Shortcuts to nonabelian braiding

0.63 MB·English

by  Torsten Karzig

Shortcuts to nonabelian braiding
Torsten Karzig,1 Falko Pientka,2 Gil Refael,1 and Felix von Oppen2
1Institute of Quantum Information and Matter, Department of Physics,
California Institute of Technology, Pasadena, California 91125, USA
2Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universita¨t Berlin, 14195 Berlin, Germany
Topologicalquantuminformationprocessingreliesonadiabaticbraidingofnonabelianquasiparti-
cles. Performingthebraidingoperationsinfinitetimeintroducestransitionsoutoftheground-state
manifoldanddeviationsfromthenonabelianBerryphase. Weshowthattheseerrorscanbeelimi-
natedbysuitablydesignedcounterdiabaticcorrectiontermsintheHamiltonian. Weimplementthe
5 resulting shortcuts to adiabaticity for simple protocols of nonabelian braiding and show that the
1 errorsuppressioncanbesubstantialevenforapproximaterealizationsofthecounterdiabaticterms.
0
2
Introduction.—It is envisaged that the information by counterdiabatic terms H (t). This shortcut to adia-
n 1
a processing of topological quantum computers relies on baticity does not apply directly to the adiabatic braid-
J adiabaticbraidingofnonabelianquasiparticles[1–5]. Ex- ingofnonabelianquasiparticlesbecauseoftheassociated

2 changing two nonabelions does not leave the quantum ground-state degeneracy. Here, we first generalize this
1 state unchanged, possibly up to a sign (as for fermions scheme to systems with degenerate manifolds of states

or bosons) or phase (as for abelian anyons) factor, but where adiabatic dynamics generates nonabelian Berry
]
l rathereffectsaunitaryrotationinadegeneratesubspace phases. Then,weapplythisgeneralizedshortcuttonon-
l
a of ground states. The ground-state degeneracy grows abelian statistics, using a simple model for braiding of
h exponentially with the number of nonabelian quasipar- Majoranaboundstates. Withinthismodel, thebraiding
-
s ticles, and quantum information processing corresponds of Majorana zero modes is based on judiciously chosen
e to manipulating the system’s ground state by braiding temporal variations of the couplings between a number
m
operations. Majorana bound states in topological su- of Majorana end states. We find that shortcuts to non-
. perconducting phases constitute the simplest example of abelian braiding can be implemented by introducing a
t
a suchnonabelions[6],andtherehasbeenconsiderableex- small number of additional local couplings.
m
perimental effort towards realizing a possible hardware Shortcuts to adiabaticity for degenerate systems.—The
- [7–13], following a series of theoretical proposals [14–22]. exactquantumdynamicsofaHamiltonianH(t)isgener-
d
n Topological quantum information processing is im- ated by the corresponding time-evolution operator U(t)
o which satisfies the Schr¨odinger equation
mune to local sources of decoherence when braiding is
c
[ performed adiabatically [1]. Quite generally, adiabatic-
i∂ U(t)=H(t)U(t). (1)
ity is protected by the gap of the underlying topological t
1
phase. Here we want to ask the question whether it is
v Thus,wecangiveanexplicitexpressionfortheHamilto-
1 possibletorealizetheexactadiabaticquantumdynamics nian H(t) generating any prescribed quantum dynamics
1 of the braiding operation, albeit in a finite time interval.
U(t),
8 There are obvious motivations why this would be desir-
2
able: First, any topological quantum computer would H(t)=i[∂ U]U†. (2)
0 t
operate at a finite clock speed which necessarily entails
.
1 possibly small, but nonzero errors. Second, a topolog- The shortcut to adiabaticity [23, 24] follows by inserting
0
ical quantum computer would presumably have to op- intothisexpressiontheadiabatictime-evolutionoperator
5
erate faster than parasitic decoherence processes such as
1
v: quasiparticlepoisoningordeviationsfromperfectground U(t)=(cid:88)e−i(cid:82)0tdt(cid:48)En(t(cid:48))+iγn(t)|ψn(t)(cid:105)(cid:104)ψn(0)| (3)
i state degeneracy originating in the finite spatial extent n
X of the Majorana quasiparticles. In both cases, such a
for the Hamiltonian H (t), with instantaneous eigenval-
r scheme could then be used to offset the incurred errors – 0
a ues E (t), instantaneous eigenstates |ψ (t)(cid:105), and Berry
enabling longer computations or higher clock speeds. n n
phase γ (t) = i(cid:82)tdt(cid:48)(cid:104)ψ (t(cid:48))|∂ ψ (t(cid:48))(cid:105). One finds that
Demirplak and Rice [23] as well as Berry [24] intro- n 0 n t(cid:48) n
H(t)=H (t)+H (t) with the so-called counterdiabatic
0 1
ducedaprotocolthatemulatestheadiabaticdynamicsof
terms [23, 24]
anynondegenerateHamiltonianH (t)astheexactquan-
0
tum dynamics in finite time. This scheme is known al- (cid:88)
H (t)=i (|∂ ψ (cid:105)(cid:104)ψ |−|ψ (cid:105)(cid:104)ψ |∂ ψ (cid:105)(cid:104)ψ |). (4)
1 t n n n n t n n
ternately as transitionless quantum driving or shortcut
n
to adiabaticity. The prize that comes with the short-
cut is that the adiabatic quantum dynamics of H (t) is Such shortcuts to adiabaticity have recently been im-
0
generated by a Hamiltonian H(t), which differs from H plemented experimentally for effective two-level systems
0
2
arisingintrappedBose-Einsteincondensates[25]andfor These counterdiabatic terms generalize the shortcut to
theelectronspinofasinglenitrogenvacancycenter[26]. adiabaticitytosystemswithdegeneratespectraandnon-
Following Wilczek and Zee [27], we now consider a abelian Berry connections.
