Lecture notes for Physics Quantum computation

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Lecture Notes for Physics 219: Quantum Computation John Preskill California Institute of Technology 14 June 2004Contents 9 Topological quantum computation 4 9.1 Anyons, anyone? 4 9.2 Flux-charge composites 7 9.3 Spin and statistics 9 9.4 Combining anyons 11 9.5 Unitary representations of the braid group 13 9.6 Topological degeneracy 16 9.7 Toric code revisited 20 9.8 The nonabelian Aharonov-Bohm effect 21 9.9 Braiding of nonabelian fluxons 24 9.10 Superselection sectors of a nonabelian superconductor 29 9.11 Quantum computing with nonabelian fluxons 32 9.12 Anyon models generalized 40 9.12.1Labels 40 9.12.2Fusion spaces 41 9.12.3Braiding: the R-matrix 44 9.12.4Associativity of fusion: the F-matrix 45 9.12.5Many anyons: the standard basis 46 9.12.6Braiding in the standard basis: theB-matrix 47 9.13 Simulating anyons with a quantum circuit 49 9.14 Fibonacci anyons 52 9.15 Quantum dimension 53 9.16 Pentagon and hexagon equations 58 9.17 Simulating a quantum circuit with Fibonacci anyons 61 9.18 Epilogue 63 9.18.1Chern-Simons theory 63 9.18.2S-matrix 64 9.18.3Edge excitations 65 9.19 Bibliographical notes 65 2Contents 3 References 679 Topological quantum computation 9.1 Anyons, anyone? A central theme of quantum theory is the concept of indistinguishable particles (also called identical particles). For example, all electrons in the world are exactly alike. Therefore, for a system with many electrons, an operation that exchanges two of the electrons (swaps their positions) is a symmetry — it leaves the physics unchanged. This symmetry is representedbyaunitarytransformationactingonthemany-electronwave function. For the indistinguishable particles in three-dimensional space that we normally talk about in physics, particle exchanges are represented in one 4 of two distinct ways. If the particles are bosons (like, for example, He atomsinasuperfluid), thenanexchangeoftwoparticlesisrepresentedby the identity operator: the wave function is invariant, and we say the par- ticles obey Bose statistics. If the particlesare fermions (like,forexample, electrons in a metal), than an exchange is represented by multiplication by (−1): the wave function changes sign, and we say that the particles obey Fermi statistics. The concept of identical-particle statistics becomes ambiguous in one spatial dimension. The reason is that for two particles to swap positions in one dimension, the particles need to pass through one another. If the wave function changes sign when two identical particles are exchanged, we could say that the particles are noninteracting fermions, but we could just as well say that the particles are interacting bosons, such that the sign change is induced by the interaction as the particles pass one an- other. More generally, the exchange could modify the wavefunction by iθ a multiplicative phase e that could take values other than +1 or −1, but we could account for this phase change by describing the particles as either bosons or fermions. 49.1 Anyons, anyone? 5 Thus, identical-particle statistics is a rather tame concept in three (or more)spatialdimensionsandalsoinonedimension. Butinbetweenthese two dull cases, in two dimensions, a remarkably rich variety of types of particle statistics are possible, so rich that we have far to go before we can give a useful classification of all of the possibilities. Indistinguishable particles in two dimensions that are neither bosons nor fermions are called anyons. Anyons are a fascinating theoretical con- struct, but do they have anything to do with the physics of real systems that can be studied in the laboratory? The remarkable answer is: “Yes” Even in our three-dimensional world, a two-dimensional gas of electrons canbe realizedby trappingthe electronsinathinlayerbetweentwoslabs ofsemiconductor, sothatatlowenergies,electronmotioninthedirection orthogonaltothelayerisfrozenout. Inasufficientlystrongmagneticfield and at sufficiently low temperature, and if the electrons in the material are sufficiently mobile, the two-dimensional electron gas attains a pro- foundly entangled ground state that is separated from all excited states byanonzeroenergygap. Furthermore,thelow-energyparticleexcitations inthesystemsdonothavethequantumnumbersofelectrons;ratherthey are anyons, and carry electric charges that are fractions of the electron charge. The anyons have spectacular effects on the transport properties of the sample, manifested as the fractional quantum Hall effect. Anyons will be our next topic. But why? True, I have already said enoughtojustifythatanyonsareadeepandfascinatingsubject. Butthis is not a course about the unusual behavior of exotic phases attainable in condensed matter systems. It is a course about quantum computation. In fact, there is a connection, first appreciated by Alexei Kitaev in 1997: anyons provide anunusual, exciting, and perhaps promising means of realizing fault-tolerant quantum computation. So that sounds like something we should be interested in. After all, I have already given 12 lectures on the theory of quantum error correc- tion and fault-tolerant quantum computing. It is a beautiful theory; I have enjoyed telling you about it and I hope you enjoyed hearing about it. But it is also daunting. We’ve seen that an ideal quantum circuit can be simulated faithfully by a circuit with noisy gates, provided the noisy gates are not too