Table Of ContentDecoherence of Majorana qubits caused by particle-hole conjugation asymmetry
R.S. Akzyanov1,2,3
1Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700 Russia
2Institute for Theoretical and Applied Electrodynamics,
Russian Academy of Sciences, Moscow, 125412 Russia
3Dukhov Research Institute of Automatics, Moscow, 127055 Russia
We investigate effects of the particle-hole asymmetry on the properties of Majorana qubtis.
6
Particle-hole asymmetric pertrubation shifts Majorana zero mode from zero-energy. This type
1
of asymmetry can act as a source of decoherence in Majorana qubits. We find that particle-hole
0
asymmetricpertrubationcausesphase-shiftsandbit-flipsdecoherenceprocessesinMajoranaqubits.
2
However,thesesourcesofdecoherencearenegligibleinuniformsystemwithspatiallyseparatedMa-
n jorana zero modes.
a
J PACSnumbers: 71.10.Pm,03.67.Lx,74.45.+c
3
1
I. INTRODUCTION cannot appear25.
] Indeed,MZMissymmetryprotectedstate thatisgov-
l
l erned by the PHC symmetry. Usually, this symmetry is
a Topological quantum computations (TQC) is promis-
hard to violate in superconductors. However,even small
h ing scheme for the fault-tolerant quantum codes. Key
- PHC asymmetry can lead to the experimentally observ-
advantageofsuchcomputationsistheirrobustnesstothe
s ablefeatures,suchasvortexcharge26. Nevertheless,pos-
e localsourcesofdecoherence. Realizationofsuchcodesre-
m quiresnon-abeliananyonsasa building blocks. Braiding sibleeffects ofviolationofPHCsymmetryandits effects
on decoherence in Majorana qubits have to be studied
. ofanyonscanbeusedforrealizationofquantumgates.1,2
t to find out optimal parameters for topological quantum
a One of the simplest realization of non-abelian anyons
m computations.
in condensed matter physics is Majorana zero modes
Aimofourworkis to study limitations ofMZMbased
- (MZM)3–6. Such modes can be realized at the ends
d quantum computations in presence of PHC asymmetric
of superconducting nanowires with spin-orbit coupling7
n pertrubation. First, we study generaleffects of the PHC
and topological insulator nanowires8; at the vortices in
o asymmetricpertrubationontheMZM.Second,westudy
c p-wave superconductors9,10 and topological insulator - decoherenceintheMZMbasedparityqubits. Finally,we
[ superconductor heterostructures11–13; as a Moore-Read
study possible origins of such PHC asymmetry in con-
state in fractional quantum Hall effect14. Recent exper-
1 densed matter systems.
iments show signatures of MZM in the superconducting
v
1 topologicalnanowires15andtopologicalinsulator-super-
1 conductor structures16.
II. PERTURBATION THEORY
2
MZM can be used for constructing parity qubits that
3
arerobustagainstthe decoherence5,17 andto disorder18.
0 We start from the general superconducting Hamilto-
Inthistypeofqubitsoccupationnumberoffermionlevel
. nian that has unitary PHC symmetry:
1 constructed from two MZM defines qubit space. How-
0
ever,robustnessofparity qubits canbe lifted by a MZM ΞHΞ= H (1)
6
coupling5,19. −
1 Ξ2 =+1 (2)
: Previously,decoherence of Majoranaqubits have been
v
studiedextensively20–22. Ithasbeenarguedthattunnel- whereH istheHamiltonian,Ξistheparticle-holeconju-
i
X ing of electrons from the environment to the topological gationoperator. WesupposethatHamiltonianhasMZM
r superconductor can destroy the coherence of the parity solution
a qubit causing bit-flip errors by changing parity20. How-
ever, recent experiments show that parity lifetime can Hγ =0 (3)
reach a large values23,24.
