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Quantum gates by periodic driving

Z. C. Shi1,2 and X. X. Yi1∗1 Center for Quantum Sciences and School of Physics,

Northeast Normal University, Changchun 130024, China2 School of Physics and Optoelectronic Technology

Dalian University of Technology, Dalian 116024 China

(Dated: January 6, 2018)

Topological quantum computation has been extensively studied due to its robustness againstdecoherence. A conventional way to realize it is by adiabatic operations—it requires relatively longtime to accomplish so that the speed of quantum computation slows down. In this work, we present amethod to realize topological quantum computation by periodic driving. Compared to the adiabaticevolution, the total operation time can be regulated arbitrarily by the amplitude and frequency ofthe periodic driving. For the sinusoidal driving, we give an expression for the total operation timein the high-frequency limit. For the square wave driving, we derive an exact analytical expressionfor the evolution operator without any approximations, and show that the amplitude and frequencyof driving field depend on its period and total operation time. This could provide a new directionin regulations of the operation time in topological quantum computation.

I. INTRODUCTION

Decoherence is an enemy of quantum computation,which is the loss of coherence due to the presence of en-vironments. As a promising avenue to deal with the de-coherence, topological quantum computations [1–8] em-ploy two-dimensional quasiparticles called anyons, whoseworld lines cross over one another to form braids in athree-dimensional spacetime. Information encoded in theanyons is robust against local perturbations and quan-tum operations can be performed by braiding the non-Abelian anyons [1, 9]. The simplest example of the non-Abelian anyons is the Majorana fermions which are pre-dicted to exist in fractional quantum Hall systems [10],topological insulators [11, 12], solid state systems [13],and semiconductor-superconductor hybrid systems [14–16]. The signatures of Majorana fermions have also beenobserved in experiments more recently [17–21], whichgives rise to an opportunity to encode a qubit by Ma-jorana fermions in these materials.

A quantum task is often accomplished by a sequenceof quantum operations rather than single quantum oper-ation [22–28]. The total operation time increases linearlywith the increasing of the number of operations. Consid-ering the limit of the coherence time of quantum systems,long operation time is not favorable, even if the quantumtopological computation is robust against perturbations.On the other hand, the time-periodic driving systemshave been extensively studied in the past few years. Es-pecially, several work [29–41] have shown that the topo-logical properties can be changed in topologically triv-ial system by time-periodic driving (e.g., the existenceof Floquet topological insulators or Floquet Majoranafermions). Recently, the Floquet Majorana fermions isrealized by periodic driving fields in the system of cou-

∗Corresponding address: yixx@nenu.edu.cn

pled quantum dots proximity to a s-wave superconductor[42]. More recently, it has been proposed to achieve thedirect coupling between the topological and conventionalqubits by periodic driving fields [43]. In this paper we ex-plore the possibility to regulate the total operation timeby periodic driving For concreteness, the physical modelof interest is the quantum dots coupled to the Majoranamodes in a topological superconductor. Of course, thismethod can also be extended to the other quantum sys-tems.

The paper is organized as follow. In Sect. II, we brieflyintroduce the topological quantum computation by adi-abatic evolution. In Sect. III, we first recall the Floquettheory, then the periodic driving fields in the form of si-nusoidal, square wave, and δ-function kick are appliedseparately to modulate the total operation time for re-alizing the quantum operations. Finally we extend thismethod to other hybrid quantum systems in Sect. IV.The discussion and conclusion are given in Sect. V.

II. QUANTUM COMPUTATION BY

ADIABATICAL EVOLUTION

Recently, the adiabatic evolution has widely applied tothe preparation and manipulation of Majorana fermions[26, 27, 44]. In particular, it has been shown that topo-logical quantum information processing becomes possiblein the one-dimensional network [28] by adiabatically con-trolling the locally tunable gates which affect the chemi-cal potential over a finite length of the wire. In followingwe describe the main idea of adiabatic evolution. Thatis, design a Hamiltonian H1 whose ground state is thetarget state |ΨT 〉 while the ground state |Ψ0〉 of Hamil-tonian H0 is easily to prepared. Assume that there existsa quantum system satisfying the following Hamiltonian

H = [1− f(t

T)]H0 + f(

t

T)H1, (1)