noisy, and we’ve seen that the overhead in cir- cuit size and depth required for the simulation to succeed is reasonable. These observationsgreatlyboost ourconfidence thatlargescale quantum computers will really be built and operated someday. Still, for fault tol- erance to be effective, quantum gates need to have quite high fidelity (by the current standards of experimental physics), and the overhead cost of achieving fault tolerance is substantial. Even though reliable quantum computation with noisy gates is possible in principle, there always will6 9 Topological quantum computation be a strong incentive to improve the fidelity of our computation by im- proving the hardware rather than by compensating for the deficiencies of the hardware through clever circuit design. By using anyons, we might achieve fault tolerance by designing hardware withan intrinsicresistance to decoherence and other errors, significantlyreducing the size and depth blowups of our circuit simulations. Clearly, then, we have ample motiva- tion for learning about anyons. Besides, it will be fun In some circles, this subject has a reputation (not fully deserved in my view)forbeingabstruseandinaccessible. Iintendtostartwiththebasics, andnottoclutterthediscussionwithdetailsthataremainlyirrelevantto ourcentralgoals. Thatway,Ihope tokeepthepresentationclearwithout really dumbing it down. Whatarethese goals? Iwillnotbe explaininghowthetheoryofanyons connects with observed phenomena in fractional quantum Hall systems. In particular, abelian anyons arise in most of these applications. From a quantum information viewpoint, abelian anyons are relevant to robust storage of quantum information (and we have already gotten a whiff of that connection in our study of toric quantum codes). We will discuss abelian anyons here, but our main interest will be in nonabelian anyons, whichaswewillseecanbeendowedwithsurprisingcomputationalpower. Kitaev (quant-ph/9707021) pointed out that a system of nonabelian anyonswithsuitablepropertiescanefficientlysimulateaquantumcircuit; thisideawaselaboratedbyOgburnandme(quant-ph/9712048),andgen- eralized by Mochon (quant-ph/0206128, quant-ph/0306063). In Kitaev’s original scheme, measurements were required to simulate some quantum gates. Freedman, Larsen and Wang (quant-ph/000110) observed that if we use the right kindof anyons, allmeasurements canbe postponed until the readout of the final result of the computation. Freedman, Kitaev, and Wang (quant-ph/0001071) also showed that a system of anyons can be simulated efficiently by a quantum circuit; thus the anyon quantum computer and the quantum circuit model have equivalent computational power. The aim of these lectures is to explain these important results. Wewillfocus ontheapplicationsofanyonstoquantumcomputing,not on the equally important issue of how systems of anyons with desirable ∗ properties canbe realizedinpractice. It willbe leftto youto figure that out ∗ Two interesting approaches to realizing nonabelian anyons — using superconduct- ing junction arrays and using cold atoms trapped in optical lattices — have been discussed in the recent literature.9.2 Flux-charge composites 7 9.2 Flux-charge composites Forthoseofus whoareput offby abstractmathematicalconstructions,it willbehelpfultobeginourexplorationofthetheoryofanyonsbythinking aboutaconcretemodel. Solet’sstartbyrecallingamorefamiliarconcept, the Aharonov-Bohm effect. Imagine electromagnetism in a two-dimensional world, where a “flux tube” is a localized “pointlike” object (in three dimensions, you may en- vision a plane intersecting a magnetic solenoid directed perpendicular to the plane). The flux might be enclosed behind an impenetrable wall, so that an object outside can never visitthe regionwhere the magnetic field is nonzero. But even so, the magnetic field has a measurable influence on charged particles outside the flux tube. If an electric chargeq is adiabat- ically transported (counterclockwise) around a flux Φ, the wave function iqΦ of the charge acquires a topological phase e (where we use units with ¯h =c=1). Here the word“topological”means that the Aharonov-Bohm phase is robust when we deform the trajectory of the charged particle — all that matters is the “winding number” of the charge about the flux. The concept of topological invariance arises naturally in the study of fault tolerance. Topological properties are those that remain invariant when we smoothlydeform a system, anda fault-tolerantquantum gate is one whose action on protected information remains invariant (or nearly so) when we deform the implementationof the gate by adding noise. The topologicalinvarianceoftheAharonov-Bohmphenomenonistheessential property that we hope to exploit in the design of quantum gates that are intrinsically robust. We usually regard the Aharonov-Bohm effect as a phenomenon that occurs in quantum electrodynamics, where the photon is exactly mass- less. But it is useful to recognize that Aharonov-Bohm phenomena can also occur in massive theories. For example, we might consider a “super- conducting” system composed of charge e particles, such that composite objects with charge ne form a condensate (where n is an integer). In this superconductor, there is a quantum of flux Φ =2π/ne, the minimal 0 nonzero flux such that a charge-(ne) particle in the condensate, when transported around the flux, acquires a trivial Aharonov-Bohm phase. An isolated region that contains a flux quantum is an island