorcorrectionprocessmayitselfbealittle
talpieceistakentobeaquantumtwostatesystemknownas noisy. More errorscould then occur duringerror correction,
a “qubit” which is the quantum analog of a bit. (Of course, andthewholeprocedurewillfailunlessthebasicerrorrateis
onecouldequallywelltakegeneral“dits”,forwhichthefun- verysmall. Estimatesofthethresholderrorrateabovewhich
damental unit is some d-state system with d not too large). error correction is impossible depend on the particular error
Whileaclassicalbit,i.e.,aclassicaltwo-statesystem,canbe correction scheme, but fall in the range 10 4 10 6 (see,
− −
either“zero”or“one”atanygiventime,aqubitcanbeinone e.g. Aharonov and Ben-Or, 1997; Knill et al.,−1998). This
oftheinfinitelymanysuperpositionsa0 +b1 .Fornqubits, meansthatwemustbeabletoperform104 106 operations
thestatebecomesavectorina2n–dim|enision|aliHilbertspace, perfectly before an error occurs. This is an−extremely strin-
inwhichthedifferentqubitsaregenerallyentangledwithone gentconstraintanditispresentlyuncleariflocalqubit-based
another. quantumcomputationcaneverbemadefault-tolerantthrough
The quantum phenomenon of superposition allows a sys- quantumerrorcorrectionprotocols.
9
Random errors are caused by the interaction between the theproblemsfacedinthetwocategoriesmightbequitediffer-
quantum computer and the environment. As a result of this ent.Forquantuminformationprocessing,unitaryerrors,such
interaction,thequantumcomputer,whichisinitiallyinapure asrotatingaqubitby90.01degreesinsteadof90,areanissue
superpositionstate,becomesentangledwithitsenvironment. ofhowpreciselyonecanmanipulatethesystem.Ontheother
This can cause errors as follows. Suppose that the quantum hand,whenaqubitissimplystoringinformation,oneislikely
computer is in the state 0 and the environment is in the tobemoreconcernedabouterrorscausedbyinteractionswith
| i
state E so that their combined state is 0 E . The in- theenvironment.Thisisinsteadanissueofhowwellisolated
0 0
| i | i| i
teraction between the computer and the environment could onecan makethe system. As we will see below, a topologi-
causethisstatetoevolvetoα0 E +β 1 E ,where E calquantumcomputerisprotectedfromproblemsinbothof
0 1 1
| i| i | i| i | i
is another state of the environment (not necessarily orthog- thesecategories.
onal to E ). The computer undergoes a transition to the
0
state 1|wiith probability β 2. Furthermore, the computer
| i | |
and the environmentare now entangled,so the reducedden-
2. Fault-TolerancefromNon-AbelianAnyons
sity matrix for the computer alone describes a mixed state,
e.g. ρ = diag(α2, β 2) if E E = 0. Since we can-
| | | | h 0| 1i Topologicalquantumcomputationis a schemefor usinga
notmeasurethestateoftheenvironmentaccurately,informa-
systemwhoseexcitationssatisfynon-Abelianbraidingstatis-
tionislost,asreflectedintheevolutionofthedensitymatrix
tics to perform quantum computation in a way that is natu-
of the computer from a pure state to a mixed one. In other
rallyimmunetoerrors.TheHilbertspace usedforquantum
words, the environment has caused decoherence. Decoher- H
computationis the subspace of the total Hilbert space of the
encecandestroyquantuminformationevenifthestateofthe
systemcomprisedofthedegenerategroundstateswithafixed
computer does not undergo a transition. Although whether
numberofquasiparticlesatfixedpositions.Operationswithin
or nota transitionoccursis basis-dependent(a bit flip in the
this subspace are carried out by braiding quasiparticles. As
σ basis is a phase flip in the σ basis), it is a useful dis-
z x wediscussedabove,thesubspaceofdegenerategroundstates
tinctionbecausemanysystemshaveapreferredbasis,forin-
is separated from the rest of the spectrum by an energygap.