Hamiltonian H (t) whose instantaneous spectrum de- Majorana systems.—In view of topological quantum
0
fined through informationprocessing,wespecificallyconsiderthecoun-
terdiabatic terms for a Bogoliubov–de Gennes Hamilto-
H (t)|ψn(t)(cid:105)=E (t)|ψn(t)(cid:105) (5)
0 α n α nian in Majorana representation,
includes one or more sets of states |ψn(t)(cid:105) which remain
α (cid:88)
H =i (cid:15) γ γ . (13)
degenerate for all t. Here, α=1,...,d labels the states 0 n n,2α−1 n,2α
n
within the degenerate subspace n of multiplicity d . nα
n
We first define |ηαn(t)(cid:105) as the adiabatic solution of the Here, both(cid:15)n andtheγn,α areexplicitlytimedependent
time-dependent Schr¨odinger equation andassociatedwiththeinstantaneousHamiltonian. The
i∂ |ηn(t)(cid:105)=H (t)|ηn(t)(cid:105) (6) instantaneous many-body spectrum of H0 contains de-
t α 0 α generacies whenever an eigenenergy (cid:15) vanishes or when
n
withinitialcondition|ηn(0)(cid:105)=|ψn(0)(cid:105). Intheadiabatic one or several nonzero (cid:15) are degenerate. The Majorana
α α n
limit, the time-evolved state need not remain parallel to eigenmodesassociatedwith(cid:15) aredenotedbyγ where
n n,α
|ψn(t)(cid:105) but will in general be a linear combination of all αtakeson2N valuesforanN-folddegenerateenergy(cid:15) .
α n
basis states within the degenerate subspace, The counterdiabatic terms H guarantee that the time
1
evolutiongeneratedbythefullHamiltonianH +H does
(cid:88) 0 1
|ηn(t)(cid:105)= Un |ψn(t)(cid:105). (7)
α αβ β not take the Majorana eigenmodes γn,α out of the sub-
β space n. At the same time, H should not alter the time
1
Inserting this expansion into the time-dependent evolution within these subspaces. In the supplementary
Schr¨odinger equation yields an equation for the coeffi- material [31] we show that these conditions yield
cient matrices Un,
i (cid:88) i (cid:88)
H = γ˙ γ − γ {γ ,γ˙ }γ . (14)
i∂ Un =Un(An−E 1), (8) 1 4 n,α n,α 8 n,α n,α n,β n,β
t n nα n,αβ
where An = i(cid:104)ψn|∂ ψn(cid:105) denotes the nonabelian Berry
αβ β t α This result complements the counterdiabatic terms in
connection [27]. This is solved by
first quantization in Eq. (12).
Un(t)=e−i(cid:82)0tdt(cid:48)En(t(cid:48))T˜ei(cid:82)0tdt(cid:48)An(t(cid:48)) (9) Applicationtononabelianbraiding.—Aminimalmodel
for nonabelian braiding starts from a Y-junction of
in terms of time-ordered exponentials. three one-dimensional topological superconductors, la-
TheadiabatictimeevolutionoftheHamiltonianH (t) beled wire 1, 2, and 3 [28–30], as illustrated in Fig. 1(a).
0
follows the time-evolution operator If all three arms are in the topological phase, there are
(cid:88) (cid:88) four Majorana bound states in this system. Three of
U(t)= |ηn(t)(cid:105)(cid:104)ηn(0)|= Un |ψn(t)(cid:105)(cid:104)ψn(0)|.