of normal material surrounded by the superconducting condensate, prevented from spreading because the magnetic flux cannot penetrate into the supercon- ductor. That is, it is a stable particle, what we could call a “fluxon.” When one of the charge-e particles is transported around a fluxon, its ieΦ 2πi/n 0 wave function acquires the nontrivial topological phase e = e . But in the superconductor, the photon acquires a mass via the Higgs mechanism, and there are no massless particles. That topological phases8 9 Topological quantum computation are compatible with massive theories is important, because massless par- ticles are easily excited, a potentially copious source of decoherence. Now,let’simaginethat,inour two-dimensionalworld,flux andelectric charge are permanently bound together (for some reason). A fluxon can be envisioned as flux Φ confined inside an impenetrable circular wall, and an electric charge q is stuck to the outside of the wall. What is the angular momentum of this flux-charge composite? Suppose that we carefully rotate the object counterclockwise by angle 2π, returning it to its original orientation. In doing so, we have transported the charge q iqΦ about the flux Φ, generating a topological phase e . This rotation by 2π is represented in Hilbert space by the unitary transformation −i2πJ iqΦ U(2π)=e =e , (9.1) whereJ is the angular momentum. We conclude, then, that the possible eigenvalues of angular momentum are qΦ J =m− (m=integer) . (9.2) 2π We can characterize this spectrum by an angular variable θ ∈ 0,2π), defined by θ = qΦ (mod 2π), and say that the eigenvalues are shifted iθ away from integer values by −θ/2π. We will refer to the phase e that represents a counterclockwise rotationby 2π asthe topological spin of the composite object. But shouldn’t a rotationby 2π act triviallyon a physical system (isn’t it the same as doing nothing)? No, we know better than that, from our experience with spinors in three dimensions. For a system with fermion number F, we have −2πiJ F e =(−1) ; (9.3) if the fermion number is odd, the eigenvalues of J are shifted by 1/2 from the integers. This shift is physically acceptable because there is a F (−1) superselection rule: no observable local operator can change the F value of (−1) (there is no physical process that can create or destroy an isolated fermion). Acting on a coherent superposition of states with F −2πiJ different values of (−1) , the effect of e is −i2πJ e (a even F i+b oddF i)=a evenF i−b oddF i . (9.4) The relative sign in the superposition flips, but this has no detectable F physical effects, since all observables are block diagonal in the (−1) basis. Similarly, in two dimensions, the shift in the angular momentum spec- −2πiJ iθ trum e = e has no unacceptable physical consequences if there is9.3 Spin and statistics 9 a θ superselection rule, ensuring that the relative phase in a superposi- tion of states with different values ofθ is physically inaccessible (not just in practice but even in principle). As for fermions, there is no allowed physical process that can create of destroy an isolated anyon. Inthreedimensions,onlyθ =0,πareallowed,because(asyouprobably know) of a topological property of the three-dimensional rotation group SO(3): a closed path in SO(3) beginning at the identity and ending at a rotation by 4π can be smoothly contracted to a trivial path. It follows that a rotation by 4π really is represented by the identity, and therefore that the eigenvalues of a rotation by 2π are +1 and −1. But the two- dimensionalrotationgroupSO(2)doesnothavethistopologicalproperty, so that any value ofθ is possible in principle. Note that the angular momentum J changes sign under time reversal (T) and also under parity (P). Unless θ = 0 or π, the spectrum of J is asymmetric about zero, and therefore a theory of anyons typically will not be T or P invariant. In our flux-charge composite model the origin of this symmetry breaking is not mysterious— it arises from the nonzero magnetic field. But in a system withno intrinsic breaking ofT andP, if anyonsoccur theneitherthesesymmetriesmustbebrokenspontaneously, or else the particle spectrum must be “doubled” so that for each anyon iθ with exchange phase e there also exists an otherwise identical particle −iθ with exchange phase e . 9.3 Spin and statistics For identical particles in three dimensions, there is a well known connec- tion between spin and statistics: indistinguishable particles with integer spin are bosons, and those with half-odd-integer spin are fermions. In two dimensions, the spin can be any real number. What does this new possibilityof “fractionalspin” implyabout statistics? The answer is that statistics, too, can be “fractionalized” What happens if we perform an exchange of two of our flux-charge composite objects,inacounterclockwisesense? Eachchargeq is adiabat- icallytransported half way aroundthe flux Φ of the other object. We can anticipate, then, that each charge will acquire an Aharonov-Bohm phase that is half of the phase generated by a complete revolutionof the charge about the flux. Adding together the phases arising from the transport of bothcharges,wefindthattheexchange ofthe twoflux-chargecomposites changes their wave function by the phase    1 1 iqΦ iθ −2πiJ exp i qΦ+ qΦ =e =e =e . (9.5) 2 2 The phase generated when the two objects are exchanged matches the10 9 Topological quantum computation phase generated when one of the two objects is rotated