Majorana Zero Modes and Topological Quantum Computation Sankar Das Sarma,1,2 Michael Freedman,2 and Chetan Nayak2,3 1Department of Physics, University of Maryland, College Park, MD 20742 2Microsoft Station Q, University of California, Santa Barbara, CA 93108 3Department of Physics, University of California, Santa Barbara, California 93106, USA WeprovideacurrentperspectiveontherapidlydevelopingfieldofMajoranazeromodesinsolid statesystems. Weemphasizethetheoreticalprediction,experimentalrealization,andpotentialuse ofMajoranazeromodesinfutureinformationprocessingdevicesthroughbraiding-basedtopological quantumcomputation. Well-separatedMajoranazeromodesshouldmanifestnon-Abelianbraiding statisticssuitableforunitarygateoperationsfortopologicalquantumcomputation. Recentexperi- 5 mental work, following earlier theoretical predictions, has shown specific signatures consistent with 1 the existence of Majorana modes localized at the ends of semiconductor nanowires in the presence 0 of superconducting proximity effect. We discuss the experimental findings and their theoretical 2 analyses, and provide a perspective on the extent to which the observations indicate the existence y of anyonic Majorana zero modes in solid state systems. We also discuss fractional quantum Hall a systems (the 5/2 state), which have been extensively studied in the context of non-Abelian anyons M and topological quantum computation.. We describe proposed schemes for carrying out braiding with Majorana zero modes as well as the necessary steps for implementing topological quantum 4 computation. 1 ] I. INTRODUCTION ations constitute the elementary gate operations for the el evolution of the topological quantum computation. r- Topological quantum computation [1, 2], is an ap- Perhaps the simplest realization of a non-Abelian st proach to fault-tolerant quantum computation in which anyon is a quasiparticle or defect supporting a Majo- . the unitary quantum gates result from the braiding rana zero mode (MZM). (The zero-mode here refers t a of certain topological quantum objects, called ‘anyons’. to the zero-energy midgap excitations that these lo- m Anyons braid nontrivially: two counter-clockwise ex- calized quasiparticles typically correspond to in a low- - changes do not leave the state of the system invariant, dimensional topological superconductor.) This is a real d unlike in the cases of bosons or fermions. Anyons can fermionicoperatorthatcommuteswiththeHamiltonian. n o arise in two ways: as localized excitations of an inter- The existence of such operators guarantees topological c acting quantum Hamiltonian [3] or as defects in an or- degeneracyand,asweexplaininSectionII,braidingnec- [ dered system [4, 5]. Fractionally-charged excitations of essarily causes non-commuting unitary transformations theLaughlinfractionalquantumHallliquidareanexam- to act on this degenerate subspace. The term “Majo- 2 v ple of the former. Abrikosov vortices in a topological su- rana” refers to the fact that these fermion operators are 3 perconductorareanexampleofthelatter. Notallanyons real, as in Majorana’s real version of the Dirac equation. 1 are directly useful in topological quantum computation; However, there is little connection with Majorana’s orig- 8 only non-Abelian anyons are useful, which does not in- inalworkoritsapplicationtoneutrinos. Rather,thekey 2 clude the anyonic excitations (sometimes referred to as concept here is the non-Abelian anyon, and MZMs are a 0 Abelian anyons, to distinguish them from the more ex- particular mechanism by which a particular type of non- . 1 otic non-Abelian anyons which are useful for topological Abelian anyons, usually called “Ising anyons” can arise. 0 quantumcomputation)thatarebelievedtooccurinmost By contrast, Majorana fermions, as originally conceived, 5 odd-denominator fractional quantum Hall states. A col- obey ordinary Fermi-Dirac statistics, and are simply a 1 lectionofnon-Abeliananyonsatfixedpositionsandwith particular type of fermion. Although the terminology : v fixedlocalquantumnumbershasanon-trivialtopological ‘Majorana fermions’ is somewhat misleading for MZMs, Xi degeneracy (which is, therefore, robust – i.e. immune to it is used extensively in the literature. weak local perturbations). This topological degeneracy If Majorana zero modes can be manipulated and r a allows quantum computation since braiding enables uni- theirstatesmeasuredinwell-controlledexperiments,this taryoperationsbetweenthedistinctdegeneratestatesof could pave the way towards the realization of a topolog- the system. The unitary transformations resulting from ical quantum computer. The subject got a tremendous braidingdependonlyonthetopologicalclassofthebraid, boost in 2012 when an experimental group in Delft pub- thereby endowing them with fault-tolerance. This topo- lishedevidencefortheexistenceofMajoranazeromodes logical immunity is protected by an energy gap in the inInSbnanowires[6],followingearliertheoreticalpredic- system and a length scale discussed below. As long as tions [7–9]. The specific experimental finding, which has the braiding operations are slow compared with the in- beenreproducedlaterinotherlaboratories,isazero-bias verse of the energy gap and external perturbations are tunneling conductance peak in a semiconductor (InSb not strong enough to close the gap, the system remains or InAs) nanowire in contact with an ordinary metallic robust to disturbances and noise. These braiding oper- superconductor (Al or Nb), which shows up only when 2 a finite external magnetic field is applied to the wire. therequiredextremehighsamplequality(mobility>107 Severalotherexperimentalgroupsalsosawevidence(i.e. cm2/V.s), very low <25mK temperature and high mag- zerobiastunnelingconductancepeakinanappliedmag- netic field > 2T. The second system is the semiconduc- netic field) for the existence of Majorana zero-modes in tor nanowire structure proposed in Refs. [7, 8], building both InSb and InAs nanowires [10–14], thus verifying uponearliertheoreticalworkontopologicalsuperconduc- the Delft finding. However, though these experiments tors [33–36]. Semiconductor nanowires are the focus of are compelling, they do not show exponential localiza- this paper, but the 5/2 fractional quantum Hall state is tionwithsystemlengthrequiredbyEq. (eqn:MZM-real- a useful point of comparison since a great deal of exper- def) or anyonic braiding behavior. As explained later in imental and theoretical work has been done on the 5/2 this article, the exponential localization of the isolated FQHS over the last 27 years. Majorana modes at wire ends and the associated non- Abelian braiding properties are the key features which enable topological quantum computation to be possible II. WHAT IS A MAJORANA ZERO MODE? in these systems. In the current article, we provide a perspective on A Majorana zero mode (MZM) is a fermionic