Majorana zero modes and topological quantum computation

Abstract

We provide a current perspective on the rapidly developing field of Majorana zero modes (MZMs) in solid-state systems. We emphasise the theoretical prediction, experimental realisation and potential use of MZMs in future information processing devices through braiding-based topological quantum computation (TQC). Well-separated MZMs should manifest non-Abelian braiding statistics suitable for unitary gate operations for TQC. Recent experimental work, following earlier theoretical predictions, has shown specific signatures consistent with the existence of Majorana modes localised at the ends of semiconductor nanowires in the presence of superconducting proximity effect. We discuss the experimental findings and their theoretical analyses, and provide a perspective on the extent to which the observations indicate the existence of anyonic MZMs in solid-state systems. We also discuss fractional quantum Hall systems (the 5/2 state), which have been extensively studied in the context of non-Abelian anyons and TQC. We describe proposed schemes for carrying out braiding with MZMs as well as the necessary steps for implementing TQC.

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Majorana zero modes and topological quantum computation

Sankar Das Sarma1,2, Michael Freedman2 and Chetan Nayak2,3

We provide a current perspective on the rapidly developing eld of Majorana zero modes (MZMs) in solid-state systems. We emphasise the theoretical prediction, experimental realisation and potential use of MZMs in future information processing devices through braiding-based topological quantum computation (TQC). Well-separated MZMs should manifest non-Abelian braiding statistics suitable for unitary gate operations for TQC. Recent experimental work, following earlier theoretical predictions, has shown specic signatures consistent with the existence of Majorana modes localised at the ends of semiconductor nanowires in the presence of superconducting proximity effect. We discuss the experimental ndings and their theoretical analyses, and provide a perspective on the extent to which the observations indicate the existence of anyonic MZMs in solid-state systems. We also discuss fractional quantum Hall systems (the 5/2 state), which have been extensively studied in the context of non-Abelian anyons and

TQC. We describe proposed schemes for carrying out braiding with MZMs as well as the necessary steps for implementing TQC.

npj Quantum Information (2015) 1, 15001; doi:http://dx.doi.org/10.1038/npjqi.2015.1

Web End =10.1038/npjqi.2015.1 ; published online 27 October 2015

INTRODUCTIONTopological quantum computation1,2 is an approach to fault-tolerant quantum computation in which the unitary quantum gates result from the braiding of certain topological quantum objects called anyons. Anyons braid non-trivially: two counterclockwise exchanges do not leave the state of the system invariant, unlike in the cases of bosons or fermions. Anyons can arise in two ways: as localised excitations of an interacting quantum Hamiltonian3 or as defects in an ordered system.4,5 Fractionally charged excitations of the Laughlin fractional quantum Hall liquid are an example of the former. Abrikosov vortices in a topological superconductor are an example of the latter. Not all anyons are directly useful in topological quantum computation (TQC); only non-Abelian anyons are useful, which does not include the anyonic excitations (sometimes referred to as Abelian anyons, to distinguish them from the more exotic non-Abelian anyons, which are useful for TQC) that are believed to occur in most odd-denominator fractional quantum Hall states. A collection of non-Abelian anyons at xed positions and with xed local quantum numbers has a non-trivial topological degeneracy (which is, therefore, robusti.e., immune to weak local perturbations). This topological degeneracy allows quantum computation as braiding enables unitary operations between the distinct degenerate states of the system. The unitary transformations resulting from braiding depend only on the topological class of the braid, thereby endowing them with fault tolerance. This topological immunity is protected by an energy gap in the system and a length scale discussed below. As long as the braiding operations are slow compared with the inverse of the energy gap and external perturbations are not strong enough to close the gap, the system remains robust to disturbances and noise. These braiding operations constitute the elementary gate operations for the evolution of the TQC.

Perhaps the simplest realisation of a non-Abelian anyon is a quasiparticle or defect supporting a Majorana zero mode (MZM). (The zero mode here refers to the zero-energy midgap excitations that these localised quasiparticles typically correspond to in a low-

dimensional topological superconductor.) This is a real fermionic operator that commutes with the Hamiltonian. The existence of such operators guarantees topological degeneracy and, as we explain in section What is a majorana zero mode?, braiding necessarily causes non-commuting unitary transformations to act on this degenerate subspace. The term Majorana refers to the fact that these fermion operators are real, as in Majoranas real version of the Dirac equation. However, there is little connection with Majoranas original work or its application to neutrinos. Rather, the key concept here is the non-Abelian anyon, and MZMs are a particular mechanism by which a particular type of non-Abelian anyons, usually called Ising anyons can arise. By contrast, Majorana fermions, as originally conceived, obey ordinary Fermi Dirac statistics, and are simply a particular type of fermion. Although the terminology Majorana fermions is somewhat misleading for MZMs, it is used extensively in the literature.

If MZMs can be manipulated and their states measured in well-controlled experiments, this could pave the way towards the realisation of a topological quantum computer. The subject got a tremendous boost in 2012 when an experimental group in Delft published evidence for the existence of MZMs in InSb nanowires,6 following earlier theoretical predictions.79 The specic experimental nding, which has been reproduced later in other laboratories, is a zero-bias tunnelling conductance peak in a semiconductor (InSb or InAs) nanowire in contact with an ordinary metallic superconductor (Al or Nb), which shows up only when a nite external magnetic eld is applied to the wire. Several other experimental groups also saw evidence (i.e., zero-bias tunnelling conductance peak in an applied magnetic eld) for the existence of MZMs in both InSb and InAs nanowires,1014 thus verifying the Delft nding. However, though these experiments are compelling, they do not show exponential localisation with system length required by Eq. (3) or anyonic braiding behaviour. As explained later in this article, the exponential localisation of the isolated Majorana modes at wire ends and the associated non-Abelian braiding properties are the key features which enable TQC to be possible in these systems.

1Department of Physics, University of Maryland, College Park, MD, USA; 2Microsoft Station Q, University of California, Santa Barbara, CA, USA and 3Department of Physics, University of California, Santa Barbara, California, USA.

Correspondence: C Nayak (mailto:[email protected]

Web End [email protected])

Received 29 December 2014; revised 12 May 2015; accepted 12 May 2015

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In the current article, we provide a perspective on where this interesting and important subject is today (at the end of 2014). This is by no means a review article for the eld of MZMs or the topic of TQC as such reviews will be too lengthy and too technical for a general readership. There are, in fact, several specialized review articles already discussing various aspects of the subject matter, which we mention here for the interested reader. The subject of TQC has been reviewed by us in great length earlier,3 and we have also written a shorter version of anyonic braiding-based TQC elsewhere.15 There are also several excellent popular articles on the braiding of non-Abelian anyons and TQC.16,17 The

theory of MZMs and their potential application to TQC have recently been reviewed in great technical depth in several articles.1821

There are essentially two distinct physical systems that have been primarily studied in the search for MZMs for TQC. The rst is the so-called 5/2-fractional quantum Hall system (5/2-FQHS) where the application of a strong perpendicular magnetic eld to a very high-mobility two-dimensional (2D) electron gas (conned in epitaxially-grown GaAsAlGaAs quantum wells) leads to the even-denominator fractional quantisation of the Hall resistance. The generic fractional quantum Hall effect leads to the quantisation with odd-denominator fractions (e.g., the original 1/3 quantisation observed in the famous experiment by Tsui et al.22). Interestingly, of the almost 100 FQHS states that have so far been observed in the laboratory, the 5/2-FQHS is the only even-denominator state ever found in a single 2D layer. It has been hypothesised that this even-denominator state supports Ising anyons. A topological qubit was proposed by us for this platform23 in 2005, building upon previous theoretical work on the 5/2 state.2428 Tantalising experimental signatures for the possible existence of the desired non-Abelian anyonic properties were reported in subsequent experiments.2932 However, these results have not been reproduced in other laboratories. Potential barriers to progress are the required extreme high sample quality (mobility 4107 cm2/V.s), very low o25 mK temperature and high magnetic eld 42 T. The second system is the semiconductor nanowire structure proposed in refs 7,8 building upon earlier theoretical work on topological superconductors.3336 Semicon

ductor nanowires are the focus of this paper, but the 5/2 fractional quantum Hall state is a useful point of comparison as a great deal of experimental and theoretical work has been done on the 5/2-FQHS over the last 27 years.