2

where f(t) is a slowly varying function of evolution timet with f(0) = 0 and f(1) = 1. According to the adia-batic theorem, the quantum system evolves adiabaticallyfrom the initial (ground) state |Ψ0〉 to the target (ground)state |ΨT 〉 at time t = T .In the present work, the physical model of interest

consists of a quantum dot coupled to a semiconductingnanowire, as shown in Fig. 1(a). In a magnetic field, byproper spin orbit interaction and proximity coupling toa superconductor, the nanowire can exist the Majoranabound states in the topological phase [15, 16, 45, 46].Then the effective Hamiltonian (in the low-energy limit)for the quantum dot coupling to the Majorana modereads [47]

H = ε(t)a†a+ (v∗a† − va)γ1, (2)

where a (a†) is the annihilation (creation) operator forthe electron in quantum dot and the on-site energy ε(t)for the quantum dot can be controlled by the gate volt-age Vg. v denotes the tunnel coupling between the quan-tum dot and the Majorana mode γ1. Without loss ofgenerality we assume v is an real number and take allphysical parameters in units of v. Since the Majorana

mode γi is Hermitian (γ†i = γi and γ2i = 1), we cannot

use the number operator γ†1 γ1 to count the occupationof the Majorana mode. Whereas, two Majorana modescan be combined to generate one ordinary fermion, e.g.,

γ1 = b+ b† and γ2 = i(b† − b). One can adopt the num-ber operator b†b of the ordinary fermion to count theMajorana modes.

FIG. 1: The setup for realizing the operation P1.

Since the total parity of the electron in quantum dotand the ordinary fermion formed by Majorana modes isconserved, the Hamiltonian is block diagonal in the basisspanned by {|0〉F |0〉D, |1〉F |1〉D, |1〉F |0〉D, |0〉F |1〉D},

H =

0 v 0 0v ε(t) 0 00 0 0 v0 0 v ε(t)

, (3)

where the state |m〉F |n〉D (m,n = 0, 1) represents m or-dinary fermion formed by Majorana mode and n electronin the quantum dot. In Ref. [47], it suggests that by adi-abatically changing the values of ε

vfrom −∞ to +∞, it

can realize the operation P1 which denotes the inversion

of the occupation in ordinary fermion combined by theMajorana mode (i.e., P1 = γ1),

P1(sin θ|0〉F + cos θ|1〉F ) = sin θ|1〉F + cos θ|0〉F . (4)

Fig. 2 shows the different dynamics behaviors for dis-tinct changing rate ε

v. It can be observed in Fig. 2(a)

that the operation P1 cannot be achieved perfectly sincethe changing of ε

vdoes not satisfy the adiabatic condition

very well (It cannot satisfy V (t) ≪ 1 all the time). Thusthe changing rate ε

vshould be small in order to meet the

adiabatic condition, along with the increasing of opera-tion time (cf. Fig. 2(c)-(d)). To implement the singlequbit rotation or non-Abelian operation, one shall suc-cessively execute the operation P1. Therefore the totaloperation time increases with the increasing of the num-ber of operation P1. In addition, it needs to point outthat this Majorana based qubits may be susceptible todecoherence due to the electron tunnel coupling process[48–51]. As a consequence, the adiabatic evolution is at adisadvantage in minimizing the influence of decoherenceas far as possible. Recently, it is overcome by short-cuts to adiabaticity for the non-Abelian braiding withY-junction structure [52]. In following it demonstratesthat the situation can be changed by periodic driving.

t

-10 0 100

0.5

1(a)

FidelityV(t)

t

-20 0 200

0.5

1(b)

FidelityV(t)

t

-50 0 500

0.5

1(c)

FidelityV(t)

t

-50 0 500

0.5

1(d)

FidelityV(t)

FIG. 2: The time evolution of the system to obtain the oper-ation P1, where the fidelity is defined as |〈ψ(t)|ψ(T )〉|2. The

expression of V (t) is defined as, V (t) = | 〈E1(t)|E2(t)〉E1(t)−E2(t)

| where

|Ei(t)〉(i = 1, 2) is the instantaneous eigenstate with corre-sponding eigenvalue Ei(t) in the even (or odd) subspace (Theadiabatic condition can be written as V (t) ≪ 1). The initialstate is |ψ(0)〉 = sin θ|0〉F + cos θ|1〉F and the target state is|ψ(T )〉 = sin θ|1〉F +cos θ|0〉F , θ = π

6. We have set the on-site

energy of the quantum dot increases with time linearly, e.g.,ε(t) = 50

Tt. The operation time is 2T and the final value of

ε(t) is 50 during the time evolution. All parameters are inunits of the tunnel coupling v. (a) T = 10. (b) T = 30. (c)T = 50. (d) T = 80. The larger T means the smaller of thechanging rate ε

v. One can find that the operation time for

perfectly achieving the operation P1 is about 2T = 100.