α α αβ β α these are located at the outer ends of the three wires,
n,α n,αβ with Bogoliubov operators labeled γ for wire j, and a
j
(10)
fourth Majorana mode γ is located at the junction of
0
Now we use Eq. (2) to derive the Hamiltonian H(t) for
the three wires. As long as the three arms have a finite
whichthisistheexacttime-evolutionoperator. Inserting
length,theseouterMajoranaboundstateshybridizewith
Eq. (10) into (2), we obtain the shortcut to adiabaticity
the central Majorana and the system is described by the
(all quantities evaluated at time t)
Hamiltonian
H =i(cid:88)(cid:88)(cid:110)[(Un)†U˙n]βα|ψαn(cid:105)(cid:104)ψβn| (cid:88)3
n αβ H0 =i ∆αγ0γα (15)
+[(Un)†Un] |∂ ψn(cid:105)(cid:104)ψn|(cid:9) (11) α=1
βα t α β
The second term in H simplifies due to unitarity of This Hamiltonian couples the central Majorana γ0 to a
Un, (Un)†Un = 1. Combining unitarity and Eq. (8), linearcombinationoftheouterthreeMajoranas. Wecan
we also have i(Un)†U˙n = (E 1−An) which simplifies thus readily bring it to the form of Eq. (13),
n
the first term. With these identities, we readily find
H =ih γ γ , (16)
H(t)=H (t)+H (t) with 0 ∆ 0 ∆
0 1
  with γ = (1/h )(cid:80)3 ∆ γ and h = [∆2 +∆2 +
(cid:88) (cid:88) (cid:88) ∆ ∆ α=1 α α ∆ 1 2
H1 =i  |∂tψαn(cid:105)(cid:104)ψαn|− |ψαn(cid:105)(cid:104)ψαn|∂tψβn(cid:105)(cid:104)ψβn|. ∆23]1/2. Foranychoiceofthecouplings∆j,therearealso
n α αβ two linearly independent combinations of the outer Ma-
(12) joranaswhichdonotappearintheHamiltonianandthus
3
(a) γ3 (b) γ1 (a) γ3 (b)
Δ
3
γ1Δ1 γ0 Δ2 γ2 γ1 γ2 γ2 γ2 γ1 γ0 γ2
(c) (b) γ1 (b) γ1 (b)
FIG. 2. (a) Minimal implementation required for braiding
γ γ γ γ γ γ
1 2 2 2 2 1 with shortcut protocol. The additional couplings needed
for the shortcut protocol are shown in blue. (b) Wire net-
FIG. 1. (a) Y-junction with central Majorana γ and three work with many Majoranas allowing for pairwise exchanges
0
outer Majoranas γ (j = 1,2,3). The outer Majoranas are of neighboring Majoranas including shortcut protocol. Im-
j
coupled to the inner Majoranas with strength ∆ . (b) Basic plementingtheshortcutmerelyrequirestheadditionoflocal
j
step of the braiding procedure, moving a zero-energy Majo- couplings within the network.
rana from the end of wire 1 to the end of wire 3 by varying
the ∆ . Dark (light) wires indicate zero (nonzero) couplings
j
∆j. Dark red circles correspond to zero-energy Majoranas, supplementary material [31], we obtain
lightbluecirclesindicateMajoranasacquiringafiniteenergy
brayncaouispdlienlgo.caInliztehdeoivneterrtmheedtiahtreeestMepa,jothraenzaesroa-leonnegrgtyheMlaigjhot- H1 = 2iγ˙∆γ∆ = 2hi2 (cid:88)(∆β∆˙α−∆α∆˙β)γαγβ. (17)
wires. (c) Three steps as described in (b) result in braiding ∆ α<β
the zero-energy Majoranas γ and γ .
1 2
Thus, the shortcut is based on additional couplings be-
tween the outer Majoranas, while the adiabatic braiding
protocolonlyuses couplingsbetweenthecentral andthe
outer Majoranas, see Fig. 2(a). Specifically, during the
remain true zero-energy Majoranas. Due to these zero-
basic step of moving a zero-energy Majorana from the
energymodes,thetwoeigenvaluesofH areeachdoubly
0 end of wire i to wire j, only the couplings ∆ and ∆
degenerate. Specifically, when just one of the couplings i j
are nonzero. According to Eq. (17), performing this step
∆ is nonzero, these two zero-energy Majoranas can be
j accurately in finite time merely requires the additional
identifiedwiththeMajoranaslocatedattheendsofthose
coupling between γ and γ .
wires with zero coupling. i j
Practical implementation.—There has been consider-
We assume that we can change the couplings ∆ as able work on how to implement braiding based on one-
j
a function of time. We can now imagine the following dimensionalsuperconductingphases[28–30,32–34]. The
braidingprocedure[28,30]. Initially,only∆ isnonzero. couplingsoftheMajoranascan,e.g.,bevariedbychang-
3
Then, γ and γ are zero-energy Majoranas. In a first ing the length of the intervening topological section.