oper- wherethisinterestingandimportantsubjectistoday(at ator γ that squares to 1 (and, therefore, is necesarily the end of 2014). This is by no means a review article self-adjoint) and commutes with the Hamiltonian H of a forthefieldofMajoranazeromodesorthetopicoftopo- system: logical quantum computation since such reviews will be too lengthy and too technical for a general readership. γ fermionic, γ2 =1, [H,γ]=0 (1) There are, in fact, several specialized review articles al- Any operator that satisfies the first two conditions is ready discussing various aspects of the subject matter called a Majorana fermion operator. If it satisfies the which we mention here for the interested reader. The third condition, as well, then it is a Majorana zero mode subject of topological quantum computation has been operatoror,simply,aMajoranazeromode[37]. Theexis- reviewed by us in great length earlier [3], and we have tence of such operators implies the existence of a degen- also written a shorter version of anyonic braiding-based erate space of ground states, in which quantum informa- topological quantum computation elsewhere [15]. There tioncanbestored. Ifthereare2nMajoranazeromodes, arealsoseveralexcellentpopulararticlesonthebraiding γ ,...γ (they must come in pairs since each MZM is, of non-Abelian anyons and topological quantum compu- 1 2n in a sense, half a fermion) satisfying tation [16, 17]. The theory of Majorana zero-modes and their potential application to topological quantum com- {γ ,γ }=2δ (2) putation has recently been reviewed in great technical i j ij depth in several articles [18–21]. then the Hamiltonian can be simultaneously diagonal- Thereareessentiallytwodistinctphysicalsystemsthat ized with the operators iγ γ , iγ γ , ..., iγ γ . The 1 2 3 4 2n−1 2n have been primarily studied in the search for Majorana ground states can be labelled by the eigenvalues ±1 of zeromodesfortopologicalquantumcomputation(TQC). these n operators, thereby leading to a 2n-fold degener- Thefirstistheso-called5/2-fractionalquantumHallsys- acy. There is a two-state system associated with each tem (5/2-FQHS) where the application of a strong per- pair of MZMs. This is to be contrasted with a collection pendicular magnetic field to a very high-mobility two- ofspin-1/2particles,forwhichthereisatwo-statesystem dimensional(2DEG)electrongas(confinedinepitaxially- associated with each spin. In the case of MZMs, we are grown GaAs-AlGaAs quantum wells) leads to the even- free to pair them however we like; different pairings cor- denominator fractional quantization of the Hall resis- respondtodifferentchoicesofbasisinthe2n-dimensional tance. The generic fractional quantum Hall effect leads ground state Hilbert space. to the quantization withodd-denominator fractions (e.g. Unfortunately,theprecedingmathematicsistooideal- the original 1/3 quantization observed in the famous ex- izedforarealphysicalsystem. Ifwearefortunate, there periment by Tsui, Stormer, and Gossard in 1982 [22]). can, instead, be self-adjoint Majorana fermion operators Interestingly, of the almost 100 FQHS states that have γ ,...γ satisfying the anti-commutation relations (2) 1 2n so far been observed in the laboratory, the 5/2-FQHS and is the only even-denominator state ever found in a sin- gle 2D layer. It has been hypothesized that this even- [H,γ ]∼e−x/ξ (3) i denominator state supports Ising anyons. A topologi- cal qubit was proposed by us for this platform [23] in where x is a length scale mentioned in the introduction 2005,buildinguponprevioustheoreticalworkonthe5/2 (which can be construed to be the separation between state [24–28]. Tantalizing experimental signatures for two MZMs in the pair) and discussed momentarily, and the possible existence of the desired non-Abelian any- ξ isacorrelationlengthassociatedwiththeHamiltonian onic properties were reported in subsequent experiments H. Inthesuperconductingsystemsthatwillbediscussed [29–32]. However,theseresultshavenotbeenreproduced in the sections to follow, ξ will be the superconducting in other laboratories. Potential barriers to progress are coherence length. All states above the 2n−1-dimensional 3 low-energy subspace have a minimum energy ∆. In or- change their signs. Moreover, fermion parity must be der for the definition (3) to approach the ideal condition conserved, which dictates that γ and γ must pick up 1 2 (1), it must be possible to make x sufficiently large that opposite signs. Hence, the transformation law is: the right-hand-side of Eq. (3) approaches zero rapidly. γ →±γ , γ →∓γ (4) This can occur if the operators γ are localized at points 1 2 2 1 i x (which we have not, so far, assumed). Then γ com- i i The overall sign is a gauge choice. This transformation mutesoranti-commutes,uptocorrections∼e−y/ξ,with, is generated by the unitary operator: respectively, alllocalbosonicorfermionicoperatorsthat can be written in terms of electron creation and annihi- U =eiθeπ4γ1γ2 (5) lation operators whose support is a minimum distance y This is the braiding transformation of Ising anyons. from some point x . The effective Hamiltonian for ener- i Strictly speaking, Ising anyons have θ =π/8. Other val- gies much lower than ∆ is a sum of local terms, which ues of θ can occur if there are additional Abelian anyons meansthatproductsofoperatorssuchasiγ γ musthave i j attached to the Ising anyons, as is believed to occur in exponentially-small coefficients ∼ e−|xi−xj|/ξ [38]. Con- theν =5/2fractionalquantumHallstate. Inthecaseof sequently, thecondition(3)thenholds[93]. Thenumber defects, rather than quasiparticles, the phase θ will not, of Majorana zero mode operators satisfying (3) must be in general, be universal, and will depend on the particu- even. Consequently,ifweaddatermtotheHamiltonian lar path through which the defects were exchanged. We that couples a single zero mode operator to the non-zero emphasize that this braiding transformation law follows mode operators, a zero mode operator will remain since from (a) the reality condition of the Majorana fermion zero modes can only be lifted in pairs. Thus, the expo- operators γ , (b) the locality of the MZMs, and (c) nential‘protection’oftheMZMsallowingtheirquantum 1,2 conservationoffermionparity. Therefore,anexperimen- degeneracyisenabledbytheenergygap,whichshouldbe tal observation consistent with such a braiding transfor- as large as possible for effective TQC operations. Thus, mation is evidence that (a)-(c) hold. This, in turn is in a loose sense, two Majoranas together give a Dirac evidence that the defects or quasiparticles support Ma- fermion, and these two MZMs must be far away from jorana zero modes satisfying the definition (3). Such a each other for the exponential topological protection to direct experimental observation of braiding has not yet apply. happened in the laboratory. It is useful to combine the two MZMs into a single In the case of