3

III. QUANTUM GATES WITH PERIODIC

DRIVING

A. Floquet theory

Let us first recall the Floquet theory briefly [53]. Pro-vided that the system Hamiltonian has a time-periodicdriving field, H(t) = H(T + t), where T is the periodand the driving frequency reads ω = 2π

T. The Floquet

theory asserts that the solutions of Schrodinger equa-tion have the form |Φn(t)〉 = eiǫnt|φn(t)〉. ǫn is quasi-energy and the Floquet state |φn(t)〉 has the property|φn(t)〉 = |φn(T + t)〉. They are satisfied the followingeigenvalue equation (~ = 1)

[H(t)− i∂

∂t]|φn(t)〉 = ǫn|φn(t)〉, (5)

where Heff = H(t)−i ∂∂t

is defined as the Floquet Hamil-tonian. To solve this equation it is very instructive tointroduce an extend Hilbert space [54] of time-periodic

functions with the inner product 〈〈·|·〉〉 = 1T

∫ T

0dt〈·|·〉.

With regard to the periodic driving system, it is nec-essary to make definite on the time-scales during theevolution. For the case of Floquet state |φn(t)〉, sinceit has the same period with the driving field, it affectsthe system dynamics on short time-scale (in the high-frequency limit). What really affects the long time-scaleof the system dynamics is the gap of the quasi-energies.Therefore, it is crucial to determine the evolution timethrough modulating the structure of quasi-energies in theperiodic driving system.

B. Sinusoidal driving

We first consider the periodic modulation of the on-site energy for quantum dot with the sinusoidal formε(t) = ε0 cos(ωt), which can be created by a waveformgenerator. In order to obtain an approximate expres-sion for the quasi-energy, we solve the time-dependentSchrodinger equation by standard perturbation theory[55, 56], where the tunneling Hamiltonian is regarded asthe perturbation. Due to the total parity conservation itis convenient to study in the even parity (or odd parity)subspace. Then the Hamiltonian reduces 2 × 2 matrix.Since Hε(t) = ε(t)a†a is diagonal, when substituting intoEq.(5), the eigenstates of [Hε(t) − i ∂

∂t] can be readily

given by

|λ1(t)〉 = (eiλ1t, 0)T ,

|λ2(t)〉 = (0, eiλ2t−iε0

ωsinωt)T , (6)

where λi (i = 1, 2) is the corresponding eigenvalue (i.e.,quasi-energy). On the other hand, due to the periodof Floquet states, one then can find that the zeroth or-der approximation of both quasi-energies are zero (mod-ulo ω). Thus the time-dependent eigenstates can be

approximately viewed as time-independent eigenstates|λ1(t)〉 = (1, 0)T and |λ2(t)〉 ≃ (0, 1)T in the high-frequency limit (ω ≫ 1). The first order approximationof quasi-energies can be obtained via diagonalizing theperturbing matrix [56]

Ht =

(

0 QQ∗ 0

)

, (7)

where the matrix element Q = vT

∫ T

0dte−i

ε0

ωsinωt af-

ter some straightforward calculations. Consequently, thequasi-energies are calculated as ǫ1,2 = ±|Q| (in the “firstBrillouin zone”) and the corresponding eigenstates be-come |ǫ1,2〉 = 1√

2(|λ1(t)〉 ± |λ2(t)〉). The gap of quasi-

energies are then given by ∆ = 2|Q|. In the light of theidentity

eiε0

ωsinωt =

∞∑

n=−∞Jn(

ε0

ω)einωt, (8)

where Jn is the n-order Bessel function, we can finally ob-tain the analytical expression for the quasi-energies gap∆ = 2|vJ0(

ε0ω)|.