by 2π. Thus the connection between spin and statistics continues to hold, in a form that is a natural generalization of the connection that applies to bosons and fermions. Theoriginofthisconnectionis fairlyclearinourflux-charge composite model,butinfactitholdsmuchmoregenerally. Why? Readingtextbooks onrelativisticquantumfieldtheory,onecaneasilygettheimpressionthat the spin-statistics connection is founded on Lorentz invariance, and has something to do with the properties of the complexified Lorentz group. Actually, this impression is quite misleading. All that is essential for a spin-statistics connection to hold is the existence of antiparticles. Special relativity is not an essential ingredient. Considerananyon,characterizedby the phaseθ, andsuppose thatthis particle has a corresponding antiparticle. This means that the particle and its antiparticle, when combined, have trivial quantum numbers (in particular,zero angularmomentum)andthereforethattherearephysical processes in which particle-antiparticle pairs can be created and annihi- lated. Draw a world line in spacetime that represents a process in which two particle-antiparticle pairs are created (one pair on the left and the other pair on the right), the particle from the pair on the right is ex- changed in a counterclockwise sense with the particle from the pair on the left, and then both pairs reannihilate. (The world line has an orien- tation; if directed forward in time it represents a particle, and if directed ◦ backwardintimeitrepresentsanantiparticle.) Turningour diagram90 , we obtain a depiction of a process in which a single particle-antiparticle pair is created, the particle and antiparticle are exchanged in a clock- ◦ wise sense, and then the pair reannihilates. Turning it 90 yet again, we have a process in which two pairs are created and the antiparticle from the pair on the right is exchanged, in a counterclockwise sense, with the antiparticle from the pair on the left, before reannihilation.  1 R R R aa aa aa What do we conclude from these manipulations? Denote by R the ab unitary operator that represents a counterclockwise exchange of particles −1 of types a andb (so that the inverse operatorR represents a clockwise ab exchange), and denote by a¯ the antiparticle ofa. We have found that −1 R =R =R . (9.6) aa a¯a¯ aa¯9.4 Combining anyons 11 iθ If a is an anyon with exchange phase e , then its antiparticlea¯ also has the same exchange phase. Furthermore, when a and a¯ are exchanged −iθ counterclockwise, the phase acquired is e . These conclusions are unsurprising when we interpret them from the perspective of our flux-charge composite model of anyons. The antipar- ticle of the object with flux Φ and charge q has flux −Φ and charge −q. Hence, when we exchange two antiparticles, the minus signs cancel and the effect is the same as though the particles were exchanged. But if we exchange a particle and an antiparticle, then the relative sign of charge −iqΦ −iθ and flux results in the exchange phasee =e . But what is the connection between these observations about statistics andthespin? Continuingtocontemplatethesamespacetimediagram,let us consider itsimplicationsregarding the orientation ofthe particles. For keeping track of the orientation, it is convenient to envision the particle world line not as a thread but as a ribbon in spacetime. I claim that our process can be smoothly deformed to one in which a particle-antiparticle pair is created, the particle is rotated counterclockwise by 2π, and then the pair reannihilates. A convenientwayto verifythis assertionis totake off your belt (or borrow a friend’s). The buckle at one end specifies an orientation;pointyour thumbtowardthe buckle, andfollowingthe right- hand rule, twist the belt by 2π before rebuckling it. You should be able tocheck thatyoucanlayoutthebelt tomatchthe spacetimediagramfor any of the exchange processes described earlier, and also for the process in which the particle rotates by 2π. Thus, ina topologicalsense, rotatinga particlecounterclockwise by 2π is really the same thing as exchanging twoparticles ina counterclockwise sense (orexchangingparticleandantiparticleinaclockwisesense), which † provides a satisfyingexplanationfor a general spin-statisticsconnection. I emphasize againthat thisargument invokesprocesses inwhichparticle- antiparticlepairs are createdandannihilated, andtherefore the existence of antiparticles is an essential prerequisite for it to apply. 9.4 Combining anyons We know that a composite object composed of two fermions is a bo- son. What happens when we build a composite object by combining two anyons? † Actually, this discussion has been oversimplified. Though it is adequate for abelian anyons, we will see that it must be amended for nonabelian anyons, because R has ab more than one eigenvalue in the nonabelian case. Similarly, the discussion in the next section of “combining anyons” will need to be elaborated because, in the nonabelian case, more than one kind of composite anyon can be obtained when two anyons are fused together.12 9 Topological quantum computation iθ Suppose thata is ananyon withexchange phasee , and that we build a “molecule” from n of these a anyons. What phase is acquired under a counterclockwise exchange of the two molecules? The answer is clear in our flux-charge composite model. Each of then iθ/2 charges inone molecule acquires a phasee when transportedhalf way 2 around each of the n fluxes in the other