stancethegroundstate 0 andexcitedstate 1 ofanionina
| i | i Hence,ifthetemperatureismuchlowerthanthegapandthe
trap. Supposethestate 0 evolvesasabove,butwithα = 1,
| i system is weakly perturbed using frequencies much smaller
β = 0 so that no transition occurs, while the state 1 E
| i| 0i thanthegap,thesystemevolvesonlywithinthegroundstate
evolves to 1 E with E E = 0. Then an initial pure
| i| 1′i h 1′| 1i subspace.Furthermore,thatevolutionisseverelyconstrained,
state(a0 +b1 ) E evolvestoamixedstatewithdensity
| i | i | 0i sinceitisessentiallythecase(withexceptionswhichwewill
matrix ρ = diag(a2, b2). The correlations in which our
| | | | discuss) that the only way the system can undergo a non-
quantuminformationresidesisnowtransferredtocorrelation
trivial unitary evolution - that is, an evolution that takes it
between the quantum computer and the environment. The
fromonegroundstatetoanother-isbyhavingitsquasipar-
quantum state of a system invariably loses coherence in this
ticlesbraided. Thereasonforthisexceptionalstabilityisthat
way over a characteristic time scale T . It was universally
coh any local perturbation (such as the electron-phonon interac-
assumed until the advent of quantum error correction (Shor,
tion and the hyperfine electron-nuclear interaction, two ma-
1995; Steane, 1996a) that quantum computation is intrinsi-
jorcausesfordecoherenceinnon-topologicalsolidstatespin-
cally impossible since decoherence-induced quantum errors
basedquantumcomputers(WitzelandDasSarma,2006))has
simplycannotbecorrectedinanyrealphysicalsystem.How-
no nontrivial matrix elements within the ground state sub-
ever,whenerror-correctingcodesareused,theentanglement
space. Thus, the system is ratherimmunefromdecoherence
is transferredfromthe quantumcomputerto ancillaryqubits
(Kitaev,2003). Unitaryerrorsarealsounlikelysincetheuni-
so that the quantum information remains pure while the en-
tary transformations associated with braiding quasiparticles
tropyisintheancillaryqubits.
aresensitiveonlytothetopologyofthequasiparticletrajecto-
Of course, even if the coupling to the environment were ries,andnottotheirgeometryordynamics.
completely eliminated, so that there were no random errors,
A model in which non-Abelian quasiparticles are utilized
therecouldstillbesystematicerrors. Theseareunitaryerrors
for quantum computation starts with the construction of
whichoccurwhilewe processquantuminformation. Forin-
qubits. In sharp contrast to most realizations of a quantum
stance,wemaywishtorotateaqubitby90degreesbutmight computer,aqubithereisanon-localentity,beingcomprised
inadvertentlyrotateitby90.01degrees.
of several well-separated quasiparticles, with the two states
Froma practicalstandpoint,itis oftenusefulto divideer- of the qubitbeingtwo differentvaluesforthe internalquan-
rorsintotwo categories: (i)errorsthatoccurwhena qubitis tumnumbersofthissetofquasiparticles.Inthesimplestnon-
beingprocessed(i.e.,whencomputationsarebeingperformed Abelian quantum Hall state, which has Landau-level filling
onthatqubit)and(ii)errorsthatoccurwhenaqubitissimply factor ν = 5/2, two quasiparticles can be put together to
storingquantuminformationandisnotbeingprocessed(i.e., form a qubit (see Sections II.C.4 and IV.A). Unfortunately,
whenitisactingasa quantummemory). Froma fundamen- aswewilldiscussbelowinSectionsIV.AandIV.C,thissys-
talstandpoint,thisisabitofafalsedichotomy,sinceonecan tem turns out to be incapable of universal topologicalquan-
think of quantuminformationstorage (or quantum memory) tum computation using only braiding operations; some un-
as being a computer that applies the identity operation over protectedoperationsarenecessaryinordertoperformuniver-
andovertothequbit(i.e.,leavesitunchanged). Nonetheless, sal quantumcomputation. The simplest system that is capa-
10
bleofuniversaltopologicalquantumcomputationisdiscussed parentproblem,however,isevadedbythenatureofthecom-
in SectionIV.B, andutilizesthreequasiparticlesto formone putations, which correspond to closed world lines that have
qubit. no loose ends: when the computation involves creation and
As mentioned above, to perform a quantum computation, annihilationofa quasiparticlequasi-holepair, theworld-line
onemustbeabletoinitializethestate ofqubitsatthebegin- is a closed curve in space-time. If the measurement occurs
ning, perform arbitrary controlled unitary operations on the by bringingtwo particles together to measure their quantum
state,andthenmeasurethestateofqubitsattheend.Wenow charge, it does not matter where precisely they are brought
addresseachoftheseinturn. together. Alternatively, when the measurement involves an
Initializationmaybeperformedbypreparingthequasipar- interferenceexperiment,the interferingparticle must close a
ticles in a specific way. For example, if a quasiparticle-anti- loop. In other words, a computation correspondsto a set of
quasiparticlepairiscreatedby“pulling”itapartfromthevac- links rather than openbraids, and the initialization and mea-
uum(e.g. paircreationfromthevacuumbyanelectricfield), surementtechniquesnecessarily involvebringingquasiparti-
thepairwillbegininaninitialstatewiththepairnecessarily clestogetherinsomeway,closingupthetrajectoriesandmak-
havingconjugatequantumnumbers(i.e.,the“total”quantum ing the full processfrom initialization to measurementcom-
numberof the pairremainsthe same asthat ofthe vacuum). pletelytopological.