1 2
step, we move a zero-energy Majorana from the end of However, this may not be easily compatible with the ge-
wire 1 to the end of wire 3, without involving the zero- ometricconstraintsimposedbytheshortcutprotocol,cf.
energy Majorana γ as shown in Fig. 1(b). To this end, Fig. 2(a). A better approach may be to vary the mag-
2
first increase ∆ to a finite value. The zero-energy Ma- nitude of the topological gap. Both methods control the
1
joranaoriginallylocatedattheendofwire1isnowdelo- overlap of the Majorana end states and hence their cou-
calized and a linear combination of γ and γ . We then pling. Physically, this can be achieved, say in quantum-
1 3
localize the Majorana zero mode at the end of wire 3 by wire based realizations, by changing the chemical poten-
reducing∆ downtozero,leavingonly∆ nonzero. The tial by means of a gate electrode [28] or a supercurrent
3 1
braiding process is completed by two completely analo- in the adjacent s-wave superconductor [35].
gousmoves(seeFig.1(c)): Wefirstmovethezero-energy More controlled variations of the Majorana couplings
Majoranafromtheendofwire2totheendofwire1,and may be possible by exploiting charging effects [30] or by
finally the zero-energy Majorana from wire 3 to wire 2. quantumdots[32]. Forsimplicity,assumethatthequan-
The combined effect of this procedure is to exchange the tum dot has a single level which is tunnel coupled to the
initial zero-energy Majoranas at the ends of wires 1 and ends of two topological wires with their Majorana end
2. One can check easily [30] that the change of the state states. When the dot level is far from the Fermi energy,
of the system under this adiabatic exchange is described thereisessentiallynocouplingbetweentheadjacentMa-
by the familiar braiding matrix U =exp(iπγ γ /4). joranas. Conversely, when the dot level is close to the
12 1 2
Fermi energy, the Majoranas become strongly coupled.
Whenperformingthisexchangeoperationoverafinite This approach modifies the coupling of the Majoranas
time interval, there will be corrections to the adiabatic by conventional gate control of a quantum-dot level and
time evolution. We can now apply one of the nonabelian is also compatible with the geometric constraints of the
shortcut formulas in Eqs. (12) or (14). As shown in the shortcut protocol.
4
So far, we have focused on the exchange of two Ma-
joranas within the minimal setting of a Y-junction. Of 1 λ =λ 0=. 07 1
cibsoouarirnnsege,nMotniareejockraaenynbaresoaacdradinlyobfimeMaraegaijnodreialyanasbscrhaaeindmdeeda.innyImwthpwioocrhtnatenhitgelhrye-, bility 10-2 λ = 0.9 phase error1100--42
a 10-6
amending this scheme to implement the counterdiabatic b
o 20 40 60 80 100
termsmerelyrequiresadditionallocalcouplingsasshown n pr duration (1/∆)
in Fig. 2(b). o
Robustness.—The manipulation of the quantum state nsiti 10-4
a
is independent of the precise braiding path as long as tr
the exchange is performed adiabatically. In contrast,
the diabatic corrections are sensitive to the details of
10-6
thebraidingprotocol. Consequently,thecounterdiabatic
10 20 30 40 50 60 70 80 90 100
terms(17)arenottopologicallyprotected,dependonthe duration (1/∆)
specifics of the braiding path, and for full effect, have to
be implemented exactly for a given H0(t). FIG. 3. Diabatic errors vs duration of braiding protocol for
However, we find that one can reach substantial re- the transition probability out of the degenerate subspace of
ductions in the diabatic errors even when the shortcut the initial state. The inset shows the phase error relative
protocol is implemented only with reasonable accuracy. to the nonabelian Berry phase. For both quantities, curves
are shown in the absence of counterdiabatic terms [λ = 0 in
We have computed the diabatic errors numerically, both
Eq.(18)]andwithcounterdiabatictermswith10%(λ=0.9)
for the bare braiding protocols and for approximate im-
and30%(λ=0.7)relativeerror. Therewouldbenodiabatic
plementations of the counterdiabatic terms. Specifically,
error if the counterdiabatic errors were implemented exactly.
we consider the diabatic errors for
H (t)=H (t)+λH (t). (18)
λ 0 1 Anexactimplementationofthecounterdiabaticterms
Forλ=1,thecounterdiabatictermsexactlycompensate fullycorrectsfortheseerrors. AscanbeseenfromFig.3,
the diabatic corrections for any duration of the braid- asuppressionbytwoordersofmagnitudemerelyrequires
ing protocol. As approximate implementations of the an implementation which is accurate at the 10% level.