molecule. Altogether then, 2n iθ/2 factors of the phase e are generated, resulting in the total phase 2 iθ in θ n e =e . (9.7) iθ 2 Said another way, the phase e occurs altogether n times because in effect n anyons in one molecule are being exchanged with n anyons in the other molecule. Contrary to what we might have naively expected, if we split a fermion (say) into two identical constituents, the constituents √ iπ 1/4 iπ/4 have, not an exchange phase of −1 =i, but rather (e ) =e . This behavior is compatible with the spin-statistics connection: the angular momentum J of then-anyon molecule satisfies 2 2 −2πiJ −2πin J in θ n e =e =e . (9.8) For example, consider a molecule of two anyons, and imagine rotating the molecule counterclockwise by 2π. Not only does each anyon in the molecule rotate by 2π; in addition one of the anyons revolves around the other. One revolution is equivalent to two successive exchanges, so that i2θ the phase generated by the revolution is e . The total effect of the two rotations and the revolution is the phase i4θ expi(θ+θ+2θ)=e . (9.9) Another wayto understand why the angularmomenta of the anyons in the molecule do not combine additively is to note that the total angular momentum of the molecule consists of two parts — the spin angular momentumS ofeachofthetwoanyons(whichisadditive)andthe orbital angular momentum L of the anyon pair. Because the counterclockwise transport of one anyon around the other generates the nontrivial phase i2θ e , the dependence of the two-anyon wavefunction ψ on the relative azimuthal angleϕ is not single-valued; instead, −i2θ ψ(ϕ+2π)=e ψ(ϕ). (9.10) This means that the spectrum of the orbital angular momentum L is shifted away from integer values: −i2πL 2iθ e =e , (9.11)9.5 Unitary representations of the braid group 13 and this orbital angular momentum combines additively with the spinS to produce the total angular momentum −2πJ =−2πL−2πS =2θ+2θ+ 2π(integer)=4θ+ 2π(integer). (9.12) What if, on the other hand, we build a molecule a¯a from an anyon a anditsantiparticlea¯? Then, as we’veseen, the spinS has the samevalue as fortheaa molecule. But the exchange phase has the opposite value, so thatthenonintegerpartoftheorbitalangularmomentumis−2πL=−2θ instead of −2πL = 2θ, and the total angular momentum J = L+S is an integer. This property is necessary, of course, if the a¯a pair is to be abletoannihilatewithoutleavingbehindanobjectthatcarriesnontrivial angular momentum. 9.5 Unitary representations of the braid group We have already noted that the angular momentum spectrum has differ- ent propertiesintwospatialdimensionsthaninthreedimensions because SO(2) has different topological properties than SO(3) (SO(3) has a com- pact simply connected covering group SU(2), but SO(2) does not). This observation provides one way to see why anyons are possible in two di- mensions but not in three. It is also instructive to observe that particle exchanges have different topological properties in two spatial dimensions than in three dimensions. Aswehavefoundinourdiscussionoftherelationbetweenthestatistics of particlesandof antiparticles,itisuseful toenvisionexchanges ofparti- cles as processes taking place in spacetime. In particular, it is convenient to imagine that we are computing the quantum transition amplitude for a time-dependent process involvingn particles by evaluating a sum over particlehistories(thoughforourpurposesitwillnotactuallybenecessary to calculate any path integrals). Consider a system ofn indistinguishable pointlikeparticles confined to a two-dimensional spatial surface (which for now we may assume is the plane), and suppose thatno twoparticlesare permittedto occupy coinci- dent positions. We maythink ofa configurationofthe particlesata fixed time as a plane with n “punctures” at specified locations — that is, we associatewitheachparticleahole inthesurface withinfinitesimalradius. The condition that the particles are forbidden to coincide is enforced by demanding that there are exactly n punctures in the plane at any time. Furthermore, just as the particles are indistinguishable, each puncture is the same as any other. Thus if we were to perform a permutation of then punctures, this would have no physical effect; all the punctures are the same anyway, so it makes no difference which one is which. All that matters is the n distinct particle positions in the plane.14 9 Topological quantum computation To evaluate the quantum amplitude for a configuration of n particles at specified initial positions at time t = 0 to evolve to a configuration of n particles at specified final positions at time t = T, we are to sum over all classical histories for then particles that interpolate between the fixed initial configuration and the fixed final configuration, weighted by iS the phasee , whereS is the classicalactionof the history. If weenvision each particle world line as a thread, each history for the n particles be- comes a braid, where each particle on the initial (t=0) time slice can be connectedby athreadtoanyone oftheparticlesonthe final(t=T)time slice. Furthermore, since the particle world lines are forbidden to cross, the braids fall into distinct topological classes that cannot be smoothly deformed one to another, and the path integral can be decomposed as a sum of contributions, with each contribution arising from a different topological class of histories. Nontrivialexchange operationsactingonthe particlesonthe finaltime slice change