WHAT IS A MZM?

A MZM is a fermionic operator that squares to 1 (and, therefore, is necessarily self-adjoint) and commutes with the Hamiltonian H of a system:

fermionic; 2 1; H; 0 1

Any operator that satises the rst two conditions is called a Majorana fermion operator. If it satises the third condition, as well, then it is a MZM operator or, simply, a MZM. (For the experts, it might be useful to comment that propagating Majorana fermions, of the type that neutrinos are hypothesised to be, can occur in any superconductor. However, localised MZMs and their concomitant non-Abelian anyonic braiding is a much more remarkable phenomenon.) The existence of such operators implies the existence of a degenerate space of ground states, in which quantum information can be stored. If there are 2n MZMs, 1,2n (they must come in pairs as each MZM is, in a sense, half a fermion) satisfying

fi; jg 2ij 2 then the Hamiltonian can be simultaneously diagonalised with the operators i12, i34, , i2n 12n. The ground states can be

labelled by the eigenvalues 1 of these n operators, thereby leading to a 2n-fold degeneracy. There is a two-state system associated with each pair of MZMs. This is to be contrasted with a collection of spin-1/2 particles, for which there is a two-state system associated with each spin. In the case of MZMs, we are free to pair them however we like; different pairings correspond to different choices of basis in the 2n-dimensional ground-state Hilbert space.

Unfortunately, the preceding mathematics is too idealised for a real physical system. If we are fortunate, there can, instead, be self-adjoint Majorana fermion operators 1,, 2n satisfying the anti-commutation relations (2) and

H; i e-x=x 3 where x is a length scale mentioned in the introduction (which can be construed to be the separation between two MZMs in the pair) and discussed momentarily, and x is a correlation length associated with the Hamiltonian H. In the superconducting systems that will be discussed in the sections to follow, x will be the superconducting coherence length. All states above the 2n1-

dimensional low-energy subspace have a minimum energy . In order for the denition (3) to approach the ideal condition (1), it must be possible to make x sufciently large that the right-hand side of Equation (3) approaches zero rapidly. This can occur if the operators i are localised at points xi (which we have not, so far, assumed). Then i commutes or anti-commutes, up to corrections ~ ey/x, with, respectively, all local bosonic or fermionic operators that can be written in terms of electron creation and annihilation operators whose support is a minimum distance y from some point xi. The effective Hamiltonian for energies much lower than

is a sum of local terms, which means that products of operators such as iij must have exponentially small coefcients ~ e|x x|/x.

(Terms that contain a single i operator (and no other fermionic operators, since none are allowed in the low-energy theory) are not allowed, due to fermion parity conservation.) Consequently, the condition (3) then holds (although much of the current interest in MZM and TQC is focused on semiconductor nanowires, as proposed in refs 79 the possibility of combining SO-coupling and spin splitting with ordinary s-wave superconductivity to articially create topological (spinless) p-wave superconductivity was actually rst considered theoretically37,38 in the context of

ultracold fermionic cold atoms.) The number of MZM operators satisfying (3) must be even. Consequently, if we add a term to the Hamiltonian that couples a single zero-mode operator to the nonzero-mode operators, a zero-mode operator will remain as zero modes can only be lifted in pairs. Thus, the exponential protection of the MZMs allowing their quantum degeneracy is enabled by the energy gap, which should be as large as possible for effective TQC operations. Thus, in a loose sense, two Majoranas together give a Dirac fermion, and these two MZMs must be far away from each other for the exponential topological protection to apply.

It is useful to combine the two MZMs into a single Dirac fermion c = 1+i2. The two states of this pair of zero modes corresponds to the fermion parities cc = 0,1. Thus, if the total fermion parity of a system is xed, then the degeneracy of 2n MZMs is 2n1-fold. This quantum degeneracy, arising from the topological nature of the MZMs, enables TQC to be feasible by braiding the MZMs around each other.

Such localised MZMs are known to occur in two related but distinct physical situations. The rst is at a defect in an ordered state, such as a vortex in a superconductor or a domain wall in a one-dimensional (1D) system. The defect does not have nite energy in the thermodynamic limit and, therefore, it is not possible to excite a pair of such defects at nite-energy cost and pull them apart. However, by tuning experimental parameters (which involves energies proportional to the system size), such

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defects can be created in pairs, thereby creating pairs of MZMs. The best example of this is a topological superconductor. Alternatively, there may be nite-energy quasiparticle excitations of a topological phase3 that support zero modes. This scenario is believed to be realized in the = 5/2 fractional quantum Hall states, where charge-e/4 excitations are hypothesised to support MZMs. Although the cases of defects in topological super-conductors and quasiparticles in true topological phases are closely related, there are some important differences, touched on later.

When two defects or quasiparticles supporting MZMs are exchanged while maintaining a distance greater than x, their

MZMs must also be exchanged. As the i operators are real, the exchange process can, at most, change their signs. Moreover, fermion parity must be conserved, which dictates that 1 and 2 must pick up opposite signs. Hence, the transformation law is:

1-72; 2- 1 4 The overall sign is a gauge choice. This transformation is generated by the unitary operator:

U eiye

412 5 This is the braiding transformation of Ising anyons. Strictly speaking, Ising anyons have = /8. Other values of can occur if there are additional Abelian anyons attached to the Ising anyons, as is believed to occur in the = 5/2 fractional quantum Hall state. In the case of defects, rather than quasiparticles, the phase will not, in general, be universal, and will depend on the particular path through which the defects were exchanged. We emphasize that this braiding transformation law follows from(i) the reality condition of the Majorana fermion operators 1,2,

(ii) the locality of the MZMs and (iii) conservation of fermion parity. Therefore, an experimental observation consistent with such a braiding transformation is evidence that (i)(iii) hold. This in turn is evidence that the defects or quasiparticles support MZMs satisfying the denition (3). Such a direct experimental observation of braiding has not yet happened in the laboratory.

In the case of quasiparticles in topological phases, braiding properties, as revealed through various concrete proposed interference experiments such as those proposed in refs 23,27,37,38 is, perhaps, the gold standard for detecting MZMs. However, in the case of defects in ordered states and, in particular, in the special case of MZMs in superconductors, a zero-bias peak (ZBP) in transport with a normal lead39 and a 4 periodic Josephson effect34 are also signatures, as discussed in section Signatures of MZMs in topological superconductors. Before discussing these in more detail in section Signatures of MZMs in topological superconductors, it may be helpful to discuss the differences between topological superconductors and true topological phases.