quasiparticles in topological phases, Diracfermionc=γ +iγ . Thetwostatesofthispairof 1 2 braiding properties, as revealed through various con- zeromodescorrespondstothefermionparitiesc†c=0,1. crete proposed interference experiments such as those Thus,ifthetotalfermionparityofasystemisfixed,then proposed in Refs. 23, 27, 40, 41, is, perhaps, the gold the degeneracy of 2n MZMs is 2n−1-fold. This quantum standard for detecting MZMs. However, in the case of degeneracy, arising from the topological nature of the defects in ordered states and, in particular, in the spe- MZMs,enablesTQCtobefeasiblebybraidingtheMZMs cial case of MZMs in superconductors, a zero-bias peak around each other. in transport with a normal lead [42] and a 4π periodic SuchlocalizedMZMsareknowntooccurintworelated Josephson effect [34] are also signatures, as discussed in butdistinctphysicalsituations. Thefirstisatadefectin SectionIV.BeforediscussingtheseinmoredetailinSec- anorderedstate,suchasavortexinasuperconductoror tion IV, it may be helpful to discuss the differences be- a domain wall in a 1D system. The defect does not have tween topological superconductors and true topological finite energy in the thermodynamic limit and, therefore, phases. it is not possible to excite a pair of such defects at finite energycostandpullthemapart. However,bytuningex- perimental parameters (which involves energies propor- III. MAJORANA ZERO MODES IN tional to the system size), such defects can be created in TOPOLOGICAL PHASES AND IN pairs, thereby creating pairs of MZMs. The best exam- TOPOLOGICAL SUPERCONDUCTORS pleofthisisatopologicalsuperconductor. Alternatively, there may be finite-energy quasiparticle excitations of a As noted in the Introduction, Ising anyons can be un- topological phase [3] that support zero modes. This sce- derstood as quasiparticles or defects that support Ma- nario is believed to be realized in the ν =5/2 fractional jorana zero modes. In the Moore-Read Pfaffian state quantum Hall states, where charge e/4 excitations are [24, 25] and the anti-Pfaffian state [43, 44], proposed as hypothesized to support MZMs. Although the cases of candidate non-Abelian states for the 5/2 FQHS, charge- defectsintopologicalsupercondcutorsandquasiparticles e/4 quasiparticles are Ising anyons [26, 45–51]. There in”true”topologicalphasesareclosely-related,thereare is theoretical [28, 52–59] and experimental [29–32, 60– some important differences, touched on later. 65] evidence that the ν = 5/2 fractional quantum Hall When two defects or quasiparticles supporting MZMs state is in one of these two universality classes. How- are exchanged while maintaining a distance greater than ever, there are also some experiments [66–69] that do ξ, their MZMs must also be exchanged. Since the γ not agree with this hypothesis. The non-Abelian statis- i operators are real, the exchange process can, at most, tics of quasiparticles at ν = 5/2 has been reviewed in 4 Ref. 3 and would require a digression into the physics acting from one of the system’s ground states. “Quasi” of the fractional quantum Hall effect. Hence, we do not permits low-energy excitations (below the gap) provided elaborate on it here, other than to note that Ising-type they are not “topological”. These subgap excitations fractional quantum Hall states are very nearly topologi- surelydoexistinrealtopologicalsuperconductors: there calphases,apartfromsomedeviationsthataresalienton will be phonons and there will be gapless excitations of higher-genussurfaces[70]. However,theelectricalcharge the superconducting order parameter - both are Gold- that is attached to Ising anyons enables their detection stonemodesofbrokensymmetries(translationinthefirst through charge transport experiments [23, 27, 40, 41]. case and U(1)-charge conservation in the second). (The Isinganyonsalsooccurinsomelatticemodelsofgapped, reader may wonder why the now-so-famous Higgs mech- topologically-orderedspinliquids[71,72]. Thesearetrue anism fails to gap the Goldstone mode of broken U(1). topological phases in which the MZM operators are as- The answer is the mismatch of dimensions, the gauge sociated with finite-energy excitations of the system and field roams 3-dimensional space while the superconduc- do not have a local relation to the underlying spin op- tor lives in either two or one dimension. In the former erators, much less the electron operators, whose charge case, the interaction with the gauge field causes super- √ degree of freedom is gapped. This limits the types of conductingphasefluctuationstohavedispersionω ∼ q effects (in comparison to the superconducting case) that while in the latter case ω ∼ q. In a bulk 3D super con- could break the topological degeneracy implied by Eqs. ductorthegaugebosonisindeedgappedout.) Themore 1 and 2. serious caveat is fermion parity protected. This is simul- taneously a blessing and a curse for any project to com- MZMs also occur at defects in certain types of super- putewithMajoranazeromodesinsuperconductors. The conductorsthatformasubsetoftheclassgenerallycalled blessing is that the basis states of the topological qubit “topological superconductors” [33, 34, 73]. We discuss havethispreciseinterpretation: fermionparity. Ifweare these in general terms in this section and then in the willing to move into an unprotected regime to measure context of specific physical realizations in Section V. them, MZMs can be brought together and their charge Topological phases have some topological features and parity detected locally. Using more sophistication, one someordinarynon-topologicalfeatures. However,thein- could keep the MZMs at topological separation and ex- terplay between these two types of physics is even more ploit the Aharonov-Casher effect to measure the charge central in topological superconductors. This is both parity encircled by a vortex. So this coupling will allow “bad” and “good.” It is bad if the nontopological fea- measurement by physics very well in hand. (It is less tures represent an opportunity for error or lead to en- clear how to do this with, for instance, the computation- ergy splittings that decohere desirable superpositions. It ally more powerful Fibonacci anyons [3].) Measurement isgoodwhentheyallowaconvenientcouplingtoconven- iscrucialforprocessingquantuminformationwithMZMs tional physics, something we had better have available if since the braid group representation for Ising anyons is weeverwishtomeasurethetopologicalsystem. Intopo- a rather modest finite group: beyond input and output, logical phases, there is a trivial tensor product situation distillation of quantum states is needed [74], and this inwhichthetopologicalandtheordinarydegreesoffree- is measurement intensive. The curse is quasi-particle- dom do not talk to each other. In this case, we do not poisoning. A nearby electron can enter the system and havetoworrythatthelatterinduceerrorsintheformer, be