Ξγ =γ (4)
Topological states of matter can be classified by their
global symmetries: time-reversal and particle-hole con- This MZM is robust to any kind of pertrubations that
jugation (PHC) symmetries and their combination, chi- do not close the gap in the spectrum. Only existence of
ral symmetry. For example, system with unitary PHC anotherMZMcanshift zero-energydegeneracyandsplit
Ξ2 = +1 and broken time-reversal symmetry T2 = 0 is zero energy. This splitting is exponentially small if the
in class D supports MZM at a point defects25. If PHC distancebetweentwoMZMislarge19. However,previous
symmetry is broken then symmetry of such a system re- analysis do not consider pertrubations that violate PHC
duces from the class D to the class A. In that class symmetry. Under this type of petrubations MZM is no
topologicalindexistrivialforpointdefects,hence,MZM longer robust.
2
In general, pertrubation can be written as the sum of (γˆ +iγˆ)/√20 = 1 . However, qubit constructed by
1 2
| i | i
two parts: two MZM cannot be rotated since parity is conserved.
Four MZM are needed to construct a qubit5. Without
V =Vs+Vas lossofgeneralitywecanassumethatnumberofelectrons
ΞV Ξ= V iseven. Then,generalstateofthequbitcanbewrittenas
s s
− superposition of two states 0 0 and 1 1 because
ΞVasΞ=+Vas (5) | i1| i2 | i1| i2
they have the same parity
where V is the pertrubation that conserve PHC sym-
s Ψ=α0 0 +β 1 1 , α2+ β 2 =1 (10)
metry, while Vas violates this type of symmetry. It | i1| i2 | i1| i2 | | | |
can be shown that if γ = Ξγ is MZM then γ V γ =
−hγ|Vs|γi = 0. So, only PHC asymmetric Vahs c|asn|sihift wonhderfeer|m.i1ionanldev|e.lis2. cTorhriessppoanridtytoqutbhiet fiisrsptroatnedcttehdefrsoemc-
singleMZMformzeroenergy. Inthiscaseenergycorrec-
the phase-shift errors,since energy of both states is zero
tion to the zero-energy of MZM can be calculated as
and the qubit do not rotate Ψ(t) = Ψ(0). Also, parity
qubit is protected against bit-flip errors that cause tran-
δE = γ V γ (6)
as
h | | i sition 0 1 because this process is forbidden by the
| i → | i
Correctionstothe wavefunctioncanbe calculatedasfol- conservedparity. Thus,parityqubitistopologicallypro-
lows tected by PHC symmetry. However, this protection can
be lifted if we consider tunneling between MZM5,19. By
δγ =δ γ+δ γ (7) analogy between MZM coupling and PHC asymmetric
s as
perturbation we can conclude that similar effects can be
ψ V γ
n s
δsγ =nX6=0h −|En| i|ψni=+Ξδsγ (8) obFseirrvste,dwine tshtuedlyastphcaassee-Vshaisft6=e0rr.or, that caused by de-
ψ V γ phasing between rotating qubits. Under PHC asymmet-
n as
δasγ = X h | | i ψn = Ξδasγ (9) ric perturbation MZM acquire non-zero energy. Thus,
E | i −
n
n6=0 − stateswith empty fermionlevel 0 whichcorrespondsto
| i
the state ψ = (γ iγ )/√2 and full fermion level 1
where δsγ is the correction of the wave function caused whichcorre−sponds1to−the2state ψ =(γ +iγ )/√2 sta|rit
by the PHC symmetric pertrubation V , while δ γ is + 1 2
s as
rotating
the correction of the wave function caused by the PHC
asymmetric pertrubation V ; E is the energy of the
as n 0 =e−iE−t 0
level with wavefunction ψn. | it | it=0
It can be shown from Eq. 7 that γ+δsγ =Ξ(γ+δsγ) |1it =e−iE+t|1it=0 (11)
and iδ γ = Ξiδ γ obey condition for MZM. So, per- (12)
as as
turbed state γ +δγ can be represented as a superposi-
tion of two terms, each of them can be formally written where E+ = ψ+ Vas ψ+ E− E. Energy E can be
h | | i ≃ ≃
as MZM: γ +δ γ and iδ γ. How we can interpret this calculated as follows
s as
result? Similar situation arises when we consider split-
tingcausedbytunnelingoftwoMZM19: Diracstatewith E =( γ1 Vas γ1 + γ2 Vas γ2 )/2 (13)
h | | i h | | i
non-zero energy is written as superposition of two sepa-
Weomitfactors γ V γ and γ V γ sincethisma-
rate MZM γ +iγ . In our case, we have one MZM γ. 2 as 1 1 as 2
1 2 h | | i h | | i
trix elements areexponentially smallifdistance between
CorrectiontotheMZMstateδ γ canbeinterpretedasa
as
MZM is much larger than a coherence length. States
’fake’MZMthatcouples to the initialMZMγ. By ’fake’
with empty and full fermion level has the same energy.