MZMS IN TOPOLOGICAL PHASES AND IN TOPOLOGICAL SUPERCONDUCTORS

As noted in the Introduction, Ising anyons can be understood as quasiparticles or defects that support MZMs. In the MooreRead Pfafan state24,25 and the anti-Pfafan state,40,41 proposed as candidate non-Abelian states for the 5/2-FQHS, charge-e/4 quasiparticles are Ising anyons.26,4248 There is theoretical28,4956

and experimental2932,5762 evidence that the = 5/2 fractional quantum Hall state is in one of these two universality classes. However, there are also some experiments6366 that do not agree with this hypothesis. The non-Abelian statistics of quasiparticles at = 5/2 has been reviewed in ref. 3 and would require a digression into the physics of the fractional quantum Hall effect. Hence, we do not elaborate on it here, other than to note that Ising-type fractional quantum Hall states are very nearly topological phases,

apart from some deviations that are salient on higher-genus surfaces.67 However, the electrical charge that is attached to Ising anyons enables their detection through charge transport experiments.23,27,37,38 Ising anyons also occur in some lattice

models of gapped, topologically-ordered spin liquids.68,69 These

are true topological phases in which the MZM operators are associated with nite-energy excitations of the system and do not have a local relation to the underlying spin operators, much less the electron operators, whose charge degree of freedom is gapped. This limits the types of effects (in comparison to the superconducting case) that could break the topological degeneracy implied by Equations (1) and (2).

MZMs also occur at defects in certain types of superconductors that form a subset of the class generally called topological superconductors.33,34,70 We discuss these in general terms in this

section and then in the context of specic physical realisations in section `Synthetic' realization of topological superconductors.

Topological phases have some topological features and some ordinary non-topological features. However, the interplay between these two types of physics is even more central in topological superconductors. This is both bad and good. It is bad if the non-topological features represent an opportunity for error or lead to energy splittings that decohere desirable superpositions. It is good when they allow a convenient coupling to conventional physics, something we had better have available if we ever wish to measure the topological system. In topological phases, there is a trivial tensor product situation in which the topological and the ordinary degrees of freedom do not talk to each other. In this case, we do not have to worry that the latter induce errors in the former, but they also will not be useful in initialising or measuring the topological degrees of freedom. (As always, in discussing topological physics, we regard effects that diminish exponentially with length, frequency or temperature as unimportant. This is somewhat analogous to computer scientists classifying algorithms as polynomial time or slower. Clearly the power and even the constants can make a difference, but such a structural dichotomy is a useful starting point.) So, for example, if there are phonons in a system, their interaction with topological degrees of freedom causes a splitting of the topological degeneracy that vanishes as eL/x at zero temperature,67 so we would consider the system as essentially a tensor product, with the phonons in a separate factor. However, a topological superconductor is not a true topological phase but, rather, following the terminology of ref. 67 a fermion parity protected quasi-topological phase. The qualier quasi permits the existence of benign gapless modes as discussed above. With slightly more precision: an excitation is topological if its local density matrices cannot be produced to high delity by a local operator acting from one of the systems ground states. Quasi permits low-energy excitations (below the gap) provided they are not topological. These subgap excitations surely do exist in real topological superconductors: there will be phonons and there will be gapless excitations of the superconducting order parameter both are Goldstone modes of broken symmetries (translation in the rst case and U(1)-charge conservation in the second). (The reader may wonder why the now-so-famous Higgs mechanism fails to gap the Goldstone mode of broken U(1). The answer is the mismatch of dimensions, the gauge eld roams three-dimensional space while the superconductor lives in either two- or one-dimension. In the former case, the interaction with the gauge eld causes superconducting phase uctuations to have dispersion ~ (q)1/2 while in the latter case ~ q. In a bulk three-dimensional superconductor the gauge boson is indeed gapped out.) The more serious caveat is fermion parity protected. This is simultaneously a blessing and a curse for any project to compute with MZMs in superconductors. The blessing is that the basis states of the topological qubit have this precise interpretation: fermion parity. If we are willing to move into an unprotected

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regime to measure them, MZMs can be brought together and their charge parity detected locally. Using more sophistication, one could keep the MZMs at topological separation and exploit the AharonovCasher effect to measure the charge parity encircled by a vortex. So this coupling will allow measurement by physics very well in hand. (It is less clear how to do this with, for instance, the computationally more powerful Fibonacci anyons.3)

Measurement is crucial for processing quantum information with MZMs as the braid group representation for Ising anyons is a rather modest nite group: beyond input and output, distillation of quantum states is needed,71 and this is measurement intensive. The curse is quasiparticle poisoning. A nearby electron can enter the system and be absorbed by a MZM, thereby ipping the fermion parityi.e., ipping a qubit. The electrons charge is absorbed by the superconducting condensate. This propensity of a topological superconductor to be poisoned (or equivalently, the fermion parity to ip in an uncontrolled manner) represents a salient distinction from the MooreRead state proposed for the = 5/2 fractional quantum Hall state. In the MooreRead state, the vortices carry electric charge ( e/4) and fermions carry charge 0 or 1/2. Consequently, there is an energy gap to bringing an electron from the outside into a = 5/2-fractional quantum Hall effect uid. Its fermion parity can be absorbed by a MZM (as in the case of a topologial superconductor), but there is no condensate to absorb its charge; instead, four disjoint charge-e/4 quasiparticles must be created with their attendant energy cost. It would be harder to poison a = 5/2 uid but also harder to discern its state and the signatures discussed in the next section are not available for non-Abelian FQHS states. Thus, one must choose between potentially better protection (5/2-fractional quantum Hall effect) or easier measurement (topological superconductor).

SIGNATURES OF MZMS IN TOPOLOGICAL SUPERCONDUCTORS Owing to the superconducting order parameter, it is possible for an electron to tunnel directly into a MZM in a superconductor. Suppose there is a MZM at the origin x = 0 in a superconductor. Then, if we bring a metallic wire near the origin, electrons can tunnel from the lead to the superconductor via a coupling of the form

Htun cy0 e-iy0=2 c0eiy0=2 6 where c(0) is the electron annihilation operator in the lead. For simplicity, we have suppressed the spin index, which is a straightforward notational choice if the superconductor and the lead are both fully spin polarised. In the more generic case, the spin index must be handled with slightly more care. Here is the phase of the superconducting order parameter. Ordinarily, we would expect that it would be impossible for an electron, which carries electrical charge, to tunnel into a MZM, which is neutral as = . However, the superconducting condensate (which is a condensate of Cooper pairs that breaks the U(1) charge conservation symmetry) can accommodate electrical charge, thereby allowing this process, which is a form of Andreev reection. In the case of the MooreRead Pfafan quantum Hall state, however, this is not possible. In order for an electron to tunnel into an MZM, four charge-e/4 quasiparticles must also be created in order to conserve electrical charge. This can only happen when the bias voltage exceeds four times the charge gap.

In the case of a topological superconductor, the coupling (6), which seems like a drawback as compared with a topological phase, can actually be an advantage as it opens up the possibility of a simple way of detecting MZMs that does not involve braiding them. For at T,V, the electrical conductivity from a 1D wire through a contact described by Equation (6) takes the form:39,7274

G V; T

2e2h h T=V; T=

7

where h(0,0) = 1 and * is a crossover scale determined by the tunnelling strength, * ~ y, where the exponent y depends on the interaction strength in the 1D normal wire so that y = 1/2 for a wire with vanishing interactions. At low voltage and low temperature, the conductivity is 2e2/h, indicative of perfect Andreev reection: each electron that impinges on the contact is reected as a hole and charge 2e is absorbed by the topological superconductor. There is vanishing amplitude for an electron to be scattered back normally. Such a conductivity can occur for other reasons (see, e.g., refs 75,76), but they are non-generic and require some special circumstances and can, in principle, be ruled out by further experiments. Thus, the observation of perfect Andreev reection, with the associated quantised conductance at zero bias, robust to parameter changes, is an indication of the presence of a MZM. In section Topological superconductors: experiments and interpretation, we discuss the extent to which this quantised tunnelling conductance associated with the zero-energy midgap Majorana modes has actually been observed in experiments.