counterdiabaticterms,weconsiderrelativeerrorsof10% Even a very rough implementation at the 30% level still
(λ=0.9)and30%(λ=0.7). Wecomputeboththetran- substantially reduces the errors. More generally, we find
sitionprobabilityoutofthedegeneratesubspaceandthe that the relative error scales approximately as (1−λ)2
accumulated deviation from the adiabatic Berry phase. with the accuracy of the implementation of H . It is
1
Implementing the basic step [shown in Fig. 1(b)] of also worth noting that the approximate counterdiabatic
the braiding protocol in Fig. 1(c) by ∆ (t) = ∆sinϕ(t) terms suppress the diabatic error, but do not modify its
1
and ∆ (t) = ∆cosϕ(t), with ϕ(t) increasing from 0 to scaling with protocol duration.
3
π/2, both the transition probability and the phase error Conclusions.—In summary, we have generalized the
exhibitapower-lawdependenceontheprotocolduration concept of shortcuts to adiabaticity to nonabelian Berry
T. Thepowerlawdependsonthespecificchoiceforϕ(t). phases and showed how this can in principle be used to
Choosing the latter such that the derivative vanishes at implement nonabelian braiding operations exactly in a
the end points yields a T−4 dependence. Corresponding finitetime. Suchprotocolscansubstantiallyimprovethe
numerical results are included with the supplementary accuracyofbraidingoperationsperformedinafinitetime
material[31]. Interestingly,wefindsimilarresultsforthe interval. It is interesting to note that our scheme bears
protocolgiveninRef.[30],inwhichoneinitiallyincreases some resemblance with the concept of quasi-adiabatic
∆ , leaving ∆ constant, and then reduces ∆ to zero in continuity for topological phases [36].
1 3 3
a second step [31]. Inthisworkwehavefocusedonasimplemodelofnon-
Exponentiallysmalltransitionratescanberealizedby abelianbraidingwhichexcludesthequasiparticlecontin-
choosing ∆ (t) = ∆sin2ϕ(t) and ∆ (t) = ∆cos2ϕ(t). uum. The current protocols are therefore useful when-
1 3
Now the gap assumes a minimum during the protocol as ever there is a separation of scales between the finite-
in the familiar Landau-Zener process. For the numerical energy subgap states and the magnitude of the topolog-
calculation presented in Fig. 3 we have chosen ϕ(t) to ical gap. Including the quasiparticle continuum is an
have a smooth derivative. The diabatic transition rate interesting problem for future research. It should also
is indeed exponential in the protocol duration which is be interesting to extend the current considerations for
somewhat conterintuitive as the transition rate actually Majorana zero modes to more exotic nonabelian quasi-
decreases relative to the previously discussed protocols particles.
althoughthegapissmaller. Thephaseerroralsoexhibits Acknowledgments.—We acknowledge useful discus-
exponential scaling as shown in the inset of Fig. 3. sionswithPietBrouwerandArisAlexandradinata. This
5
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6
SUPPLEMENTARY MATERIAL
Derivation of Eq. (14) in the main text
Here we derive general expressions for the counterdiabatic correction for non-interacting systems in terms of the
(instantaneous) eigenmodes γ of the original (time-dependent) Hamiltonian in a Majorana representation,
n,α
(cid:88)
H =i (cid:15) γ γ . (19)
0 n n,2α−1 n,2α
nα
Note that both the eigen-Majoranas γ and the eigenvalues (cid:15) are time dependent. Degeneracies of the many-body
n,α n
spectrumcanarisewhenoneormoreoftheeigenvalues(cid:15) vanishorwhensomenonzero(cid:15) isdegenerate,independent
n n
of time. The various Majorana operators associated with each single-particle eigenvalue (cid:15) are labeled by the index
n
α. If the single-particle eigenvalue (cid:15) is N-fold degenerate, α takes on 2N different values, α=1,...,2N.
n
AdirectderivationofthecounterdiabatictermsbasedonthegeneralEq.(12)inthemaintextiscumbersome. Here,
we choose to proceed as follows. The counterdiabatic terms suppress transitions out of the degenerate subspace but
leavethedynamicswithinthedegeneratesubspaceasgovernedbythenonabelianBerryconnectionoftheoriginaltime
evolution unchanged. Thus, we can determine the counterdiabatic terms H uniquely from the following constraints:
1
(a) H has no matrix elements which act within the degenerate eigenspaces of H . This ensures that H affects
1 0 1
only transitions between states with different energies.
(b) Thetimeevolutionoftheγ withrespecttothe full shortcutHamiltonianH =H +H doesnotincludeany
n,α 0 1
transitions between different degenerate subspaces.