the topological class of the braid. Thus we see that the elementsofthe symmetrygroupgeneratedbyexchangesareinone-to-one correspondence withthetopologicalclasses. This(infinite)groupiscalled B , the braidgrouponnstrands; the groupcompositionlawcorresponds n to concatenation of braids (that is, followingone braid with another). In thequantumtheory,thequantumstateofthenindistinguishableparticles belongs to a Hilbert space that transforms as a unitary representation of the braid groupB . n The group can be presented as a set of generators that obey particular defining relations. To understand the defining relations, we may imag- ine that then particles occupy n ordered positions (labeled 1,2,3,...,n) arranged on a line. Let σ denote a counterclockwise exchange of the 1 particles that initially occupy positions 1 and 2, let σ denote a counter- 2 clockwise exchange of the particles that initially occupy positions 2 and 3, and so on. Any braid can be constructed as a succession of exchanges of neighboring particles; hence σ ,σ ,...,σ are the group generators. 1 2 n−1 The defining relations satisfied by these generators are of two types. The first type is σ σ =σ σ , j−k≥2 , (9.13) j k k j whichjustsaysthatexchangesofdisjointpairsofparticlescommute. The second, slightly more subtle, type of relation is σ σ σ =σ σ σ , j =1,2,...,n−2 , (9.14) j j+1 j j+1 j j+1 which is sometimes called the Yang-Baxter relation. You can verify the Yang-Baxter relation by drawing the two braids σ σ σ and σ σ σ on 1 2 1 2 1 2 a piece of paper, and observing that both describe a process in which the particlesinitiallyinpositions1 and 3 are exchanged counterclockwise9.5 Unitary representations of the braid group 15 about the particle labeled 2, which stays fixed — i.e., these are topologi- cally equivalent braids. VV 1 2 VV 2 1 VV 1 2 Since the braid group is infinite, it has an infinite number of unitary irreduciblerepresentations,andinfactthereareaninfinitenumberofone- dimensionalrepresentations. Indistinguishableparticlesthattransformas a one-dimensionalrepresentationofthe braidgroupare saidtobe abelian anyons. Intheone-dimensionalrepresentations,eachgeneratorσ ofB is j n iθ j represented by a phaseσ =e . Furthermore, the Yang-Baxter relation j iθ iθ iθ iθ iθ iθ iθ iθ iθ j j+1 j j+1 j j+1 j j+1 becomes e e e =e e e , which implies e =e ≡e — all exchanges are represented by the same phase. Of course, that makes sense; if the particles are really indistinguishable, the exchange phase ought not to depend on which pair is exchanged. For θ = 0 we obtain bosons, and forθ =π, fermions The braid group also has many nonabelian representations that are of dimension greater than one; indistinguishable particles that transform as such representations are said to be nonabelian anyons (or, sometimes, nonabelions). Tounderstandthephysicalpropertiesofnonabeliananyons we will need to understand the mathematical structure of some of these representations. In these lectures, I hope to convey some intuition about nonabelian anyons by discussing some examples in detail. For now, though, we can already anticipate the main goal we hope to fulfill. For nonabelian anyons, the irreducible representation of B real- n izedbynanyonsactsona“topologicalvectorspace”V whosedimension n D increases exponentially with n. And for anyons with suitable prop- n erties, the image of the representation may be dense in SU(D ). Then n braiding of anyons can simulate a quantum computation— any (special) unitary transformationacting on the exponentially large vector space V n can be realized witharbitrarilygood fidelity by executing a suitably cho- sen braid. Thus we are keenly interested in the nonabelian representations of the braid group. But we should also emphasize (and will discuss at greater16 9 Topological quantum computation length later on) that there is more to a model of anyons than a mere rep- resentationof the braid group. In our flux tube model of abelian anyons, wewereabletodescribe notonlytheeffectsofanexchangeofanyons,but also the types of particlesthat canbe obtainedwhen two or more anyons are combined together. Likewise, in a general anyon model, the anyons are of various types, and the model incorporates “fusionrules” that spec- ify what types can be obtained when two anyons of particular types are combined. Nontrivial consistency conditions arise because fusion is asso- ciate (fusing a with b and then fusing the result with c is equivalent to fusingb withc and then fusing the result witha), and because the fusion rules must be consistent with the braiding rules. Though these consis- tency conditions are highly restrictive, many solutions exist, and hence many different models of nonabelian anyons are realizable in principle. 9.6 Topological degeneracy But before moving on to nonabelian anyons, there is another important idea concerning abelian anyons that we should discuss. In any model of anyons (indeed, inany localquantum system witha mass gap), there is a ground state or vacuum state, the state in whichno particles are present. Ontheplanethegroundstateisunique,butforatwo-dimensionalsurface withnontrivialtopology,thegroundstateisdegenerate,withthedegreeof degeneracy depending onthe topology. We havealreadyencounteredthis phenomenon of “topological degeneracy” in the model of abelian anyons that arose in our study of a particular quantum error-correcting code, Kitaev’s toric code. Now we will observe that topologicaldegeneracy is a general feature of any model of (abelian) anyons. We can arrive at the concept of topological degeneracy by examining the representationsof a simpleoperator algebra. Consider the case of the torus, represented as a square withopposite sides identified, and consider the two fundamental 1-cycles of the torus: C , which winds around the 1 squareinthex direction,andC whichwindsaroundinthex direction. 