A second probe of MZMs that is special to topological superconductors is the so-called fractional Josephson effect. When two normal superconductors are in electrical contact, separated by a thin insulator or a weak link, the dominant coupling between them at low temperatures is

H - J cos y 8 where is the difference in the phases of the order parameters of the two superconductors. It is periodic in with period 2. The Josephson current is the derivative of this coupling with respect to ; it, too, is periodic in with period 2. The Josephson coupling is proportional to the square of the amplitude for an electron to tunnel from one superconductor to the other, Jt2. However, when two topologial superconductors are in contact and there are MZMs on both sides of the Josephson junction, the leading coupling is:

H - itLR cos y=2

9 So long as iLR = 1 remains xed during the measurement, the

Josephson current now has period 4, rather than 2 as in non-topological superconductors. An observation of the 4 fractional Josephson effect in alternating current (AC) measurements would be compelling evidence in favour of the existence of MZMs in a superconducting system. However, if iLR = 1 can vary in order to nd the minimum energy at each value of , then it will ip when cos(/2) changes sign. Consequently, the current will have period 2. The value of iLR = 1 can change if a fermion is absorbed by one of the zero modes L or R. Such a fermion may come from a localised low-energy state or an out-of-equilibrium fermion excited above the superconducting gap. In order to use the Josephson effect to detect MZMs, an alternating current measurement must be done at frequencies higher than the inverse of the timescale for such processes.

This can be done through the observation of Shapiro steps.10

When an ordinary Josephson junction is subjected to electromagnetic waves at frequency , a direct current (DC) voltage develops and passes through a series of steps VDC = n(h)/(2e) as

the current is increased. However, when there are MZMs at the junction, then the 4 periodicity discussed above translates to Shapiro steps VDC = n(h)/(e). In essence, charge transport across a junction with MZMs is due to charge e rather than charge 2e objects, so the ux periodicity and voltage steps are doubled. In terms of conventional Shapiro steps, the odd steps should be missing,10 but the experiment actually observes only one missing odd step. This simple picture of missing odd Shapiro steps, although physically plausible, may not be complete, and a complete theory for Shapiro steps in the presence of MZMs has not yet been formulated (see, however, ref. 77).

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SYNTHETIC REALISATION OF TOPOLOGICAL SUPERCONDUCTORS

Before further discussing experimental probes of Ising anyons, we pause to discuss synthetic realisations of topological super-conductors because it will be useful to have concrete device structures in mind when we describe procedures for braiding non-Abelian anyons. Synthetic systems are important because there is no known natural system that spontaneously enters a topological superconducting phase. The A-phase of superuid He-3 (ref. 78) and superconducting Sr2RuO4 (ref. 79) are hypothesised to possess some topological properties, but it is not known precisely how to bring these systems into topological superconducting phases that support MZMs, nor is it known precisely how to detect and manipulate MZMs in these systems.80 There are also specic proposals for converting ultracold superuid atomic fermionic gases into topological superuids,81 but experimental progress has been slow in the atomic systems because of inherent heating problems. However, topological superconductivity can occur in synthetic systems7,8,35,36,8284 that combine ordinary non-

topological superconductors with other materials, thereby facilitating interplay between superconductivity and other (explicitly, rather than spontaneously) broken symmetries.

The following single-particle Hamiltonian is a simple toy model for a topological superconducting wire,34 which illustrates how MZMs can arise at the ends of a 1D wire:

H X

i

the ends of the chain and satisfy Equation (3). Thus, the 1D toy model describes a system with localised zero-energy Majorana excitations at the wire ends, which serve as the defects.

Very similar ideas hold in 2D,33,70 where an hc/2e vortex in a

fully spin-polarised p+ip superconductor supports a MZM. The 1D edge of such a 2D superconductor supports a chiral Majorana fermion:

S Z dx dt it vx

10

Here the electrons are treated as spinless fermions that hop along a wire composed of a chain of lattice sites labelled as i = 1, 2,, N. It is assumed that a xed pair eld = ||ei is induced in the wire by contact with a three-dimensional superconductor through the proximity effect. To analyse this Hamiltonian, it is useful to absorb the phase of the superconducting pair eld into the operators cj and then to express them in terms of their real and imaginary parts: ei()/(2)cj = a1,j+ia2,j, ei()/(2)cj = a1,j ia2,j. The operators a1,j,

a2,j are self-adjoint fermionic operatorsa1,j = a1,j, a2,j = a2,ji.e., they are Majorana fermion operators. They are (generically) not zero modes as they do not commute with the Hamiltonian but they enable us to elucidate the physics of this Hamiltonian as it can be written as:

H

i 2

13

where (x,t) = (x,t) and {(x,t),(x,t)} = 2(x x). When an odd number of vortices penetrate the bulk of the superconductor, the eld has periodic boundary conditions, (x,t) = (x+L,t), where L is the length of the boundary. Then, the allowed momenta are k = 2n/L with n = 0,1,2, and the corresponding energies are En = vk. The k = 0 mode is a MZM. If an even number of vortices penetrate the bulk of the superconductor, has anti-periodic boundary conditions, (x,t) = (x+L,t) and there is no zero mode because the allowed momenta are k = (2n+1)/L. A vortex may be viewed as a very short edge in the interior of the superconductor, so that there is a large energy splitting between the n = 0 mode and the n 1 modes.

Although the toy model described above is not directly experimentally relevant, we can realise either a 1D or a 2D topological superconductor in an experiment, if we somehow induce spinless p-wave superconductivity in a metal in which a single spin-resolved band crosses the Fermi energy. This can be done with a Zeeman splitting that is large enough to fully spin polarise the system, but superconductivity has never been observed in such a system; if induced through the super-conducting proximity effect, it is likely to be very weak as the amplitude of Cooper pair tunnelling from the superconductor into the ferromagnet would be very small. However, the surface state of a three-dimensional topological insulator8587 has such a band

that can be exploited for these purposes.36 Moreover, a doped semiconductor with a combination of spin-orbit coupling and Zeeman splitting leads, for a certain range of chemical potentials, to a single low-energy branch of the electron excitation spectrum in both 2D (ref. 36) and 1D systems.79 In the former case, the Zeeman eld must generically be in the direction perpendicular to the 2D system. In the presence of a superconductor, such a Zeeman splitting must be created by proximity to a ferromagnetic insulator, rather than with a magnetic eld. The exception is a system in which the Rashba and Dresselhaus spin-orbit couplings balance each other.82 In 1D, however, the Zeeman eld can be created with an applied magnetic eld, thus making a 1D semiconducting nanowire with strong spin-orbit coupling and superconducting proximity effect particularly attractive as an experimental platform for investigating MZMs. This idea79 has

been adapted by several experimental groups.6,1014

In all of these cases, the electrons spin is locked to its momentum, rendering it effectively spinless. Such a situation has the added virtue that an ordinary s-wave superconductor can induce topological superconductivity79,35,36,88,89 as the spin-orbit coupling mixes s-wave and p-wave components. An effective model for this scenario takes the following form:

H Z dx y -

- t cyi1ci cyici1

h i

- cyici cici1 cyi1cyi

Xj- a1;ja2;j t j j a2;ja1;j1 - t j j a1;ja2;j1

11

Now, it is clear that there is a trivial gapped phase (an atomic insulator) centred about the point || = t = 0, o0. The Hamiltonian is a sum of on-site terms i||a1,ja2,j/2, each of which has

eigenvalue ||/2 in the ground state, with minimum excitation energy ||. However, there is another gapped phase that includes the points t = ||, = 0. At these points, the Hamiltonian is a sum of commuting terms, but they are not on site. Consider, for the sake of concreteness, the point t = ||, = 0. Then the Hamiltonian couples each site to its neighbours by coupling a2,j to a1,j+1. As a

result, we can form a set of independent two-level systems on the links of the chain. Each link is in its ground state ia2,ja1,j+1 = 1.