ε00 20 40 60 80 100

∆

0

1

2(a)

t

0 20 40

Fid

elity

0

0.5

1(c)

t

0 5 100

0.5

1(e)

t

0 500

0.5

1 (d)

ω10 20 30 40 50 60

∆

0

1

2 (b)

FIG. 3: The gap of quasi-energies versus (a) the amplitude ε0when ω = 10, (b) the frequency ω when ε0 = 50. The gap ofquasi-energies approaches 2 when the driving frequency tendsto 60 in panel (b). After that the gap increases slowly withthe increasing of driving frequency. The dynamics evolutionof periodic driving system with (c) ω = 10, (d) ω = 20.8, (e)ω = 50. All parameters are in units of the tunnel coupling v.

Fig. 3(a)-(b) demonstrate the relation between thedriving field and the quasi-energies gap, while Fig. 3(c)-(e) show that the evolution time reduces with the increas-ing of the quasi-energies gap. Therefore, we can choosespecial evolution time for the periodic driving system byappropriately selecting the frequency and amplitude ofdriving field. An inspection of Fig. 3(a) also showsthat the operation time varies with the decreasing of theamplitude of driving field when we fix a high frequency.Interestingly, there exists a special case that the quasi-energies of periodic driving system vanishes (namely the

4

two quasi-energies approach degeneracy) by choosing theamplitude ε0 and the driving frequency ω properly, whichis known as coherent destruction of tunneling [57, 58]. Asa consequence the state is localization so that it is invalidto achieve the operation P1, as shown in Fig. 3(d).Since the initial state can be approximately given by

|ψ(0)〉 ≃ 1√2(|ǫ1〉 − |ǫ2〉), the time evolution of periodic

driving system approximately reads

|ψ(t)〉 ≃ 1√2(e−iǫ1t|ǫ1〉 − e−iǫ2t|ǫ2〉)

≃ cos [J0(ε0

ω)vt]|λ2〉 − i sin [J0(

ε0

ω)vt]|λ1〉. (9)

Hence the total operation time for realizing the operationP1 approximately equals

T ≃ π

2v|J0(ε0ω)| . (10)

FIG. 4: The total operation time T versus the amplitude ε0and the frequency ω of the driving field.

Fig. 4 depicts the relation between the total operationtime and the amplitude as well as the frequency of thedriving field. It suggests that one shall avoid the param-eter regions with the coherent destruction of tunneling,since it takes long operation time to realize the operationP1. Apart from this regions, the total operation time canbe regulated within proper range. In addition, from theEq. (9), one can find readily that the expression of fi-delity is

F = sin2 [J0(ε0

ω)vt] =

1

2(1− cos [2J0(

ε0

ω)vt]). (11)

Fig. 5 plots the relation between the gap of quasi-energiesand the coefficients of fidelity in the exact and pertur-bation regime, respectively. It demonstrates that theperturbation results work extremely well in the high-frequency limit.In order to check the validity of the perturbation the-

ory, we plot the dynamics of the system with different

∆( ε0ω)

0 0.5 1 1.5 2

a0

0.4

0.45

0.5

0.55

0.6

(a) ε0=50

ε0=40

ε0=30

∆( ε0ω)

0 0.5 1 1.5 2

a1

-0.6

-0.55

-0.5

-0.45

-0.4

(b) ε0=50

ε0=40

ε0=30

∆( ε0ω)

0 0.5 1 1.5 2

b1

-0.1

-0.05

0

0.05

0.1

(c) ε0=50ε0=40ε0=30

∆( ε0ω)

0 0.5 1 1.5 2

ω'

0

0.5

1

1.5

2

(d)

ε0=50ε0=40ε0=30

FIG. 5: The coefficients (a) a0, (b) a1, (c) b1, (d) ω′ of fidelity

versus the gap of quasi-energies, where the expression of fi-delity for curve-fitting is F ′ = a0 + a1 cosω

′t+ b1 sin ω′t. The

lines represent the analytical solutions given by Eq. (11) withperturbation theory while the circles, squares, and stars rep-resent exact results obtained by curve-fitting. Note that thecurve-fitting has high degree of precision for the exact resultssince the values of R-square and Adjusted R-square approachunit (≥ 99.37%) in MATLAB.

driving frequencies. The results are given in Fig. 6. Weobserve that the dynamics is well in agreement with theresults by perturbation theory when ω > 10 (in units ofv), while it deviates seriously from the perturbation re-sults when ω < 10 (in units of v, see the pink dot-dashline in Fig. 6). As a result, one can employ the perturba-tion theory safely when the frequency of the driving fieldis at least an order of magnitude larger than the tunnelcoupling.

t

0 5 6.14 10 15 20

Fid

elity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1ω=5ω=10ω=20ω=40

FIG. 6: The evolution of the fidelity with different frequenciesof driving field. We have set the quasi-energies gap ∆ =2|vJ0(

ε0

ω)| = 1.0236, thus the period of system dynamics is

approximated to T ≃ 2π∆

= 6.1383 in the high-frequency limit,which is confirmed by the green solid line and red dash line.