we means that δ γ do not correspond to any physical
as
So, qubit evolves in time by the law
statealone. However,thisanalogygivesusatooltoana-
lyzedecoherenceeffectsofPHCasymmetricpertrubation
Ψ =e−it(E1+E2)Ψ (14)
in topological quantum computations. t t=0
and rotating with period
III. PHC ASYMMETRY ON TOPOLOGICAL
T =2π/(E +E ) (15)
QUANTUM COMPUTATION 1 2
where E is defined by Eq. 13 and belongs to the two
1,2
In this section we investigate stability of TQC versus spatially separated MZM pairs. Phase-shift error oc-
PHC asymmetry pertrubation. curswhendifferentqubits havedifferentrotatingperiod.
We start with the case without PHC asymmetric Then, typical time of phase-shift is proportional to the
pertrubation Vas = 0. Two separate MZM γˆ1 with difference between rotating periods of different qubits a
wavefunction γ1 and γˆ2 with wavefunction γ2 define one and b
fermion level which can be empty 0 or full 1 , lad-
| i | i
der operators acts as follows (γˆ iγˆ)/√21 = 0 and t T T , (16)
1 2 phase-shift a b
− | i | i ∼ −
3
where T is defined by Eq. 15. For uniform system asymmetry in the pseudogap state36 and in the Mott
H(r ,r )=H(r ,r ), where r denote by the MZM po- state37. Also, PHC symmetry implies Ξψ = ψ , so
i j j i i,j E −E
sitions. In case of uniform system and uniform pertru- everystate hasits partnerwithopposite energy. It leads
bation all qubit states have the same rotation period, so to the symmetric differential bias conductance near the
no phase-shift errors occur. It means that phase-shift of Fermi level. However, experiments with cuprates show
the parity qubits occurs only in non-uniform system. that this picture is not symmetric with respect to the
Second, we deal with the bit-flip errors. This type Fermi level35, meaning that PHC symmetry is broken.
of error correspond to the transitions of the state with 2. Particle-hole asymmetry in fractional quantum hall
empty fermion level to the state with full fermion level effect. Usual fractional quantum hall effect have PHC
0 1 and viceversa. This process is allowed now symmetryandMZMcanberealizedasMoore-Readstate
| i → | i
sincePHCconjugationisbroken. Wecanfindamplitude with filling factor ν = 5/214. However, three body in-
of this process by calculating the matrix element teraction causes Landau levels mixing and break PHC
symmetry38–40. Such Landau level mixing removes the
0Vas 1 ( γ1 Vas γ1 γ2 Vas γ2 )/2. (17) degeneracy between Pfaffian and its PHC anti-Pfaffian.
h | | i≃ h | | i−h | | i
This is the samethat PHC asymmetricpertrubationV
as
Once again, we omit factors γ V γ and γ V γ
h 2| as| 1i h 1| as| 2i do with MZM and its particle-hole conjugation. Recent
since this matrix elements are exponentially small if
experiments shows the presence of PHC asymmetry and
MZM are spatially separated. In case of uniform sys-
Landaumixinginthe high-qualitysamplesofGaAs41,42.
tem this matrix element is zero. It means that bit-flips
0 1 can occur only in non-uniform system with
| i → | i
typical time of the one bit-flip
V. DISCUSSION
t 1/( γ V γ γ V γ ). (18)
bit-flip 1 as 1 2 as 2
∼ h | | i−h | | i
Previous studies of decoherence in Majorana qubits
shows that stability of the MZM is not imply that TQC
IV. ORIGINS OF PHC ASYMMETRY using MZM are stable. Indeed, tunneling of quasipar-
ticles in the system do not affect MZM but it destroys
In this section we will study possible physical ori- coherence in the system20,22. In our case, we have sur-
gin of PHC asymmetry in superconductors. We start prisingly different result: TQC are stable versus PHC
with general form of PHC operator for the spinless asymmetric pertrubation even if MZM are not stable.