1 2 2 A unitary operator T can be constructed that describes a process in 1 which an anyon-antianyon pair is created, the anyon propagates around C , and then the pair reannihilates. Similarly a unitary operator T can 1 2 be constructed that describes a process in which the pair is created, and the anyon propagates around the cycle C before the pair reannihilates. 2 Each of the operatorsT andT preserves the ground state of the system 1 2 (thestatewithnoparticles);indeed,eachcommuteswiththeHamiltonian H of the system and so either can be simultaneously diagonalized with H (T andT are both symmetries). 1 2 However, T and T do not commute with one another. If our torus 1 2 has infinite spatial volume, and there is a mass gap (so that the only9.6 Topological degeneracy 17 interactions among distantly separated anyons are due to the Aharonov- Bohm effect), then the commutator of T andT is 1 2 −1 −1 −i2θ T T T T =e I , (9.15) 2 1 2 1 iθ where e is the anyon’s exchange phase. The nontrivial commutator arises because the process in which (1) an anyon winds around C , (2) 1 an anyon winds around C (3) an anyon winds around C in the reverse 2 1 direction, and (4) an anyon winds around C in the reverse direction, is 2 topologically equivalent to a process in which one anyon winds clockwise −1 −1 around another. To verify this claim, view the action of T T T T 2 1 2 1 as a process in spacetime. First note that the process described by the −1 operatorT T , inwhichan anyonworldline first sweeps thoughC and 1 1 1 then immediately traverses C in the reverse order, can be deformed to 1 a process in which the anyon world line traverses a topologically trivial loop that can be smoothly shrunk to a point (in keeping with the prop- −1 erty that T T is really the identity operator). In similar fashion, the 1 1 −1 −1 process described by the operator T T T T can be deformed to one 2 1 2 1 where the anyon world lines traverse two closed loops, but such that the world lines link once with one another; furthermore, one loop pierces the surface bounded by the other loop in a direction opposite to the orien- tation inherited by the surface via the right-hand rule from its bounding loop. This process can be smoothly deformed to one in which two pairs are created, one anyon winds clockwise around the other, and then both pairs annihilate. The clockwise winding is equivalent to two successive clockwise exchanges, represented in our one-dimensional representation −i2θ of the braid group by the phase e . We conclude that T and T are 1 2 noncommuting, except in the cases θ =0 (bosons) and θ =π (fermions). 2 118 9 Topological quantum computation SinceT andT both commute withthe HamiltonianH, both preserve 1 2 the eigenspaces of H, but since T and T do not commute with one 1 2 another, they cannot be simultaneouslydiagonalized. SinceT is unitary, 1 its eigenvalues are phases; let us use the angular variable α ∈ 0,2π) to iα label an eigenstate ofT with eigenvaluee : 1 iα T αi=e αi . (9.16) 1 Then applyingT to the T eigenstate advances the value of α by 2θ: 2 1 i2θ i2θ iα T (T αi)=e T T αi =e e (T αi) . (9.17) 1 2 2 1 2 Suppose that θ is a rational multiple of 2π, which we may express as θ =πp/q , (9.18) whereqandp(p2q)arepositiveintegerswithnocommonfactor. Then we conclude that T must have at least q distinct eigenvalues; T acting 1 1 onα generates an orbit withq distinct values:   2πp α+ k (mod 2π), k =0,1,2,...,q−1 . (9.19) q SinceT commutes withH, on the torus the ground state of our anyonic 1 system (indeed, any energy eigenstate)must havea degeneracy thatis an integer multiple of q. Indeed, generically (barring further symmetries or accidental degeneracies), the degeneracy is expected to be exactlyq. Foratwo-dimensionalsurfacewithgenusg(aspherewithg “handles”), g the degree of this topological degeneracy becomes q , because there are operators analogous to T and T associated with each of the g handles, 1 2 and all of theT -like operators can be simultaneously diagonalized. Fur- 1 thermore, we can apply a similar argument to a finite planar medium if singleanyonscanbecreatedanddestroyedattheedgesofthesystem. For example, consider an annulus in which anyons can appear or disappear at the inner and outer edges. Then we could define the unitary opera- tor T as describing a process in which an anyon winds counterclockwise 1 around the annulus, and a unitary operatorT as describing a process in 2 which an anyon appears at the outer edge, propagates to the inner edge, and disappears. These operators T and T have the same commutator 1 2 as the corresponding operators defined on the torus, and so we conclude as before that the ground state on the annulus is q-fold degenerate for θ = πp/q. For a disc with h holes, there is an operator analogous to T that winds an anyon counterclockwise around each of the holes, and 1 an operator analogous to T that propagates an anyon from the outer 2 h boundary of the disk to the edge of the hole; thus the degeneracy is q .9.6 Topological degeneracy 19 What we have described here is a robust topological