absorbed by a Majorana zero mode, thereby flipping but they also will not be useful in initializing or mea- the fermion parity – i.e. flipping a qubit. The electrons’ suring the topological degrees of freedom. (As always, charge is absorbed by the superconducting condensate. in discussing topological physics, we regard effects that This propensity of a topological superconductor to be diminish exponentially with length, frequency, or tem- poisoned (or equivalently, the fermion parity to flip in peratureasunimportant. Thisissomewhatanalogousto an uncontrolled manner) represents a salient distinction computer scientists classifying algorithms as polynomial fromtheMoore-Readstateproposedfortheν =5/2frac- timeorslower. Clearlythepowerandeventheconstants tional quantum Hall state. In the Moore-Read state, the can make a difference, but such a structural dichotomy vortices carry electric charge (±e/4) and fermions carry is a useful starting point.) So, for example, if there are charge 0 or ±1/2. Consequently, there is an energy gap phonons in a system, their interaction with topological to bringing an electron from the outside into a ν = 5/2 degreesoffreedomcausesasplittingofthetopologicalde- FQHEfluid. ItsfermionparitycanbeabsorbedbyaMa- generacythatvanishesase−L/ξ atzerotemperature[70], joranazeromode(asinthecaseofatopologialsupercon- so we would consider the system as essentially a tensor ductor), but there is no condensate to absorb its charge; product,withthephononsinaseparatefactor. However, instead, four disjoint charge-e/4 quasiparticles must be a topological superconductor is not a true topological created, with their attendant energy cost. It would be phase but, rather, following the terminology of Ref. 70 harder to poison a ν = 5/2 fluid but also harder to dis- a fermion parity protected quasi topological phase. The cern its state and the signatures discussed in the next qualifier “quasi” permits the existence of benign gapless section are not available for non-Abelian FQHS states. modes as discussed above. With slightly more precision: Thus, one must choose between potentially better pro- an excitation is topological if its local density matrices tection (5/2 FQHE) or easier measurement (topological cannot be produced to high fidelity by a local operator 5 superconductor). can, in principle, be ruled out by further experiments. Thus the observation of perfect Andreev reflection, with theassociatedquantizedconductanceatzerobias,robust IV. SIGNATURES OF MZMS IN to parameter changes, is an indication of the presence of TOPOLOGICAL SUPERCONDUCTORS aMajoranazeromode. InSectionVI,wediscusstheex- tenttowhichthisquantizedtunnelingconductanceasso- Duetothesuperconductingorderparameter,itispos- ciatedwiththezero-energymidgapMajoranamodeshas sible for an electron to tunnel directly into a MZM in a actually been observed in experiments. superconductor. SupposethereisaMZMγ attheorigin A second probe of Majorana zero modes that is spe- x = 0 in a superconductor. Then, if we bring a metallic cial to topological superconductors is the the so-called wire near the origin, electrons can tunnel from the lead fractional Josephson effect. When two normal supercon- to the superconductor via a coupling of the form ductors are in electrical contact, separated by a thin in- sulator or a weak link, the dominant coupling between H =λc†(0) γe−iθ(0)/2+λ∗γ c(0) eiθ(0)/2 (6) them at low temperatures is tun where c(0) is the electron annihilation operator in the H =−Jcosθ (8) lead. For simplicity, we have suppressed the spin index, which is a straightforward notational choice if the su- where θ is the difference in the phases of the order pa- percondutor and the lead are both fully spin-polarized. rameters of the two superconductors. It is periodic in θ In the more generic case, the spin index must be han- with period 2π. The Josephson current is the derivative dled with slightly more care. Here, θ is the phase of the of this coupling with respect to θ; it, too, is periodic in θ superconducting order parameter. Ordinarily, we would with period 2π. The Josephson coupling is proportional expectthatitwouldbeimpossibleforanelectron,which to the square of the amplitude for an electron to tunnel carries electrical charge, to tunnel into a Majorana zero from one superconductor to the other, J ∝t2. However, mode,whichisneutralsinceγ =γ†. However,thesuper- when two topologial superconductors are in contact and conducting condensate (which is a condensate of Cooper thereareMZMsonbothsidesoftheJosephsonjunction, pairs that breaks the U(1) charge conservation symme- the leading coupling is: try) can accomodate electrical charge, thereby allowing this process, which is a form of Andreev reflection. In H =−itγLγRcos(θ/2) (9) the case of the Moore-Read Pfaffian quantum Hall state, Solongasiγ γ =±1remainsfixedduringthemeasure- however, this is not possible. In order for an electron to L R ment, the Josephson current now has period 4π, rather tunnelintoanMZM,fourcharge-e/4quasiparticlesmust than2π asinnontopologicalsuperconductors. Anobser- also be created in order to conserve electrical charge. vation of the 4π ‘fractional’ Josephson effect in AC mea- Thiscanonlyhappenwhenthebiasvoltageexceedsfour surements would be compelling evidence in favor of the times the charge gap. existence of MZMs in a superconducting system. How- In the case of a topological superconductor, the cou- ever, if iγ γ = ±1 can vary in order to find the min- pling (6), which seems like a drawback as compared to a L R imum energy at each value of θ, then it will flip when topological phase, can actually be an advantage since it cos(θ/2)changessign. Consquently,thecurrentwillhave opensupthepossibilityofasimplewayofdetectingMa- period 2π. The value of iγ γ = ±1 can change if a jorana zero modes that does not involve braiding them. L R fermion is absorbed by one of the zero modes γ or γ . For at T,V (cid:28) ∆, the electrical conductivity from a 1D L R Such a fermion may come from a localized low-energy wire through a contact described by Eq. (6) takes the state or an out-of-equilibrium fermion excited above the form [42, 75–77]: supercondcuting gap. In order to use the Josephson ef- 2e2 fect to detect MZMs, an AC measurement must be done G(V,T)= h(T/V,T/Λ∗) (7) at frequencies higher than the inverse of the time scale h for such processes. where h(0,0)=1 and Λ∗ is a crossover scale determined This can be done through the observation of Shapiro by the tunneling strength, Λ∗ ∼λy, where the exponent steps [10]. When an ordinary Josephson junction is sub- y depends on the interaction strength in the 1D normal jected to electromagnetic waves at frequency ω, a DC wire so that y = 1/2 for a wire with vanishing interac- voltage develops and passes through a series of steps tions. At low voltage and low temperature, the conduc- V =nhω as the