case Ξ = τ K. So, pertrubation that violates PHC If the system is uniform, then negative effects of the
x
symmetry ΞV (k)Ξ = +V ( k) can be written as PHC asymmetry on decoherence of parity qubits are ex-
as as
−
V (k) = τ ,τ ,τ ,τ k. For the spinful case PHC op- ponentially small, according to Eqs. 16, 18. However,
as x y 0 z
erator can be written as Ξ = σ τ K. So, petruba- if we perform braiding operation using pairwise interac-
y y
tion that violates PHC symmetry can be written as tions43 then the system becomes non-unform and deco-
V (k)=σ τ ,τ ,σ k,τ k. Hereσ actsinthespinspace, herence effects occurs.
as i j 0 i i i
τ acts in the particle-hole space, i,j = x,y,z, τ is the Recent works on Majorana qubits consider tunneling
j 0
unitymatrix,K isthecomplexconjugation,k isthemo- between MZM as one of the major source of decoher-
mentum. ence44. Decoherencedue toPHCasymmetrycanbe eas-
Terms σ τ and σ τ corresponds to the supercon- ily includedto their results: wecanuse analogybetween
i x i y
ducting order parameter with non-trivial spin structure. PHC asymmetryandcoupling to the ’fake’MZM.Inthe
Terms σ τ correspondsto the asymmetric Zeemanfield. formulas we just need to replace energy splitting caused
i z
So, PHC asymmetry can have either magnetic or su- bytunnelingbetweenseparateMZMwithfirstordercor-
perconducting origin, or its combination. We want to rectiontotheenergycausedbyPHCasymmetricpertru-
note that in standard mean-field Fermi liquid theory bation.
such PHC asymmetric terms do not appear, since we In the Sec. IV it is shown that the PHC asymmetry
cannot write them down in the standard second quan- canoccurinthestronglycorrelatedsystemswithastrong
tization form. Nevertheless, PHC asymmetry can arise many-bodyinteractions,suchasfractionalquantumHall
in the Gutzwiller-projected mean-field theory, which is effect and cuprates. It means that systems with strong
non-Fermi liquid theory that arrived from the Hubbard many-bodyinteraction(forexample,three-bodyinterac-
Hamiltonian29–31. So, we can assume that PHC asym- tion) may not be favorable for TQC. However, general
metry could manifest itself in the states with strong descriptionofthe PHC asymmetry insuch a systemstill
electron-electron correlations. lacks.
1. Intrinsic particle-hole asymmetry in high- Simplest case of topological superconductor is p-wave
temperature superconductors. It has been argued by superconductors. If PHC symmetry is exact, then this
Hirsch that some high-temperature superconductors can systemisinD symmetryclass45. Inthisclassnon-trivial
have intrinsic PHC asymmetry due to electron-electron Z topologicalindex canbe defined, consequently,MZM
2
interactions32–34. Recent experiments show that several can appear at a point defects in such a system25. PHC
high-temperature superconductors have intrinsic PHC asymmetric pertrubation reduces symmetry class D to
4
the symmetry class A, where topological index for point putations. WefoundthatPHCasymmetricpertrubation
defects is trivial, so there is no MZM 25. In our case shift MZM from zero energy and add correction to the
it means that PHC asymmetric perturbation converts wavefunctions, which can be interpreted as an appear-
MZM to the single Dirac fermion with nonzero energy. ance of ’fake’ MZM. We studied stability of MZM based
However, even in this case quantum computations using parity qubits. We get that phase-shift and bit-flip er-
Majorana qubits can be efficiently done. It arises inter- rors occur due to PHC asymmetric pertrubation. We
esting question: what properties of the system make it discussed possible origins of such asymmetry and argue
suitable for topological quantum computations? thatthisasymmetrycanariseinfractionalquantumhall
states with Landau mixing and high-temperature super-
conductors.
VI. CONCLUSION WeacknowledgepartialsupportbytheDynastyFoun-
dation and ICFPM (MMK), the Ministry of Educa-
In this work we investigated the influence of PHC tion and Science of the Russian Federation (Grant No.
asymmetric pertrubation on MZM based quantum com- 14Y26.31.0007).
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