To implement these constraints, we note that the time evolution of the Majorana operators is governed by the
Heisenberg equation of motion
dγ ∂γ
n,α = n,α +i[H +H ,γ ], (20)
dt ∂t 0 1 n,α
wherethefirsttermontheright-handsideaccountsfortheexplicittimedependenceoftheMajoranaoperators. This
term allows for the expansion
∂γn,α =(cid:88)Cαβγ . (21)
∂t nm m,β
m,β
Here, the coefficients Cαβ can in principle be expressed in terms of the instantaneous eigenfunctions and their time
nm
derivatives. Itturnsout,however,thatwedonotneedtheseexplicitexpressionsforthepresentpurpose. Toproceed,
we also write the counterdiabatic terms in the general form
(cid:88)(cid:88)
H =i hα,βγ γ . (22)
1 n,m n,α m,β
n,mα,β
Thus, it is our goal to derive the coefficients hα,β which satisfy the antisymmetry relation hα,β =−hβ,α.
n,m n,m m,n
Implementing the contraints (a) and (b), we demand that the γ satisfy the time evolution
n,α
dγn,α =(cid:88)Cαβγ +i[H ,γ ]. (23)
dt nn n,β 0 n,α
β
Here, the first term of the right-hand side contains only those terms of ∂γn,α that belong to the same subspace n. All
∂t
terms in ∂γn,α which belong to different subspaces must be cancelled by the counterdiabatic terms H .
∂t 1
The desired shortcut time evolution in Eq. (23) satisfies
(cid:26) (cid:27)
dγ
n,α,γ =0 (24)
dt m,β
for m(cid:54)=n. Inserting the Heisenberg equation of motion (20) into this condition, we obtain
(cid:26) (cid:27)
∂γ
4hαβ −4hβα + n,α,γ =0 (25)
nm mn ∂t m,β
7
for m(cid:54)=n. Using the antisymmetry property of the hαβ yields
nm
(cid:26) (cid:27)
H =−i (cid:88) (cid:88) ∂γn,α,γ γ γ . (26)
1 8 ∂t m,β n,α m,β
n,m αβ
(n(cid:54)=m)
Finally, we write this as
(cid:26) (cid:27) (cid:26) (cid:27)
H = i (cid:88)(cid:88)γ γ ,∂γn,α γ − i (cid:88)(cid:88)γ γ ,∂γn,α γ . (27)
1 8 m,β m,β ∂t n,α 8 n,β n,β ∂t n,α
n,m αβ n αβ
and use the relation
(cid:26) (cid:27)
(cid:88)(cid:88)γ γ ,∂γn,α =2∂γn,α (28)
m,β m,β ∂t ∂t
m β
to obtain Eq. (14) of the main text.
In the following, we derive the counterdiabatic terms for the braiding procedure given in Eq. (17) of the main text
using this general result as well as a more basic approach starting with Eq. (12).
Derivation of Eq. (17) using the general Majorana counterdiabatic terms
Here, we derive Eq. (17) using H in the Majorana operator representation as given in Eq. (14). The braiding
1
Hamiltonian in Eq. (16) is
H =ih (t)γ γ (t). (29)
0 ∆ 0 ∆
Thesystemcomprisesamodeofenergyh (t)associatedwiththetwoMajoranaoperatorsγ andγ (t)asdefinedin
∆ 0 ∆
the main text. In addition there are two Majoranas γ (t) and γ (t) which remain uncoupled by Eq. (29) and form a
A B
zero-energy mode. Using the identity
{γ˙ ,γ }=−{γ˙ ,γ } (30)
α β β α
(withtheshorthandγ˙ = ∂γ)wecanwritethetimederivatives(suppressingtimearguments)inthemostgenericform
∂t
as
γ˙ =η γ +η γ ,
A 1 B 2 ∆
γ˙ =−η γ +η γ ,
B 1 A 3 ∆
γ˙ =−η γ −η γ . (31)
∆ 2 A 3 B
with real coefficients η . The first term in H can be written as
i 1
(cid:88)
γ˙ γ =γ˙ γ +γ˙ γ +γ˙ γ =2γ˙ γ +2η γ γ (32)
n,α n,α A A B B ∆ ∆ ∆ ∆ 1 B A
nα
ThesecondcontributiontoH subtractsalltermswithinadegeneratesubspaceandthuseliminatestheterm∼γ γ .
1 B A
Thus we obtain
H = iγ˙ γ = i (cid:88)∆˙ ∆ γ γ (33)
1 2 ∆ ∆ 2h2 α β α β
∆ αβ
as given in Eq. (17).