However, there are dangling Majorana fermion operators at the ends of the chain because a1,1 and a2,N do not appear in the

Hamiltonian. They are MZM operators:

a1;1; a2;N

H; a1;1

1

2m2x - iyx Vxx mk h:c:

H; a2;N

14

This model is in the topological superconducting phase when the following condition holds:79 Vx4(||2+2)1/2, i.e., when the

Zeeman spin splitting Vx is larger than the induced super-conducting gap and the chemical potential a situation that presumably can be achieved by tuning an external magnetic eld B to enhance the Zeeman splitting. (Although much of the current interest in MZM and TQC is focused on semiconductor nanowires,

0 12 If we move away from the point t = ||, = 0, a1,1 and a2,N will

appear in the Hamiltonian and, as a result, they will no longer commute with the Hamiltonian. However, there will be a more complicated pair of operators that are exponentially localised at

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(k)

(k)

(k)

k

k

k

Figure 1. The electron energy (k) as a function of momentum k for a 1D wire modelled by the Hamiltonian in Equation (14) for (a) vanishing spin-orbit coupling and Zeeman splitting; (b) nonzero spin-orbit splitting but vanishing Zeeman splitting; (c) non-zero spin-orbit and Zeeman splitting. In the situation in the c, if the Fermi energy is close to = 0, then there is effectively a single band of spinless electrons at the Fermi energy.

Figure 2. The experimental differential conductance spectrum in an InSb nanowire in the presence of a variable magnetic eld showing the theoretically predicted Majorana ZBP at nite magnetic eld (taken from ref. 6). See the text for a more detailed discussion of the experiment.

as proposed in refs 79 the possibility of combining spin-orbit-coupling and spin splitting with ordinary s-wave superconductivity to articially create topological (spin less) p-wave superconductivity was actually rst considered theoretically91,92 in the context of

ultracold fermionic cold atoms.) (In principle, the system can be tuned by changing the chemical potential as well using an external gate to control the Fermi level in a semiconductor nanowire, thus adding considerable exibility to the set up for eventual TQC braiding manipulations of the MZMs.) When the two sides of this equation are equal, the system is gapless in the bulk and is at a quantum phase transition between ordinary and topological superconducting phases. The emergence of an effectively spinless band of electrons in this model is summarised by Figure 1. Here for simplicity, we have assumed that there is a single sub-band, i.e., a single transverse mode, in the wire. If there are more modes, then the requirement is that there must be an odd number of modes described by Equation (14) in the topological superconducting phase.7,90,91 (In addition, there can

be any number of modes in the non-topological phase; recall from section MZMs in topological phases and in topological super-conductor that non-topological physics, here in the form of normal bands, may coexist with the topological bands.) From the preceding analysis, we see that there is a minimum magnetic eld that must be exceeded in order for the system to be in a topological superconducting phase. In a real system in which there will be multiple sub-bands, there is a maximum applied magnetic eld, too, beyond which the lowest empty sub-band

crosses the Fermi energy. (Also, at high applied elds, the topological superconducting gap decreases inversely with increasing spin splitting, thus requiring very low temperatures to study the MZMs.9) It is important that the magnetic eld be perpendicular to the spin-orbit eld. If the latter is in the y-direction, as in Equation (14), then the applied magnetic eld must be in the x z plane. In practise, this angular dependence on the magnetic eld can be and has been used to study the MZMs in the laboratory.6

TOPOLOGICAL SUPERCONDUCTORS: EXPERIMENTS AND INTERPRETATION

A number of experimental groups6,1014 have fabricated devices

consisting of an InSb or InAs semiconductor nanowire in contact with a superconductor, beginning with the Mourik et al.6 experiment. Both InSb and InAs have appreciable spin-orbit coupling and large Land g-factor so that a small applied magnetic eld can produce large Zeeman splitting. The experiments of refs 6,12 used the superconductor NbTiN, which has very high critical eld, while the experiments of refs 11,13,14 used Al. All of these experiments observed a ZBP, consistent with the MZM expectation. The ZBP of Mourik et al.6 is shown in Figure 2. Meanwhile, the experiment of ref.10 observed Shapiro steps in the alternating current Josephson effect in an InSb nanowire in contact with Nb.

According to the considerations of the previous two sections, once the magnetic eld is sufciently large that Vx4(||2+2)1/2, where Vx = gBB, the conductance through the wire between a normal lead and a superconducting one will be 2e2/h at vanishing bias voltage and temperature,39,7274 provided that the wire is

much longer than the induced coherence length in the wire(i.e., the typical size of the localised MZMs). The ve experiments of refs 6,1114 observe a ZBP at magnetic elds B0.1 T, provided that the eld is perpendicular to the putative direction of the spin-orbit eld. The peak conductance is, however, signicantly smaller than 2e2/h in all of these experiments. Moreover, the wires appear to be short, as compared with the inferred coherence length in the wires, raising the question of why the MZM peak is not split into two peaks away from zero-bias voltage due to the hybridisation of the two end MZMs overlapping with each other (although some signatures of ZBP splitting are indeed observed in some of the data6,1114). In addition, the subgap background conductance is

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not very strongly suppressed at low non-zero voltages, i.e., the gap appears to be soft. Finally, the appearance of the peak at B ~ 0.1 T does not appear to be accompanied by a closing of the gap, as expected at a quantum phase transition.

However, the peak conductance is expected to be suppressed by non-zero temperature in conjunction with nite tunnel barrier, and in short wires (see, e.g., refs 92,93). Some of the experiments do appear to nd that the ZBP sometimes splits1214 and that this

splitting oscillates with magnetic eld, as predicted,94 although a detailed quantitative comparison between experimental and theoretical ZBP splittings has not yet been carried out in depth, and such a comparison necessitates detailed knowledge about the experimental set ups (e.g., whether the system is at constant density or constant chemical potential94) unavailable at the current time. The softness of the gap may be due to disorder, especially inhomogeneity in the strength of the superconducting proximity effect95 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.96 The softness of the gap may also help explain why the zero-bias conductance is suppressed from its expected quantised peak value, although other factors (e.g., nite wire length, nite temperature, nite tunnel barrier, etc.) are likely to be playing a role too. Very recent experimental efforts97,98 using epitaxial

superconductor (Al)-semiconductor (InAs) interfaces have led to hard proximity gaps. The absence of a visible gap closing at the putative quantum phase transition may be due to the vanishing amplitude of bulk states near the ends of the wire;92 a tunnelling probe into the middle of the wire would then observe a gap closing (but presumably no MZM peaks that should decay exponentially with distance from the ends of the wires). Such a gap closing has been tentatively identied in the experiments on InAs nanowires in ref. 13.

In the experiment of ref. 10 it was observed that the n = 1 Shapiro step was suppressed for magnetic elds larger than B = 2 T. If this is the critical eld beyond which gBBx = Vx4

(||2+2)1/2 in this device, then all of the odd Shapiro steps should be suppressed. However, one could argue that the fermion parity of the MZMs uctuates more rapidly at higher voltages so that only the n = 1 step is suppressed. More theoretical work is necessary to understand Shapiro step behaviour in the presence of MZMs (see, however, ref. 77).