This gives us a known initial state to start with. It is also Due to its special characteristics, then, topological quan-
possible to use measurement and unitary evolution (both to tum computation intrinsically guarantees fault-tolerance, at
bediscussedbelow)asaninitializationscheme—ifonecan the level of “hardware”, without “software”-basederror cor-
measurethequantumnumbersofsomequasiparticles,onecan rectionschemesthataresoessentialfornon-topologicalquan-
then perform a controlled unitary operationto put them into tumcomputers. Thisimmunitytoerrorsresultsfromthesta-
anydesiredinitialstate. bilityofthegroundstatesubspacewithrespecttoexternallo-
Once the system is initialized, controlled unitary opera- calperturbations.Innon-topologicalquantumcomputers,the
tions are then performed by physically dragging quasiparti- qubits are local, and the operations on them are local, lead-
clesaroundoneanotherinsomespecifiedway. Whenquasi- ing to a sensitivity to errors induced by local perturbations.
particles belonging to different qubits braid, the state of the In a topologicalquantum computer the qubits are non-local,
qubitschanges. Sincetheresultingunitaryevolutiondepends andtheoperations—quasiparticlebraiding—arenon-local,
only on the topology of the braid that is formed and not on leadingtoanimmunitytolocalperturbations.
thedetailsofhowitisdone,itisinsensitivetowigglesinthe Suchimmunitytolocalperturbationgivestopolgicalquan-
path,resulting,e.g.,fromthequasiparticlesbeingscatteredby tum memories exceptional protection from errors due to the
phononsorphotons.Determiningwhichbraidcorrespondsto interaction with the environment. However, it is crucial to
which computation is a complicated but eminently solvable note thattopologicalquantumcomputersare also exception-
task,whichwillbediscussedinmoredepthinsectionIV.B.3. allyimmunetounitaryerrorsduetoimprecisegateoperation.
Oncetheunitaryevolutioniscompleted,therearetwoways Unlikeothertypesofquantumcomputers,theoperationsthat
tomeasurethestate ofthequbits. Thefirstreliesonthefact canbeperformedonatopologicalquantumcomputer(braids)
thatthedegeneracyofmulti-quasiparticlestatesissplitwhen naturally take a discrete set of values. As discussed above,
quasiparticlesarebroughtclosetogether(withinsomemicro- whenonemakesa90degreerotationofaspin-basedqubit,for
scopic length scale). When two quasiparticles are brought example,itispossiblethatonewillmistakenlyrotateby90.01
closetogether,forinstance,ameasurementofthisenergy(or degreesthusintroducingasmallerror. Incontrast,braidsare
ameasurementoftheforcebetweentwoquasiparticles)mea- discrete: eitheraparticleistakenaroundanother,oritisnot.
suresthethetopologicalchargeofthepair. Asecondwayto There is no way to make a small error by having slight im-
measurethetopologicalchargeofagroupofquasiparticlesis precision in the way the quasiparticles are moved. (Taking
bycarryingoutanAharanov-Bohmtypeinterferenceexperi- a particleonlypartofthe wayaroundanotherparticlerather
ment.Wetakea“beam”oftestquasiparticles,senditthrough than all of the way does not introduce errors so long as the
abeamsplitter,sendonepartialwavetotherightofthegroup topologicalclassofthelinkformedbytheparticletrajectories
to be measured and another partial wave to the left of the –asdescribedabove–isunchanged.)
group and then re-interfere the two waves (see Figure 2 and Given the exceptional stability of the ground states, and
the surrounding discussion). Since the two different beams their insensitivity to local perturbations that do not involve
makedifferentbraidsaroundthetestgroup,theywillexperi- excitationstoexcitedstates,onemayaskthenwhichphysical
encedifferentunitaryevolutiondependingonthetopological processesdocauseerrorsinsuchatopologicalquantumcom-
quantumnumbersofthetestgroup. Thus,there-interference puter. Duetothetopologicalstabilityoftheunitarytransfor-
ofthesetwobeamswillreflectthetopologicalquantumnum- mationsassociated with braids, the only error processes that
berofthegroupofquasiparticlesenclosed. we must be concerned about are processes that might cause
Thisconcludesa roughdescriptionof the way a topologi- us to form the wrong link, and hence the wrong computa-
calquantumcomputationisto beperformed. Whilethe uni- tion.Certainly,onemustkeepcarefultrackofthepositionsof
tary transformation associated with a braid depends only on allofthequasiparticlesinthesystemduringthecomputation
the topology of the braid, one may be concerned that errors andassure thatonemakesthe correctbraidto dothe correct
could occur if one does not return the quasiparticles to pre- computation. Thisincludesnotjustthe“intended”quasipar-
ciselythecorrectpositionattheendofthebraiding. Thisap- ticleswhichwe needtomanipulateforourquantumcompu-