Derivation of Eq. (17) using the spin construction
WecanalternativelyderiveEq.(17)usingthegeneralformulationofthecounterdiabatictermsinEq.(12). Tothis
end, we introduce conventional fermionic operators through
1 1
c = (γ −iγ ) ; c = (γ −iγ ). (34)
1 2 1 2 2 2 0 3
8
Using the inverse relations
γ =c +c† ; γ =i(c −c†) ; γ =i(c −c†) ; γ =c +c†, (35)
1 1 1 2 1 1 3 2 2 0 2 2
we can write H in terms of c and c
0 1 2
3
(cid:88)
H =i ∆ γ γ
0 j 0 j
j=1
=i∆ (c†c −c†c +c c −c†c†)−∆ (c†c +c†c +c c +c†c†)−∆ (2c†c −1). (36)
1 2 1 1 2 2 1 1 2 2 2 1 1 2 2 1 1 2 3 2 2
Specifically, we write the Hamiltonian in the basis {|00(cid:105),|11(cid:105),|10(cid:105),|01(cid:105)}, where the basis states are defined as
|11(cid:105)=c†c†|00(cid:105) , |10(cid:105)=c†|00(cid:105) , |01(cid:105)=c†|00(cid:105) (37)
1 2 1 2
with c |00(cid:105)=c |00(cid:105)=0. This yields
1 2
 
∆ i∆ −∆ 0 0
3 1 2
H0 =−i∆10−∆2 −∆0 3 ∆03 −i∆10−∆2 . (38)
0 0 i∆ −∆ −∆
1 2 3
The block-diagonal structure originates from the conservation of fermion-number parity. In fact, it is easy to show
that the Hamiltonian H commutes with the parity operator
P =γ γ γ γ . (39)
0 1 2 3
The top-left block H = ∆ τ −∆ τ −∆ τ corresponds to even fermion parity, while the bottom-right block
even 3 z 1 y 2 x
H =∆ τ +∆ τ −∆ τ has odd fermion parity. Here we have defined Pauli matrices τ within the even and odd
odd 3 z 1 y 2 x i
subspaces. If we also define Pauli matrices π in the even-odd subspace, then we can write
j
H =∆ τ −∆ τ π −∆ τ (40)
0 3 z 1 y z 2 x
for the overall Hamiltonian H. Expressing H and H in terms of Pauli matrices makes it obvious that
even odd
these Hamiltonians take the form of a spin Hamiltonian in magnetic fields B = (−∆ ,−∆ ,∆ ) and B =
even 2 1 3 odd
(−∆ ,∆ ,∆ ), respectively. The degeneracy due to the presence of the Majorana modes implies that the two sub-
2 1 3
spaces have the same eigenvalues. At the same time, the spectrum for each subspace by itself is non-degenerate.
In order to evaluate the counterdiabatic terms, it is useful to eliminate the time derivatives of the states from Eq.
(12). To achieve this, we first multiply the first term on the right-hand side of Eq. (12) by 1 = (cid:80) (cid:80) |ψm(cid:105)(cid:104)ψm|
m β β β
from the left and obtain
(cid:88) (cid:88)
H =i |ψm(cid:105)(cid:104)ψm|∂ ψn(cid:105)(cid:104)ψn|. (41)
1 β β t α α
m(cid:54)=n αβ
Taking the time derivative of Eq. (5) and multiplying from the left by (cid:104)ψm|, one finds
β
(cid:104)ψm|∂ H |ψn(cid:105)
(cid:104)ψm|∂ ψn(cid:105)= β t 0 α (42)
β t α E −E
n m
for n(cid:54)=m. Inserting this into Eq. (41) yields
H =i (cid:88) (cid:88)|ψm(cid:105)(cid:104)ψβm|∂tH0|ψαn(cid:105)(cid:104)ψn|. (43)
1 β E −E α
n m
m(cid:54)=n αβ
Using this expression, the counterdiabatic terms H can be conveniently derived.
1
Todo so, we temporarily performrotations withinthe even andodd subspacessuchthat H (t) inEq. (40)involves
0
only the τ term. Then, the eigenstates in the even and odd subspaces are simply the “spin-up” and the “spin-down”
z
states. Using
∂ H =∆˙ τ −∆˙ τ π −∆˙ τ , (44)
t 0 3 z 1 y z 2 x
9
1 λ = 0 1
λ = 0.7
bility 10-2 λ = 0T.-94 phase error1100--42
a 10-6
b
o 20 40 60 80 100
n pr duration (1/∆)
o
nsiti 10-4
a
tr
10-6
10 20 30 40 50 60 70 80 90 100
duration (1/∆)
FIG. 4. Diabatic errors vs duration of the braiding protocol defined by ∆ (t) = ∆sinϕ(t) and ∆ (t) = ∆cosϕ(t) for the
1 3
transition probability out of the degenerate subspace of the initial state. The inset shows the phase error relative to the
nonabelian Berry phase. The excitation gap remains unchanged during the entire braiding protocol, which results in a power-
law dependence on the duration. The function ϕ(t) is chosen to have zero derivative at both endpoints. For both transition
probability and phase error, curves are shown in the absence of counterdiabatic terms and with counterdiabatic terms with
10% and 30% relative error. There would be no diabatic error if the counterdiabatic errors were implemented exactly.