current is increased. However, when DC 2e tivity is 2e2/h, indicative of perfect Andreev reflection: there are Majorana zero modes at the junction, then the eachelectronthatimpingesonthecontactisreflectedas 4πperiodicitydiscussedabovetranslatestoShapirosteps aholeandcharge2eisabsorbedbythetopologicalsuper- V = nhω. In essence, charge transport across a junc- DC e conductor. There is vanishing amplitude for an electron tion with MZMs is due to charge e rather than charge to be scattered back normally. Such a conductivity can 2e objects, so the flux periodicity and voltage steps are occur for other reasons (see, e.g. [78, 79]), but they are doubled. IntermsofconventionalShapirosteps,theodd non-generic and require some special circumstances and stepsshouldbemissing[10],buttheexperimentactually 6 observes only one missing odd step. This simple picture it can be written as: of missing odd Shapiro steps, although physically plau- sible, may not be complete, and a complete theory for i (cid:88)(cid:2) H = −µa a +(t+|∆|)a a Shapiro steps in the presence of MZMs has not yet been 2 1,j 2,j 2,j 1,j+1 j formulated (see, however, Ref. 80). (cid:3) +(−t+|∆|)a a (11) 1,j 2,j+1 Now, it is clear that there is a trivial gapped phase (an V. ‘SYNTHETIC’ REALIZATION OF atomic insulator) centered about the point |∆| = t = 0, TOPOLOGICAL SUPERCONDUCTORS µ < 0. The Hamiltonian is a sum of on-site terms i|µ|a a /2,eachofwhichhaseigenvalue−|µ|/2inthe 1,j 2,j Before further discussing experimental probes of Ising groundstate,withminimumexcitationenergy|µ|. How- anyons, we pause to discuss ‘synthetic’ realizations of ever, there is another gapped phase that includes the topological superconductors because it will be useful to points t = ±|∆|, µ = 0. At these points, the Hamil- have concrete device structures in mind when we de- tonian is a sum of commuting terms, but they are not scribeproceduresforbraidingnon-Abeliananyons. ‘Syn- on-site. Consider, for the sake of concreteness, the point thetic’ systems are important because there is no known t = |∆|, µ = 0. Then the Hamiltonian couples each site ‘natural’ system that spontaneously enters a topologi- to its neighbors by coupling a to a . As a result, 2,j 1,j+1 cal superconducting phase. The A-phase of superfluid we can form a set of independent two-level systems on He-3[81]andsuperconductingSr2RuO4 [82]arehypoth- the links of the chain. Each link is in its ground state esized to possess some topological properties, but it is ia a = −1. However, there are ”dangling” Majo- 2,j 1,j+1 not known precisely how to bring these systems into rana fermion operators at the ends of the chain because topological superconducting phases that support MZMs, a and a do not appear in the Hamiltonian. They 1,1 2,N nor is it known precisely how to detect and manipu- are Majorana zero mode operators: late Majorana zero modes in these systems [83]. There are also specific proposals for converting ultracold su- {a ,a }=[H,a ]=[H,a ]=0 (12) 1,1 2,N 1,1 2,N perfluid atomic fermionic gases into topological super- fluids [84], but experimental progress has been slow in If we move away from the point t=|∆|, µ=0, a1,1 and the atomic systems because of inherent heating prob- a2,N willappearintheHamiltonianand,asaresult,they lems. However, topological superconductivity can occur will no longer commute with the Hamitonian. However, in‘synthetic’systems[7,8,35,36,85–87]thatcombineor- there will be a more complicated pair of operators that dinarynon-topologicalsuperconductorswithothermate- are exponentially-localized at the ends of the chain and rials,therebyfacilitatinginterplaybetweensuperconduc- satisfy Eq. (3). Thus, the 1D toy model describes a tivity and other (explicitly, rather than spontaneously) system with localized zero-energy Majorana excitations broken symmetries. at the wire ends, which serve as the defects. Very similar ideas hold in 2D [33, 73], where an hc/2e The following single-particle Hamiltonian is a simple vortexinafullyspin-polarizedp+ipsuperconductorsup- toy model for a topological superconducting wire [34] portsaMZM.The1Dedgeofsucha2Dsuperconductor which illustrates how MZMs can arise at the ends of a supports a chiral Majorana fermion: 1D wire: (cid:90) H =(cid:88)(cid:16)−t[c† c +c†c ]−µc†c S = dxdtχ(i∂t+v∂x)χ (13) i+1 i i i+1 i i i +∆c c +∆∗c† c†(cid:17) (10) where χ(x,t) = χ†(x,t) and {χ(x,t),χ(x(cid:48),t)} = 2δ(x− i i+1 i+1 i x(cid:48)). When an odd number of vortices penetate the bulk of the superconductor, the field χ has periodic bound- Here, the electrons are treated as spinless fermions that ary conditions, χ(x,t) = χ(x + L,t), where L is the hop along a wire composed of a chain of lattice sites la- length of the boundary. Then, the allowed momenta are belled by i=1,2,...,N. It is assumed that a fixed pair k = 2πn/L with n = 0,1,2,... and the corresponding field ∆=|∆|eiθ is induced in the wire by contact with a energies are E = vk. The k = 0 mode is a MZM. If n 3D superconductor through the proximity effect. To an- an even number of vortices penetrate the bulk of the su- alyzethisHamiltonian,itisusefultoabsorbthephaseof perconductor, χ has anti-periodic boundary conditions, the superconducting pair field into the operators c and j χ(x,t)=−χ(x+L,t)andthereisnozeromodebecause thentoexpressthemintermsoftheirrealandimaginary the allowed momenta are k = (2n+1)π/L. A vortex parts: eiθ2cj =a1,j+ia2,j, e−iθ2c†j =a1,j−ia2,j. Theop- may be viewed as a very short edge in the interior of the erators a , a are self-adjoint fermionic operators – 1,j 2,j superconductor, so that there is a large energy splitting a†1,j = a1,j, a†2,j = a2,j – i.e. they are Majorana fermion between the n=0 mode and the n≥1 modes. operators. They are (generically) not zero modes since Althoughthetoymodeldescribedaboveisnotdirectly theydonot commutewiththe Hamiltonianbuttheyen- experimentally relevant, we can realize either a 1D or a ableustoelucidatethephysicsofthisHamiltoniansince 2D topological superconductor in an experiment, if we 7 somehow induce spinless p-wave superconductivity in a Εk Εk Εk metal in which a single spin-resolved band crosses the (cid:72)(cid:76) (cid:72)(cid:76) (cid:72)(cid:76) Fermi energy. This can be done with a Zeeman splitting thatislargeenoughtofullyspin-polarizethesystem,but k k k superconductivityhasneverbeenobservedinsuchasys- tem; if induced through the superconducting proximity FIG.1: Theelectronenergy(cid:15)(k)asafunctionofmomentum effect, it is likely to be very weak since the amplitude k for a 1D wire modeled by the Hamiltonian in Eq. (14) for of Cooper pair tunneling from the superconductor into (left panel) vanishing spin-orbit coupling and Zeeman split- the ferromagnet would be very small. However, the sur- ting;(centerpanel)non-zerospin-orbitsplittingbutvanishing face state of a 3D topological insulator [88–90] has such Zeeman splitting; (right panel) non-zero