quantum memory. i2θ i2πp/q The phase e = e ≡ ω acquired when one anyon winds counter- clockwise around another is a primitiveqth root of unity, and in the case of a planar system withholes, the operatorT can be regarded as the en- 1 ¯ coded Pauli operator Z acting on a q-dimension system associated with s ¯ a particularhole. Physically, the eigenvalueω ofZ just counts the num- ber s of anyons that are “stuck” inside the hole. The operator T can 2 ¯ be regarded as the complementary Pauli operatorX that increments the value of s by carrying one anyon from the boundary of the system and depositing it in the hole. Since the quantum information is encoded in a nonlocal property of the system, it is well protected from environmental decoherence. By the same tokendepositing a quantum state in the mem- ory, and reading it out, might be challenging for this system, though in ¯ principleZ couldbe measured by, say, performing an interference experi- ment in which an anyon projectile scatters off of a hole. We will see later that by using nonabelian anyons we will be able to simplify the readout; in addition, with nonabelian anyons we can use topological properties to process quantum informationas well as to store it. Just how robust is this quantum memory? We need to worryabout er- rors due to thermal fluctuations and due to quantum fluctuations. Ther- mal fluctuations might excite the creation of anyons, and thermal anyons might diffuse around one of the holes in the sample, or from one bound- ary to another, causing an encoded error. Thermal errors are heavily −Δ/T suppressed by the Boltzman factor e , if the temperature T is suffi- ciently small compared to the energy gap Δ (the minimal energy cost of creating a single anyon at the edge of the sample, or a pair of anyons in the bulk). The harmful quantum fluctuations are tunneling processes in which a virtual anyon-antianyon pair appears and the anyon propagates around a hole before reannihilating, or a virtual anyon appears at the edge of a hole and propagates to another boundary before disappearing. These errorsdue toquantumtunnelingareheavilysuppressedifthe holes are sufficiently large and sufficiently well separatedfrom one another and ‡ from the outer boundary. Notethatourconclusionthatthetopologicaldegeneracyisfinitehinged on the assumption that the angle θ is a rational multiple of π. We may say that a theory of anyons is rational if the topological degeneracy is finite for any surface of finite genus (and, for nonabelian anyons, if the ‡ If you are familiar with Euclidean path integral methods, you’ll find it easy to verify that in the leading semiclassical approximation the amplitudeA for such a tunneling −L/L 0 process in which the anyon propagates a distance L has the form A = Ce , ∗ −1/2 ∗ where C is a constant and L = ¯h(2m Δ) ; here ¯h is Planck’s constant and m 0 is the effective mass of the anyon, defined so that the kinetic energy of an anyon ∗ 2 1 traveling at speed v is m v . 220 9 Topological quantum computation topological vector space V is finite-dimensional for any finite number of n anyonsn). Wemayanticipatethattheanyonsthatariseinanyphysically reasonable system will be rational in this sense, and therefore should be expected to have exchange phases that are roots of unity. 9.7 Toric code revisited Ifthese observationsabouttopologicaldegeneracyseem hauntinglyfamil- iar, it may be because we used quite similar arguments in our discussion of the toric code. The toric code can be regarded as the (degenerate) ground state of a system of qubits that occupy the links of a square lattice on the torus, with Hamiltonian X X 1 H =− Δ Z + X , (9.20) P S 4 P S where the plaquette operator Z =⊗ Z is the tensor product of Z’s P `∈P ` acting on the four qubits associatedwiththe links contained inplaquette P, and the site operator X ⊗ X is the tensor product of X’s acting S `3S ` onthefourqubitsassociatedwiththelinksthatmeetatthe siteS. These plaquetteandsiteoperatorsarejustthe(commuting)stabilizergenerators for the toric code. The ground state is the simultaneous eigenstate with eigenvalue 1 of all the stabilizer generators. This model has two types of localized particle excitations — plaquette excitationswhereZ =−1, whichwemightthinkofas magneticfluxons, P and site excitations where X =−1, which we might think of as electric S charges. A Z error acting on a link creates a pair of charges on the two site joined by the link, and an X error acting on a link creates a pair of fluxons on the two plaquettes that share the link. The energy gap Δ is the cost of creating a pair of either type. The charges are bosons relative to one another (they have a trivial iθ exchange phase e = 1), and the fluxons are also bosons relative to one another. Since the fluxons are distinguishable from the charges, it does not make sense to exchange a charge with a flux. But what makes this an anyon model is that a phase (−1) is acquired when a charge is carried around a flux. The degeneracy of the ground state (the dimension of the code space) can be understood as a consequence of this property of the particles. For this model on the torus, because there are two types of particles, there are twotypes ofT operators: T , whichpropagatesa charge (site 1 1,S defect) around the 1-cycleC , and T , which propagates a fluxon (pla- 1 1,P quette defect) around C . Similarly there are two types of T operators, 1 2

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