As mentioned, the perturbation theory is not valid in

5

the low-frequency limit, this gives rise to a question howthe system behaviors in this limit. Now we go to explorethis issue. When the driving frequency is small such thatthe adiabatic condition approximately holds (since theon-site energy changes slowly), we expected that the sys-tem dynamics, e.g., the fidelity, at the long time-scalewould be periodic with period T = 2π

ω. As expectation,

we find from the dash line in Fig. 7(a)-(c) that this isexact the case. Besides, one can observe that the highfidelity lasts a long time within a period when the driv-ing frequency is small, see Fig. 7(a)-(d). The fidelitychanges fast when the frequency of driving field is large(see Fig. 7(a), where the yellow lines and blue lines alterfrequently). It also affects the dynamics when the ampli-tude of driving field is large, which is shown in Fig. 7(b).Fig. 7(c) illustrates how the offset energy affects the fi-delity. Interestingly, the high fidelity lasts longer time(see the yellow region) when the offset on-site energy islarger.

FIG. 7: The fidelity as a function of the evolution time andthe distinct parameters of driving field. The expression ofdriving field is ε(t) = ε0 cosωt−ξ, where ξ is the offset energyof driving field. (a) ε0 = 40, ξ = 20. (b) ω = 0.05, ξ = 0.5ε0.(c) ε0 = 40, ω = 0.05, ξ = δε0. (d) ε0 = 40, ω = 0.04, ξ = 20.The dash lines are ploted by the function t = 2π

ωn, where n

is integer.

C. Square wave driving

It is believed that the periodic square wave drivingfields are easily achieved in practice. In fact these driv-ing fields have been studied extensively in time-periodicdriving system. In particular, it has been shown in ex-periment [59] that the Stuckelberg interference in a su-perconducting qubit is driven by the square wave formwhich we use in following. We first give the exact ana-lytical expressions for system evolution operator withoutany approximations. The square wave driving for the

on-site energy is expressed as

ε(t) =

{

ε1, nT ≤ t < t1 + nT,ε2, t1 + nT ≤ t < (n+ 1)T,

(12)

where n = 1, ..., N and t1 ∈ [0, T ]. Thus the systemevolution operator U of one period can be written asU(T, 0) = e−iH2t2e−iH1t1 with t2 = T − t1. After somelengthy algebra, one can obtain the expression of the evo-lution operator

U(T, 0) =1

x1x2e−

i

2D

(

A B + CB − C A

)

, (13)

where

A = −4v2 sinx1t1

2sin

x2t2

2+ (x1 cos

x1t1

2+ iε1 sin

x1t1

2)

·(x2 cosx2t2

2+ iε2 sin

x2t2

2),

B = −2iv(x2 cosx2t2

2sin

x1t1

2+ x1 cos

x1t1

2sin

x2t2

2),

C = 2v(ε1 − ε2) sinx1t1

2sin

x2t2

2,

D = ε1t1 + ε2t2, x1 =√

ε21 + 4v2, x2 =√

ε22 + 4v2. (14)

At first we design the driving time t1 (t2) of the on-siteenergy ε1 (ε2) to satisfy x1t1 = π (x2t2 = π), that is,

t1 =π

√

4v2 + ε21, t2 =

π√

4v2 + ε22, t1 + t2 = T. (15)

Consequently, the period of the square wave driving isconfirmed. According to Eq. (15), the evolution operatorcan be further simplified,

U(T, 0) =−1

x1x2e−

i

2D

(

x3 −x4x4 x3

)

, (16)

where x3 = 4v2 + ε1ε2 and x4 = 2v(ε1 − ε2). After Nevolution periods, the final evolution operation becomes

U(T , 0) = UN (T, 0) =1

2(−1

x1x2)Ne−

iND

2

·(

rN1 + rN2 −i(rN1 − rN2 )i(rN1 − rN2 ) rN1 + rN2

)

, (17)

where r1 = x3−ix4 = |r|e−iθ, r2 = x3+ix4 = |r|eiθ , |r| =√

x23 + x24, and tan θ = x4

x3

. From the expression in Eq.