spin-orbit and Zee- a band which can be exploited for these purposes[35]. mansplitting. Inthesituationintheright-panel,iftheFermi Moreover, a doped semiconductor with a combination of energyiscloseto(cid:15)=0,thenthereiseffectivelyasingleband spin-orbitcouplingandZeemansplittingleads,foracer- of spinless electrons at the Fermi energy. tain range of chemical potentials, to a single low-energy branch of the electron excitation spectrum in both 2D [36]and1Dsystems[7–9]. Intheformercase,theZeeman is a single sub-band, i.e. a single transverse mode, in the field must generically be in the direction perpendicular wire. If there are more modes, then the requirement is to the 2D system. In the presence of a superconductor, that there must be an odd number of modes described suchaZeemansplittingmustbecreatedbyproximityto by Eq. (14) in the topological superconducting phase a ferromagnetic insulator, rather than with a magnetic [7, 94, 95]. (In addition, there can be any number of field. The exception is a system in which the Rashba modes in the non-topological phase; recall from Sec. III and Dresselhaus spin-orbit couplings balance each other that non-topological physics, here in the form of normal [85]. In 1D, however, the Zeeman field can be created bands, may coexist with the topological bands.) From with an applied magnetic field, thus making a 1D semi- the preceding analysis, we see that there is a minimum conductingnanowirewithstrongspin-orbitcouplingand magnetic field that must be exceeded in order for the superconducting proximity effect particularly attractive system to be in a topological superconducting phase. In as an experimental platform for investigating Majorana a real system in which there will be multiple sub-bands, zero-modes. This idea [7–9] has been adapted by several there is a maximum applied magnetic field, too, beyond experimental groups [6, 10–14]. which the lowest empty sub-band crosses the Fermi en- In all of these cases, the electron’s spin is locked to ergy. (Also, at high applied fields, the topological super- its momentum, rendering it effectively spinless. Such a conducting gap decreases inversely with increasing spin situation has the added virtue that an ordinary s-wave splitting, thus requiring very low temperatures to study superconductorcaninducetopologicalsuperconductivity the MZMs [9].) It is important that the magnetic field [7–9, 35, 36, 91, 92] since the spin-orbit coupling mixes beperpendiculartothespin-orbitfield. Ifthelatterisin s-wave and p-wave components. An effective model for they-direction,asinEq. (14),thentheappliedmagnetic this scenario takes the following form: fieldmustbeinthex−z plane. Inpractice, thisangular dependence on the magnetic field can be and has been H =(cid:90) dx(cid:104)ψ†(cid:0)− 1 ∂2−µ+iασ ∂ +V σ (cid:1)ψ used to study the MZMs in the laboratory [6]. 2m x y x x x (cid:105) +∆ψ ψ +h.c. (14) ↑ ↓ VI. TOPOLOGICAL SUPERCONDUCTORS: This model is in the topological superconducting phase EXPERIMENTS AND INTERPRETATION when the following condition holds [7–9]: V > x (cid:112) |∆|2+µ2, i.e. when the Zeeman spin splitting V is A number of experimental groups [6, 10–14] have fab- x larger than the induced superconducting gap ∆ and the rcateddevicesconsistingofanInSborInAssemiconduc- chemical potential µ – a situation which presumably can tor nanowire in contact with a superconductor, begin- beachievedbytuninganexternalmagneticfieldB toen- ning with the Mourik et al. experiment of Ref 6. Both hancetheZeemansplitting[93]. (Inprinciple,thesystem InSb and InAs have appreciable spin-orbit coupling and can be tuned by changing the chemical potential as well large Land´e g-factor so that a small applied magnetic usinganexternalgatetocontroltheFermilevelinasemi- field can produce large Zeeman splitting. The experi- conductor nanowire, thus adding considerable flexibility ments of Ref. 6, 12 used the superconductor NbTiN, to the set up for eventual TQC braiding manipulations which has very high critical field, while the experiments of the MZMs.) When the two sides of this equation are of Refs. 11, 13, 14 used Al. All of these experiments ob- equal, thesystemis gaplessinthe bulkandis ataquan- servedazero-biaspeak(ZBP),consistentwiththeMZM tum phase transition between ordinary and topological expectation. Meanwhile, the experiment of Ref. 10 ob- superconductingphases. Theemergenceofaneffectively served Shapiro steps in the AC Josephson effect in an spinlessbandofelectronsinthismodelissummarizedby InSb nanowire in contact with Nb. Fig. 1. Here, for simplicity, we have assumed that there According to the considerations of the previous two 8 at the current time. The softness of the gap may be duetodisorder,especiallyinhomogeneityinthestrength of the superconducting proximity effect [99] or perhaps an inverse proximity effect at the tunnel barriers where normal electrons could tunnel in from the metallic leads into the superconducting wire, leading to subgap states [100]. The softness of the gap may also help explain why the zero-bias conductance is suppressed from its ex- pectedquantizedpeakvalue,althoughotherfactors(e.g. finite wire length, finite temperature, finite tunnel bar- rier, etc.) are likely to be playing a role too. Very recent experimental efforts [101, 102] using epitaxial supercon- ductor (Al)-semiconductor (InAs) interfaces have led to hardproximitygaps. Theabsenceofavisiblegapclosing at the putative quantum phase transition may be due to the vanishing amplitude of bulk states near the ends of FIG. 2: The experimental differential conductance spectrum the wire [96]; a tunneling probe into the middle of the in an InSb nanowire in the presence of a variable magnetic wire would then observe a gap closing (but presumably field showing the theoretically predicted Majorana zero bias peakatfinitemagneticfield(takenfromRef. 6). Seethetext no MZM peaks which should decay exponentially with for a more detailed discussion of the experiment. distance from the ends of the wires). Such a gap clos- ing has been tentatively identified in the experiments on InAs nanowires in Ref. 13. sections, once the magnetic field is sufficently large that In the experiment of Ref. 10, it was observed that the V > (cid:112)|∆|2+µ2, where V = gµ B, the conductance n = 1 Shapiro step was suppressed for magnetic fields x x B through the wire between a normal lead and a supercon- larger than B = 2T. If this is the critical field beyond (cid:112) ducting one will be 2e2/h at vanishing bias voltage and whichgµBBx =Vx > |∆|2+µ2 inthisdevice,thenall temperature [42, 75–77], provided that the wire is much oftheoddShapirostepsshouldbesuppressed. However, longerthantheinducedcoherencelengthinthewire(i.e. one could argue that the fermion parity of the MZMs the typical size of the localized MZMs). The five experi- fluctuates more rapidly at higher voltages so that only mentsofRefs. 