ZBPs can occur for other reasons, which must be ruled out before one can conclude that the experiments of refs 6,1114 have observed a MZM, particularly as the expected conductance quantisation associated with the perfect Andreev reection has not been seen. The Kondo effect leads to a ZBP.75 In the presence

of spin-orbit coupling and a magnetic eld, the two-level system may not be the two states of a spin-1/2, but may be a singlet state and the lowest state of a triplet, which become degenerate at some non-zero magnetic eld.75 Alternatively, the ZBP may be due to resonant Andreev scattering. Of course, a MZM is a type of resonant Andreev bound state so this alternative really means that there may be an Andreev bound state at the end of the wire that is not due to topological superconductivity but is accidentally(i.e., at one point in parameter space, rather than across an entire phase) at zero energy. ZBPs could also arise simply due to strong disorder due to antilocalization at zero energy in 1D systems without time reversal, charge conservation or spin-rotational symmetry, usually called class D superconductors.76

The multiple observations of a ZBP in different laboratories, occurring only in parameter regimes consistent with theory99102

substantiate these interesting observations in semiconductor nanowires and show that they are, indeed, real effects and not experimental artifacts. Although these experiments are broadly consistent with the presence of MZMs at the ends of these wires, there is still room for scepticism, which can be answered by showing that the ZBPs evolve as expected when the wires are

made longer, the soft gap is hardened (which has happened recently97,98), and the expected gap closing observed at the

quantum phase transition. Finally, experiments that demonstrate the fractional alternating current Josephson effect and the expected non-Abelian braiding properties of MZMs would settle the matter.

Very recently, there has been an interesting new development: the claim of an observation of MZMs in metallic ferromagnetic (specically, Fe) nanowires on superconducting (specically, Pb) substrates where ZBPs appear at the wire ends without the application of any external magnetic eld, presumably because of the large exchange spin splitting already present in the Fe wire.103

There have been several theoretical analyses of this ferromagnetic nanowire Majorana platform104108 showing that such a system is indeed generically capable of supporting MZMs without any need for ne tuning of the chemical potential, i.e., the system is always in the topological phase as the spin splitting Vx is always much larger than and . Although potentially an important development, more data (particularly, at lower temperatures, higher induced superconducting gap values and longer wires) would be necessary before any rm conclusion can be drawn about the experiment of ref. 103 as the current experiments, which are carried out at temperatures comparable to the induced topological superconducting energy gap in wires much shorter than the Majorana coherence length, only manifest very weak (34 orders of magnitude weaker than 2e2/h) and very broad (broader than the energy gap) ZBPs. If validated as MZMs, this new metallic platform gives a boost to the study of non-Abelian anyons in solid-state systems.

NON-ABELIAN BRAIDINGAs noted in the introduction, the primary signicance of MZMs is that they are a mechanism for non-Abelian braiding statistics, arising from their ground-state topological quantum degeneracy. The braiding of non-Abelian anyons provides a set of robust quantum gates with topological protection (although, of course, this only applies if the temperature is much lower than the energy gap and all anyons are kept much further apart than the correlation length, so that the system is in the exponentially small Majorana energy-splitting regime). These braiding properties are also the most direct and unequivocal way to detect non-Abelian anyonsincluding, as a special case, those supporting MZMs.

It is useful, at this point, to make a distinction between the two computational uses of braiding, for unitary gates and for projective measurement. Braiding-based gates can operate in essentially the same way for quasiparticles in a topological phase and for defects in an ordered (quasi-topological) state. However, braiding-based measurement procedures rely on interferometry, which is only possible if the motional degrees of freedom of the objects being braided are sufciently quantum mechanical. This will be satised by quasiparticles at sufciently low temperatures, but the motion of defects is classical at any relevant temperature except, possibly, in some special circumstances.

Consider, rst, braiding-based gates. As noted above, braiding two anyons that support MZMs (either quasiparticles or defects) causes the unitary transformation in Equation (5). But how are we actually supposed to perform the braid? Here quasi-topological phases have an advantage over topological phases (which no one has presently proposed to build). In a true topological phase, it may be very difcult to manipulate a quasiparticle because it need not carry any global quantum numbers. However, in an Ising-type quantum Hall state, the non-Abelian anyons carry electrical charge, and one can imagine moving them by tuning electrical gates.23 In the case of a 2D topological superconductor, MZMs are localised at vortices, and one can move vortices quantum mechanically through an array of Josephson junctions by tuning

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uxes. In a 1D topological superconducting wire MZMs are localised at domain walls between the topological superconductor and a non-topological superconductor or an insulator (e.g., at the wire ends). These domain walls can be moved by tuning the local chemical potential or magnetic eld. In short, it is easier to grab quasiparticles when they are electrically charged and, potentially, easier still to grab a defect when it occurs at a boundary between two phases between which the system can be driven by varying the electric or magnetic eld.109 The latter scenario is exemplied in Figure 3a. There are in fact many theoretical proposals on how to braid the end-localised MZMs using electrical gates in various T junctions made of nanowires, all of which depend on the ability of external gates in controlling semiconductor carriers. The potential to manipulate MZMs through external electrical gating is, in fact, one great advantage of semiconductor-based Majorana platforms.

In both cases, quasiparticles and defects, it turns out not to be necessary to move quasiparticles to braid them. Instead, one can effectively move non-Abelian anyons via a measurement-only scheme.110,111 Through the use of ancillary Einstein-Podolsky-

Rosen (EPR) pairs and a sequence of measurements, quantum states can be teleported from one qubit to another. Similarly, a measurement involving an ancillary quasiparticlequasihole or defectanti-defect pair can be used to teleport a non-Abelian anyon. A sequence of such teleportations can be used to braid quasiparticles. The required sequence of measurements can be performed without moving the anyons at all, as illustrated by the ux-based scheme of refs 112114. By tuning Josephson couplings (which can be done by varying the ux through SQUID loops), pairs of MZMs can be measured electrostatically, as depicted in Figure 3b. The fermion parity of a pair of MZMs is measured by isolating that pair on a small superconducting island so that the two parity states differ by an electrostatic charging energy. When the Josephson coupling between the island and a large superconductor is non-zero, that pair of MZMs is not measured, and a different pair (possibly involving one member of the rst pair of MZMs) can be measured. Thereby, a measurement-only braiding scheme can be implemented without moving any defects at all; all that is necessary is to teleport their quantum information.

The second use of braiding is for interferometry-based measurement. This can only be done when the non-Abelian anyons are light so that two different braiding paths can be interfered. This can be done with charge-e/4 quasiparticles in Ising-type = 5/2 fractional quantum Hall states. The two-point contact interferometer depicted in Figure 4a measures the ratio between the unitary transformations associated with the two paths. In the case of non-Abelian anyons, this is not merely a phase. For Ising anyons, there is no interference at all when an odd number of MZMs is in the interference loop. When an even

number is in the interference loop, the interference pattern is offset by a phase of 0 or , depending on the fermion parity of the MZMs in the loop. The experiments of refs 2932 are consistent with these predictions, but their interpretation has been questioned.115

Domain walls in nanowires are always classical objects whose position is determined by gate voltages. Abrikosov vortices in 2D topological superconductors are similarly classical in their motion. However, Josephson vortices, whose cores lie in the insulating barriers between superconducting regions, may move quantum mechanically, thereby making possible an interferometer such as that depicted in Figure 4. Moreover, the fermionic excitations at the edge of a superconductor are light and can be used to detect

Figure 4. (a) With a two-point contact interferometer in a quantum Hall state, it is possible to detect topological charge and, thereby, read out a qubit by measuring electrical conductance (taken from ref. 3). (b) In a long Josephson junction with two arms, different paths for Josephson vortices can interfere, thereby enabling the detection of topological charge through electrical measurement (taken from ref. 18). Conversely, if two MZMs, 71 and 72, are brought close together, then the right-hand-side of Equation (3) may no longer be small.

Figure 3. (a) MZMs localised at domain walls between topological superconducting (TS) and normal superconducting (NS) phases can be moved by tuning regions between these phases to move the domain walls.109 (b) As explained in the text, a measurement-only scheme can replace actual movement of MZMs. A pair of MZMs can be measured by tuning the ux through a SQUID loop to decouple the superconducting island on which the pair resides. This causes the island and nanowire to be in a superselection sector of xed electrical charge.112

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the presence or absence of a MZM (but not to detect the quantum information encoded in a collection of MZMs).