(17), it clearly requires rN1 + rN2 = 0 in order to realizethe operation P1 perfectly (up to a global phase factor).By making the vectors rN1 and rN2 produce a π-phasedifference, that is, Nθ − (−Nθ) = π, one can readilyobtain the number of evolution periods

N =π

2 arctan 2v(ε2−ε1)4v2+ε1ε2

. (18)

Note that according to Eq. (18) the number of evolu-tion periods N is not integer generally. Nevertheless it

6

does not affect the main results because we can just takean integer nearest to N , as the fidelity increases slowlywhen it approaches 1. In turn, if one designates the pe-riod T and the number of evolution periods N in theperiodic square wave driving system, the values of on-site energy ε1 and ε2 can be determined by Eq. (15) andEq. (18) as well. It demonstrates the system evolutionwith distinct values of ε1 and ε2 in Fig. 8 (a)-(b), as wellas the special period and the number of evolution periodsin Fig. 8 (c)-(d). As expected, it can also realize the op-eration P1 by square wave driving and we can modulatethe period and the total operation time in this case.

t

0 5 10

Fid

elity

0

0.5

1(a)

t

0 2

Fid

elity

0

0.5

1(b)

t0 10

Fid

elity

0

0.5

1(c)

t0 5

Fid

elity

0

0.5

1(d)

FIG. 8: The system dynamics of distinct forms for squarewave. (a) ε1 = 40, ε2 = 60. (b) ε1 = 10, ε2 = 90. (c)T = 0.3, N = 60. (d) T = 0.2, N = 30. The other twoparameters are calculated by Eq.(15) and Eq.(18).

D. δ-function kick

When ε2 → +∞, one can readily find in Eq.(15) thatt2 → 0. Then the square wave driving field reduces toperiodic δ-function kick, i.e.,

ε(t) = ε1 + ε2

N∑

n=1

δ(t− nT ), (19)

where T is the driving period and can be calculated ap-proximately as T ≃ π√

4v2+ε21

. Therefore the total oper-

ation time for realizing the operation P1 is approxima-tively

T ≃ π2

2√

4v2 + ε21 arctan2vε1

. (20)

Note that the dynamics behavior is quite different fromthe absence of δ-function kick, i.e., under a static drivingfield. In the static case, the evolution operator U reads

U(t, 0) = e−iε1t

2

(

cos xt2 + i ε1

xsin xt

2 −i 2vxsin xt

2−i 2v

xsin xt

2 cos xt2 − i ε1

xsin xt

2

)

,

x =√

ε21 + 4v2. (21)

In absence of δ-function kick, the expression of fidelityfor realizing the operation P1 becomes | 2v

xsin xt

2 |2, wherethe maximum of fidelity is | 2v

x|2. One easily observes that

it cannot obtain the operation P1 when 2vx

≃ 0, i.e., theon-site energy ε1 ≫ 1. However the situation changes inthe presence of δ-function kick and it can realize the op-eration P1 regardless of the large value of on-site energyε1 (the value of ε1 only determines the driving period).Especially, when we take away the δ-function kick if theoperation P1 has been completed, the system is still sta-tionary.

IV. APPLICATION TO OTHER SYSTEMS

The periodic driving method can be applied to theother structure of hybrid quantum dot-topological sys-tem. Here we apply it to a system described by the fol-lowing Hamiltonian [61], as illustrated in Fig. 9,

H = ε(t)(a†↑a↑ + a†↓a↓) + V a

†↑a↑a

†↓a↓ + (v∗1 a

†↑ − v1a↑)γ1

+(v∗2 a†↓ − v2a↓)γ2. (22)

ε(t) is the on-site energy of the quantum dot. vi(i = 1, 2)denotes the tunnel coupling between the quantum dotand the Majorana mode γi. In particular, the spin-up(labeled as ↑) and spin-down (labeled as ↓) electrons canonly tunnel into the Majorana mode γ1 and γ2, respec-tively. V represents the energy contributed by doubleoccupation on the quantum dot. In the situation of largeV , the quantum dot can only hold single electron.Since the total parity (the electrons in quantum

dot and the ordinary fermions formed by Majoranamodes) of the hybrid system is conserved, we canrestrict ourself in the even-parity subspace spannedby {|0〉F1

|0〉F2|0〉D, |1〉F1

|1〉F2|0〉D, |0〉F1

|1〉F2|1↑〉D,

|0〉F1|1〉F2

|1↓〉D, |1〉F1|0〉F2

|1↑〉D, |1〉F1|0〉F2

|1↓〉D, },where the subscript Fi(i = 1, 2) represents the ordinaryfermions formed by Majorana modes. The matrix formof Hamiltonian then can be written as