6,11–14observeazero-biaspeakatmag- the n = 1 step is suppressed. More theoretical work netic fields B >∼0.1 T, provided that the field is perpen- is necessary to understand Shapiro step behavior in the dicular to the putative direction of the spin-orbit field. presence of MZMs (see, however, Ref. 80). The peak conductance is, however, significantly smaller ZBPscanoccurforotherreasons,whichmustberuled than 2e2/h in all of these experiments. Moreover, the outbeforeonecanconcludethattheexperimentsofRefs. wiresappeartobeshort,ascomparedtotheinferredco- 6,11–14haveobservedaMZM,particularlysincetheex- herence length in the wires, raising the question of why pectedconductancequantizationassociatedwiththeper- the MZM peak is not split into two peaks away from fectAndreevreflectionhasnotbeenseen. TheKondoef- zero bias voltage due to the hybridization of the two end fectleadstoaZBP[78]. Inthepresenceofspin-orbitcou- MZMs overlapping with each other (although some sig- pling and a magnetic field, the two-level system may not natures of ZBP splitting are indeed observed in some of bethetwostatesofaspin-1/2,butmaybeasingletstate thedata[6,11–14]). Inaddition,thesubgapbackground and the lowest state of a triplet, which become degener- conductance is not very strongly suppressed at low non- ate at some non-zero magnetic field [78]. Alternatively, zero voltages, i.e. the gap appears to be ‘soft’. Finally, the ZBP may be due to ‘resonant Andreev scattering’. theappearanceofthepeakatB ∼0.1Tdoesnotappear Of course, a MZM is a type of resonant Andreev bound to be accompanied by a closing of the gap, as expected state so this alternative really means that there may be at a quantum phase transtion. anAndreevboundstateattheendofthewirethatisnot However, the peak conductance is expected to be sup- duetotopologicalsuperconductivitybutis‘accidentally’ pressed by non-zero temperature in conjunction with fi- (i.e. at one point in parameter space, rather than across nite tunnel barrier, and in short wires (see, e.g. Refs. an entire phase) at zero energy. ZBPs could also arise 96, 97). Some of the experiments do appear to find that simply due to strong disorder due to antilocalization at thezero-biaspeaksometimessplits[12–14]andthatthis zero energy in 1D systems without time-reversal, charge splitting oscillates with magnetic field, as predicted [98], conservation, orspin-rotationalsymmetry, usuallycalled althoughadetailedquantitativecomparisonbetweenex- class D superconductors [79]. perimental and theoretical zero bias peak splittings has The multiple observations of a zero-bias peak in dif- not yet been carried out in depth, and such a compar- ferent laboratories, occuring only in parameter regimes ison necessitates detailed knowledge about the experi- consistent with theory [103–106] substantiate these in- mental set ups (e.g. whether the system is at constant teresting observations in semiconductor nanowires and density or constant chemical potential [98]) unavailable show that they are, indeed, real effects and not experi- 9 mentalartifacts. Althoughtheseexperimentsarebroadly can operate in essentially the same way for quasiparti- consistent with the presence of Majorana zero modes at cles in a topological phase and for defects in an ordered theendsofthesewires, thereisstillroomforskepticism, (quasi-topological) state. However, braiding-based mea- which can be answered by showing that the ZBPs evolve surementproceduresrelyoninterferometry,whichisonly asexpectedwhenthewiresaremadelonger,thesoftgap possibleifthemotionaldegreesoffreedomoftheobjects is hardened (which has happened recently [101, 102]), being braided are sufficiently quantum-mechanical. This and the expected gap closing observed at the quantum will be satisfied by quasiparticles at sufficiently low tem- phase transition. Finally, experiments that demonstrate peratures, but the motion of defects is classical at any thefractionalACJosephsoneffectandtheexpectednon- relevant temperature except, possibly, in some special Abelian braiding properties of MZMs would settle the circumstances. matter. Consider, first, braiding-based gates. As noted above, Veryrecently,therehasbeenaninterestingnewdevel- braiding two anyons that support MZMs (either quasi- opment: theclaimofanobservationofMZMsinmetallic particlesordefects)causestheunitarytransformationin ferromagnetic (specifically, Fe) nanowires on supercon- Eq. (5). But how are we actually supposed to perform ducting (specifically, Pb) substrates where ZBPs appear thebraid? Here,quasi-topologicalphaseshaveanadvan- at the wire ends without the application of any exter- tage over topological phases (which no one has presently nal magnetic field, presumably because of the large ex- proposed to build). In a true topological phase, it may changespinsplittingalreadypresentintheFewire[107]. be very difficult to manipulate a quasiparticle because it Therehavebeenseveraltheoreticalanalysesofthisferro- need not carry any global quantum numbers. However, magneticnanowireMajoranaplatform[108–112]showing in an Ising-type quantum Hall state, the non-Abelian that such a system is indeed generically capable of sup- anyonscarryelectricalcharge,andonecanimaginemov- porting MZMs without any need for fine-tuning of the ing them by tuning electrical gates [23]. In the case of chemical potential, i.e. the system is always in the topo- a 2D topological superconductor, MZMs are localized at logical phase since the spin splitting Vx is always much vortices, and one can move vortices quantum mechani- larger than ∆ and µ. Although potentially an impor- cally through an array of Josephson junctions by tuning tantdevelopment, moredata(particularly, atlowertem- fluxes. In a 1D topological superconducting wire MZMs peratures, higher induced superconducting gap values, arelocalizedatdomainwallsbetweenthetopologicalsu- and longer wires) would be necessary before any firm perconductor and a non-topological superconductor or conclusion can be drawn about the experiment of Ref. an insulator (e.g. at the wire ends). These domain walls 107 since the current experiments, which are carried out can be moved by tuning the local chemical potential or at temperatures comparable to the induced topological magnetic field. In short, it is easier to ‘grab’ quasiparti- superconducting energy gap in wires much shorter than cles when they are electrically-charged and, potentially, the Majorana coherence length, only manifest very weak easier still to grab a defect when it occurs at a bound- (3−4 orders of magnitude weaker than 2e2/h) and very ary between two phases between which the system can broad (broader than the energy gap) ZBPs. If validated be driven by varying the electric or magnetic field [113]. asMZMs,thisnewmetallicplatformgivesaboosttothe The latter scenario is exemplified in Fig. 3a. There are study of non-Abelian anyons in