QUANTUM INFORMATION PROCESSING WITH MZMSThere are two primary approaches to storing quantum information in MZMs: dense and sparse encodings. In the dense encoding, n qubits are stored in 2n+2 MZMs 1, 2,, 2n+2. The

two basis states of the kth qubit correspond to the eigenvalues i2k 12k = 1. The last pair, 2n+1, 2n+2 is entangled with the total fermion parity of the n qubits so that the state of the system is always an eigenstate of the total fermion parity of all 2n+2 MZMs. The advantage of this encoding is that it is easy to construct gates that entangle qubits. The disadvantage is that the last pair of MZMs is always highly entangled with the rest of the system, so errors in that pair (even if rare) can infect all of the qubits. In the sparse encoding, n qubits are stored in 4n MZMs 1,2,,4n. For all k, we enforce the condition 4k 34k 24k 14k = 1, i.e., the total fermion parity of the set of four MZMs is even in the computational subspace. The two basis states of the kth qubit correspond to the two eigenvalues i4k 34k 2 = 1. (Note that, in the computational subspace, i4k 34k 2 = i4k 14k.) As each quartet of MZMs has xed fermion parity, it is easier to keep errors isolated. However, there are no entangling gates resulting from braiding alone. In order to entangle qubits, we need to perform measurements in order to pass from one encoding to the other.

The gates H,T,(z) form a universal gate set, where H is the Hadamard gate, T is the /8-phase gate and (z) is the controlled-Z gate:

H 1

2

p

1 0 0 - 1

:

To apply the Hadamard gate to the kth qubit, we perform a counterclockwise exchange of the MZMs 4k 2 and 4k 1. In order to apply (z) to two qubits encoded in eight MZMs, we rst change to the dense encoding in which the two qubits are encoded in six MZMs. This involves a measurement. In this encoding, a braid implements (z). Finally, we introduce an ancillary pair of MZMs and perform a measurement in order to return to the sparse encoding. To be more precise, suppose that our two qubits are associated with MZMs 1,8 in the sparse

encoding, with the rst four encoding the rst qubit and the second four the second qubit. First, we measure i45. If it is equal

to +1, then the remaining MZMs form a dense encoding of the two qubits. If the measurement returns 1, a straightforward correction will be needed. Then we perform a counterclockwise exchange 3 and 6 (which are the middle two of the remaining MZMs) followed by clockwise exchanges of 1 and 2 and of 7 and 8. Finally, we return to the sparse encoding by introducing an ancillary pair of MZMs, which we will call 4 and 5, which are in the known state i45 = 1. Then a measurement of 5678 returns the system to the sparse encoding.

A single-qubit phase gate can be performed by bringing two MZMs close together for a period of time, t, so that their two states will be split in energy by E, and then pulling them apart again:

U

; T

1 0 0 ei=4

; Z

1 1 1 - 1

1 0 0 eiEt

15

This is a completely unprotected operation. Topology does not help us here. If we had perfect control over our system, then we would be able to control E and t precisely so that we could set Et = /4 and obtain a T gate. (Indeed, this is the type of control on which conventional qubits rely.) However, we do not expect to have such perfect control, so some error correction will be needed. In the case of the T gate, for example, we can use magic state distillation71 to provide a higher delity T gate. Fortunately,

the availability of topologically protected operations, namely protected Clifford operations, to perform error correction and distillation means fewer physical qubits should be required in the topological case compared with the conventional case.

The basic idea behind distillation is as follows. If we can produce the state |a = |0+ei/4|1 on demand, this is as good as being able to apply the T gate as we can perform a controlled-NOT (CNOT) gate with |a as the control qubit and our data qubit as the target. This is followed by a measurement of the latter and a correction by a Clifford operation if the measurement returns a+1. Therefore, the goal is to produce a high-delity copy of |a. This can be done in a variety of ways and has become, now, highly optimised.116120

The original distillation protocol71,121 proceeds by taking 15 approximate copies of |a: 91; ; 915, each with delity at least 1 . The tensor product of these 15 states is projected on the code subspace of the1,3,15 ReedMuller code. This stabiliser

code has the following properties: it encodes 1 logical qubit in 15 physical qubits; it can detect up to two phase (Z) errors and up to six bit (X) errors; and, remarkably, the logical state |a is the product of 15 copies of |a. Consequently, given 15 noisy copies of |a, we can check 14 stabilisers to see if it is consistent with being in the ReedMuller code subspace. If it is, we can decode the resulting 15 physical qubits into a logical qubit, which will be a puried version of the state |a, with delity 1 out1 353, in

the limit that is small. Distillation improves the delity so long as the initial delity exceeds the threshold found by solving out() = . The threshold is roughly 00.141 ref. 121. The distillation protocol can be applied recursively to achieve even higher delities on the state |a. Practically, the delity of the Clifford operations implementing the stabiliser checks dictates the minimum out achievable using the distillation protocol. For example, to achieve out1012, a reasonable value for quantum algorithms, the Clifford operations must also have delity of 1012

(ref. 122) Conventional qubit systems will require, e.g., the surface code to achieve such delities on the Clifford operations, while topological qubit systems may achieve this delity naturally. Thus, a potential advantage of MZM-based TQC would be the need for fewer qubits and fewer gate operations than in conventional quantum computation.

A given quantum algorithm must be decomposed into a circuit consisting of gates drawn from a fault-tolerant universal gate set, such as the set consisting of H,T,(z). Quantum algorithm decomposition methods based on algebraic number theory have recently dramatically reduced the number of T gates required to implement a given quantum algorithm.123125 By additionally

allowing an ancilla qubit and measurement to be used during decomposition, another constant factor reduction in the number of T gates can be achieved.126128 The latter techniques are

referred to as probabilistic Repeat-until-Success circuits. These aforementioned methods, as well as, e.g., techniques to produce Fourier angle states,129 may be ultimately hybridised to more efciently and fault-tolerantly implement a quantum algorithm using Majorana anyons.

Before concluding this section, we briey mention some of the potential problems in carrying out TQC with the current Majorana nanowire systems. First, the soft gap problem alluded to above indicates the presence of considerable non-thermal subgap fermionic states that would cause quasiparticle poisoning of the MZM as the Majorana will hybridise with the subgap fermions and decay (and thereby lose its non-Abelian anyonic character). Thus, poisoning by stray subgap non-thermal quasiparticles puts an absolute upper bound on the effective Majorana coherence time as poisoning will directly destroy the fermion parity at the heart of the proposed non-Abelian TQC. Recent experimental work has suppressed quasiparticle poisoning considerably, leading to possible coherence times as long as 1 min.130,131 Another

issue is that the current experimental topological gap is rather small (a few K), whereas the Majorana splitting due to the overlap

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of the MZMs from the two ends of the nanowire are likely to be in the range of 100200 mK (as the current nanowires are rather short). The lack of a large separation between these two energy scales introduces complications as the TQC braiding operations must be slow (adiabatic) compared with the topological gap energy and fast (so that one is in the topologically protected regime) compared with the Majorana splitting energy. Improvement in materials should lead to larger (smaller) gap (splitting), making this issue go away eventually. Finally, the current ZBPs, even assuming that they are indeed the predicted MZM conductance peaks, are much smaller (by more than an order of magnitude) than the quantised MZM conductance value of 2e2/h

associated with the Majorana-induced perfect Andreev reection, perhaps because of nite temperature, short wire length and nite tunnel barrier at the interfaces. This could lead to severe visibility problem during Majorana braiding with very weak signal to noise ratio, necessitating considerable measurement averaging. Only future braiding experiments could actually decisively establish whether the observed ZBPs in the nanowire tunnelling measurements are indeed the predicted MZMs or not.