we then find that
1 (cid:110) (cid:111)
H = (∆ ∆˙ −∆ ∆˙ )τ π +(∆ ∆˙ −∆ ∆˙ )τ −(∆ ∆˙ −∆ ∆˙ )τ π . (45)
1 2(∆2+∆2+∆2) 3 1 1 3 x z 2 3 3 2 y 1 2 2 1 z z
1 2 3
This can be readily expressed in terms of the original Majorana operators. Indeed, we have the identities
i∆ γ γ =−∆ (2c†c −1)
12 1 2 12 1 1
i∆ γ γ =∆ (c c +c†c† −c†c −c†c ) (46)
13 1 3 13 2 1 1 2 2 1 1 2
i∆ γ γ =i∆ (c c −c†c† +c†c −c†c ),
23 2 3 23 2 1 1 2 1 2 2 1
or, in the basis specified above,
i∆ γ γ =∆ τ π
12 1 2 12 z z
i∆ γ γ =∆ τ π (47)
13 1 3 13 x z
i∆ γ γ =−∆ τ .
23 2 3 23 y
Thus, we finally find
i (cid:110) (cid:111)
H = (∆ ∆˙ −∆ ∆˙ )γ γ +(∆ ∆˙ −∆ ∆˙ )γ γ +(∆ ∆˙ −∆ ∆˙ )γ γ (48)
1 2(∆2+∆2+∆2) 2 1 1 2 1 2 3 1 1 3 1 3 3 2 2 3 2 3
1 2 3
in terms of the original Majorana operators.
Numerical calculation of the robustness
In this section we provide details of the numerical calculations. For completeness we also include numerical results
for the transition probability and Berry phase errors of the non-exponential protocols mentioned in the main text.
Due tothe conservation of fermion-number paritythe nonabelianBerry phase takes the form exp(iγτ ), where τ is a
z z
Pauli matrix in parity space. Performing the braiding protocol adiabatically yields γ =π/4. For finite durations we
numerically compute the Berry phase as γ = arg[(cid:104)Ψ (T)Ψ (0)(cid:105)/(cid:104)Ψ (T)Ψ (0)(cid:105)]/2, where Ψ (t) denotes the ground
e e o o e/o
state wavefunction at time t with even (odd) parity. The diabatic phase error is |γ−π/4|.
We first consider the protocol with the basic step ∆ (t) = ∆sinϕ(t) and ∆ (t) = ∆cosϕ(t). When ϕ(t) has zero
1 3
derivative at both endpoints (ϕ = 0 for t = 0 and ϕ = π/2 for t = T/3), the transition probability scales as T−4
10
(a) ((bb))
∆h∆ 1 λ =λ 0=. 07 1
1 ∆∆123 bility 10-2 λ = 0T.-92 phase error1100--42
a
b
o 10-6
pr 20 40 60 80 100
n duration (1/∆)
o
nsiti 10-4
a
tr
0 10-6
T/3 2T/3 T 10 20 30 40 50 60 70 80 90 100
time duration (1/∆)
FIG.5. (a)Couplings∆ andgaph duringthebraidingprocessfortheprotocolinRef.[30]. (b)Diabaticerrorsvsduration
j ∆
of braiding protocol. Similar to the protocol in Fig. 4 the error has a power-law dependence on the duration. The derivative
of φ(t) jumps at the end points and therefore the error scales as T−2.
with the protocol duration T. This is shown in Fig. 4. Specifically, we have chosen ϕ(t)=(π/2)[3(3t/T)2−2(3t/T)3]
between 0 and t = T/3 for this calculation (as well as for the one in the main text). When the derivative jumps at
either (or both) end points (as for the simplest choice ϕ(t)=3πt/2T), we find the errors to decay even more slowly,
namely as T−2. Note that both the transition probability out of the degenerate subspace and the phase error of the
topological qubit scale in the same manner with T, see inset of Fig. 4.
Interestingly, the same dependences are found for the protocol given in Ref. [30] and displayed in Fig. 5(a). The
initial step of the braiding operation is effected by increasing ∆ first at constant ∆ . The latter is reduced to zero
1 3
only subsequently. In this protocol, the gap increases and takes on a maximum halfway through this basic step.
Nevertheless,thediabaticerrorsstillvaryasapowerlawofT. Fig.5(b)showscorrespondingnumericalresults. Here
we chose a linear protocol, ϕ(t)=3πt/2T, in which the derivatives of ϕ(t) do not vanish at the end points.