Majorana Zero Modes and Topological Quantum Computation

3.2 MB·English

by  Sankar Das Sarma

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