OUTLOOKIt does not seem fanciful to compare Majorana systems and non-

Abelian topological quantum systems in general with the eld-effect transistor (FET). Both are sweet theoretical solutions to the problem of efcient processing of signals and the information they carry. (For FETs, of course, this theoretical solution has turned out, through Moores law, to be an astounding practical engineering success as well, leading to the modern information technology universe we live in.) The kinds of information (classical versus quantum) and the energy scales (eV versus meV) are different, just as the two ideas are temporally separated by more than 50 years, but each proposes a radical solution to an information processing roadblock. In each case, the roadblock was not absolute but sufciently daunting to inspire serious and sustained effort. There were pretransistor electronic computers, and it may well be possible to build a pretopological quantum computer through an extraordinary investment in error correction using ordinary non-topological qubits.132 As with our current

efforts to build MZM systems, the history of the FET was anchored in materials development and required a rethinking of solid-state physics (involving substantial and continuous developments in surface science, semiconductor physics, materials growth and lithography). Today, building topological materials will push the frontiers of purity and precision in materials growth and force us to extend our ability to model exotic bulk materials, interfaces and, nally, devices. As our entire civilisation now turns around the transistor, it would be grandiloquent to claim any untested technology as the new transistor, and we make no such claim. No one can see the future. However, we have arrived at a gateway where, in the next few years, our ability to process information may explode disruptively; there is certainly a large heterogeneous international effort in this direction of building quantum information processing devices and circuits. In such a world the topological route is the analogue of the FET.

Edgar Lilienfeld led the rst FET patent in 1925. It was in an entirely metallic system in which the required electronic depletion was too difcult to accomplish reliably. It took roughly four decades and the advent of semiconductor devices to realise the initial FET vision. Where do we stand with MZM systems today? Experimentalists have picked the most promising materials: high Land g-factor (to keep the applied B-elds moderate), high spin-orbit coupling (to strongly lock the spin and momentum bands in order to produce a large topological superconducting gap), low Schottky barriers and good epitaxial contact (to facilitate induced superconductivity), and high mobility (for coherent transport), among what was known, i.e., lying around, and predicted by the

theorists. Incremental improvements in nanowire design, pacication of interfaces and transparency to contacting superconductors, may take us into the regime of workable devicesthe transistor of the 1950s. But one may expect now that the concepts are clear, that systematic study of materials and their growth and interface properties could easily lead to new choices. A lesson already emerging from experiments in Copenhagen97 can radically reduce subgap states. Their data shows a remarkably crisp Bardeen-Cooper-Schrieffer (BCS) spectrum in epitaxially coated nanowires.98 We cannot of course be sure that the appropriate materials for the future TQC devices have already been developed after all, the rst transistors were made of germanium although silicon now rules the electronics worldbut there is now a clear path for progress toward the eventual building of TQC using Majorana anyons.

The materials frontier discussed above addresses delity and lifetimes the numerator of the expression dening computational power. The denominator is the clock rate. In the case of a MZM system, the key timescale is that of measurement. As explained in section Quantum information processing with majorana zero modes, measurement of fermion parity is essential to the distillation of magic states and is the leading candidate even for braiding operations. To compute well we must be able to measure quickly and accurately. The two gures of merit are in fact related: if we can make n measurements within the qubit lifetime, it does us no good if the delity is o1 1/n, for with less delity the qubit state will be forgotten long before we make the nth

measurement. For computations in parallel (as will be the norm), the demands on delity are proportionately greater because the appropriate n is the total number of measurements during the computation, not the number on any particular qubit. This tells us that there will be a second measurement frontier in which accuracy and speed will be the gures of merit. The leading measurement ideas today involve coupling to superconducting qubits living in an optical cavity and using a shift in the resonant frequency of the microwaves to read out fermion parity.112,114 This

is certainly a good starting point, but the typical number are photon frequencies ~ 6 GHz and, with beat frequencies recording the energy spitting of tens of MHz, read out would be limited to perhaps a MHz clock speed. The inherent energy scales of present MZM systems are on the order of 1 K20 GHz so there is room to do much better. In fact, to combine the two frontiers one might envision exploiting exotic superconductors with very large (~100 K) energy gap, pnictides or cuprates,133,134 in conjuction

with semiconductor wires to increase the gap protecting Majorana systems and clock rates by an order of magnitude.

Lifetimes/clock rate are hardware specs, but equally important is the scaling of the algorithms that we will run. There have been roughly three epochs: (i) Circa 1982, Feynman135 told us that if we could build a quantum computer, its resource requirements would scale in precisely the same way as the quantum mechanical problems, e.g., quantum chemistry problems, we wished to solve replacing the exponential scaling of a classical computer (in which memory must double to account for each new spin-1/2 degree of freedom); (ii) in the 1990s and 2000s, many key quantum algorithms were developed, including Shors factoring algorithm,136 and a detailed analysis of Feynmans idea; (iii) recent papers have focused on realistic regimes for quantum chemistry, rather than asymptotics. A straightforward estimate for gate counts of quantum chemistry Hamiltonians found that the number of computational steps for near equilibration to the ground-state scaled rather disastrously; polynomially by very high powers ~ 11 so that to obtain the energy of FeO2 to a milliHartree with a GHz clock rate would take the age of the universe.137

However, improved estimates,138 combined with some algorithmic improvement,139 has this time down now to a few minutes (with the most recent polynomial scaling ~ 5th power). This is one example; now that quantum computers appear to be increasingly

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realistic, computer scientists and physicists will nd efcient quantum algorthims for an array of problems. Many of these will be physical (e.g., quantum eld theory140 and many-body

localisation are attractive targets141), but even areas distant from

physics are seeing quantum advances. Deep learning has had a dramatic impact on machine learning in the last few years,142145

but there is a computational bottleneck: computation of the true gradient of L, where L is the log-likelihood function, is classically intractable, leading to classical methods that can efciently only approximate L. In physical terms, L is an entropy of a transverse eld Ising model on a union of complete bipartite graphs. It is now known146 that quantum computers may be used to estimate L efciently by emulating the corresponding Ising model, which leads to improved deep learning models using a quantum computer.

But when do we get to the analogue of the silicon FET? Presumably we will eventually do better than MZMs. Even as we anticipate great breakthroughs in the physics and engineering of Majorana systems, we can anticipate their eventual eclipse by anyonic systems (e.g., Fibonacci) that have topologically protected universal quantum operation. For many years, that phrase primarily meant a dense braid group representation. MZMs mirror the topological phase associated with SU(2)2 (see, e.g., ref. 3 for an explanation of this notation). Fibonacci anyons are present in SU(2)3 and have dense braid group representations. Furthermore, there is a hint of a potential path towards physical realisation147

through a combination of fractional quantum Hall effect (at the = 2/3 plateau) and superconductivity. SU(2) and all levels 5 and higher also have dense braiding but seem physically impractical. SU(2)4 is an anomaly; it is potentially related to metaplectic anyonic systems148 with a proposed realisation,5,149151 but braiding alone does not furnish a dense gate set. However, recent unpublished work152 has demonstrated that SU(2)4 becomes universal when braiding is combined with interferometric measurement.

We are poised on the brink of a revolution in our ability to control quantum systems. Topological systems, initially Majorana systems, will have a role. How wide the technological impact will be outside of physics is not foreseeable, but we can say that we are standing at a transitionwe are about to learn to process informationto think, so to speakin the manner that we know the universe operates: quantum mechanically. The rst steps in this intellectual journey have been taken with the potential realisation of MZMs in the laboratory,6,1014 but we still have a

long way to go.

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