solid state systems. in fact many theoretical proposals on how to braid the end-localized MZMs using electrical gates in various T junctions made of nanowires, all of which depend on the VII. NON-ABELIAN BRAIDING abilityofexternalgatesincontrollingsemiconductorcar- riers. ThepotentialtomanipulateMZMsthroughexter- As noted in the introduction, the primary significance nal electrical gating is, in fact, one great advantage of ofMajoranazeromodesisthattheyareamechanismfor semiconductor-based Majorana platforms. non-Abelianbraidingstatistics,arisingfromtheirground In both cases, quasiparticles and defects, it turns out state topological quantum degeneracy. The braiding of nottobenecessarytomovequasiparticlestobraidthem. non-Abelian anyons provides a set of robust quantum Instead,onecaneffectivelymovenon-Abeliananyonsvia gates with topological protection (although, of course, a “measurement-only” scheme [115, 116]. Through the this only applies if the temperature is much lower than use of ancillary EPR pairs and a sequence of measure- the energy gap and all anyons are kept much further ments, quantum states can be teleported from one qubit apart than the correlation length, so that the system to another. Similarly, a measurement involving an ancil- is in the exponentially-small Majorana energy splitting laryquasiparticle-quasiholeordefect-anti-defectpaircan regime). These braiding properties are also the most di- be used to teleport a non-Abelian anyon. A sequence rect and unequivocal way to detect non-Abelian anyons of such teleportations can be used to braid quasiparti- –including,asaspecialcase,thosesupportingMajorana cles. Therequiredsequenceofmeasurementscanbeper- zero modes. formed without moving the anyons at all, as illustrated Itisuseful,atthispoint,tomakeadistinctionbetween bytheflux-basedschemeofRefs. [114,117,118]. Bytun- thetwocomputationalusesofbraiding,forunitarygates ing Josephson couplings (which can be done by varying and for projective measurement. Braiding-based gates the flux through SQUID loops), pairs of MZMs can be 10 FIG. 4: (Left panel) With a two-point contact interferom- FIG.3: (a)MZMslocalizedatdomainwallsbetweentopolog- eter in a quantum Hall state, it is possible to detect topo- ical superconducting (TS) and normal superconducting (NS) logical charge and, thereby, read-out a qubit by measuring phasescanbemovedbytuningregionsbetweenthesephases electrical conductance (taken from Ref. 3. (Right panel) tomovethedomainwalls[113]. (b)Asexplainedinthetext, In a long Josephson junction with two arms, different paths a measurement-only scheme can replace actual movement of forJosephsonvorticescaninterfere,therebyenablingthede- MZMs. A pair of MZMs can be measured by tuning the flux tection of topological charge through electrical measurement Φ through a SQUID loop to decouple the superconducting (taken from Ref. 18). island on which the pair resides. This causes the island and nanowire to be in a superselection sector of fixed electrical charge [114]. of MZMs). measured electrostatically, as depicted in Fig. 3b. The VIII. QUANTUM INFORMATION fermionparityofapairofMZMsismeasuredbyisolating PROCESSING WITH MAJORANA ZERO MODES that pair on a small superconducting island so that the two parity states differ by an electrostatic charging en- Therearetwoprimaryapproachestostoringquantum ergy. When the Josephson coupling between the island information in MZMs: “dense” and “sparse” encodings. a large superconductor is non-zero, that pair of MZMs Inthedenseencoding,nqubitsarestoredin2n+2MZMs is not measured, and a different pair (possibly involving γ ,γ ,...,γ . The two basis states of the kth qubit onememberofthefirstpairofMZMs)canbemeasured. 1 2 2n+2 correspond to the eigenvalues iγ γ = ±1. The last Thereby,ameasurement-onlybraidingschemecanbeim- 2k−1 2k pair, γ ,γ is entangled with the total fermion plemented without moving any defects at all; all that is 2n+1 2n+2 parity of the n qubits so that the state of the system necessary is to teleport their quantum information. is always an eigenstate of the total fermion parity of The second use of braiding is for interferometry-based all 2n + 2 MZMs. The advantage of this encoding is measurement. This can only be done when the non- that it is easy to construct gates that entangle qubits. Abeliananyons are“light” so that two different braiding The disadvantage is that the last pair of MZMs is al- pathscanbeinterfered. Thiscanbedonewithchargee/4 ways highly entangled with the rest of the system, so quasiparticles in Ising-type ν = 5/2 fractional quantum errors in that pair (even if rare) can infect all of the Hall states. The two point contact interferometer de- qubits. In the sparse encoding, n qubits are stored in 4n picted in Fig. 4a measures the ratio between the unitary MZMsγ ,γ ,...,γ . Forallk,weenforcethecondition 1 2 4n transformations associated with the two paths. In the γ γ γ γ = −1, i.e. the total fermion parity 4k−3 4k−2 4k−1 4k case of non-Abelian anyons, this is not merely a phase. ofthesetoffourMZMsiseveninthecomputationalsub- For Ising anyons, there is no interference at all when an space. The two basis states of the kth qubit correspond odd number of MZMs is in the interference loop. When to the two eigenvalues iγ γ =±1. (Note that, in 4k−3 4k−2 an even number is in the interference loop, the interfer- the computational subspace, iγ γ = iγ γ .) 4k−3 4k−2 4k−1 4k ence pattern is offset by a phase of 0 or π, depending on Since each quartet of MZMs has fixed fermion parity, it the fermion parity of the MZMs in the loop. The experi- is easier to keep errors isolated. However, there are no mentsofRefs. 29–32areconsisentwiththesepredictions, entangling gates resulting from braiding alone. In order but their interpretation has been questioned [119]. to entangle qubits, we need to perform measurements in Domainwallsinnanowiresarealwaysclassicalobjects order to pass from one encoding to the other. whosepositionisdeterminedbygatevoltages. Abrikosov ThegatesH,T,Λ(σ )formauniversalgateset, where z vortices in 2D topological superconductors are similary H is the Hadamard gate, T is the π/8-phase gate, and classical in their motion. However, Josephson vortices, Λ(σ ) is the controlled-Z gate: z whose cores lie in the insulating barriers between su- (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) perconductingregions,maymovequantummechanically, 1 1 1 1 0 1 0 H = √ , T = , Z = . thereby making possible an interferometer such as that 2 1 −1 0 eiπ/4 0 −1 depicted in Fig. 4. Moreover, the fermionic excitations attheedgeofasuperconductorarelightandcanbeused In order to apply the Hadamard gate to the kth qubit, to detect the presence or absence of a MZM (but not to we perform a counter-clockwise exchange of the MZMs detect the quantum information encoded in a collection γ and γ . In order to apply Λ(σ ) to two qubits 4k−2 4k−1 z