H =

0 0 0 v2 v1 00 0 v1 0 0 v20 v∗1 ε(t) 0 0 0v∗2 0 0 ε(t) 0 0v∗1 0 0 0 ε(t) 00 v∗2 0 0 0 ε(t)

. (23)

This quantum dot-Majorana system [60, 61] can beused to prepare entanglement between spin and topologi-cal qubits or quantum information transfer between spinand the topological qubits (even for the quantum logicgates) by the adiabatic evolution. We take the prepa-ration of entanglement (denoting as the operation P2)between the electron spin and Majorana modes as an ex-ample to exemplify how to manipulate the operation timeby periodic square wave driving given in Eq. (12). The

7

FIG. 9: The setup for realizing entanglement between theconvention qubit and topological qubit.

operation P2 reads,

P2 (sin θ|0〉F1|0〉F2

+ cos θ|1〉F1|1〉F2

)|0〉D=

v1

v(sin θ|1〉F1

|0〉F2+ cos θ|0〉F1

|1〉F2)|1↑〉D

+v2

v(sin θ|0〉F1

|1〉F2+ cos θ|1〉F1

|0〉F2)|1↓〉D.(24)

where v =√

v21 + v22 . As the Hamiltonian is a 6× 6 ma-trix, the analytical expression of the evolution operatorU(T, 0) = e−iH2t2e−iH1t1 is involved. Here we only givethe equations that determine the period of the drivingfield and the total number of evolution periods, i.e.,

T = t1 + t2,

N =π

2 arctanx′

4

x′

3

, (25)

where t1 = π√ε21+4(v2

1+v2

2), t2 = π√

ε22+4(v2

1+v2

2), x′3 =

ε1ε2 + 4v21 + 4v22 , and x′4 = 2(ε2 − ε1)√

v21 + v22 . Fig.10 plots the fidelity of realizing operation P2 as a func-tion of the evolution time by the adiabatic evolution andthe periodic square wave driving, respectively. Again,we find that the operation time for adiabatic evolutionrequires relatively long time since it must satisfy the adi-abatic condition while the operation time and the periodof the square wave driving can be regulated.

t

-40 -20 0 20 40

Fid

elity

0

0.2

0.4

0.6

0.8

1(a)

t

0 5 10

Fid

elity

0

0.2

0.4

0.6

0.8

1(b)

FIG. 10: (a) Realizing the operation P2 by the adiabaticevolution. The on-site energy of the quantum dot increaseswith time slowly, ε(t) = 1.25t. (b) Realizing the operation P2

by the periodic square-wave driving. ε1 = 30, ε2 = 50. θ = π

6,

v1 = v2 = 1√2. All parameters are in units of v.

V. DISCUSSION AND CONCLUSION

In the last section, we have studied how to implementthe operation P1 by periodically driving the on-site en-ergy of the quantum dot. For a single operation P1, it isfar from sufficient to permit quantum computation. Wenext briefly discuss how to realize an arbitrary rotationfor a qubit by successively executing the operation P1

twice.As shown in Fig. 1(b), the system Hamiltonian of in-

terest reads

H = ε(t)a†a+ (|v1|e−iθ1 a† − |v1|eiθ1 a)γ1+(|v2|e−iθ2 a† − |v2|eiθ2 a)γ2, (26)

where we have introduced a phase θ1 (θ2) into the tunnelcoupling v1 (v2) in the Hamiltonian. Defining the opera-

tor c = aeiθ1 , z =√

|v1|2 + |v2|2, z1 = |v1|z, and z2 =

|v2|z,

the Hamiltonian (26) becomes,

H = ε(t)c†c+ z(c† − c)(z1γ1 + z2γ2), (27)

where the phase difference θ1−θ2 equals 2nπ (n is integerand the phase can be modulated by the magnetic flux φ).The form of Eq.(27) is the same as Eq.(2) if we redefine anew Majorana mode γ12 = z1γ1+ z2γ2, where the tunnelcoupling is denoted by z. Clearly, the operation P1 = γ12in this notation. Consider the two level system spannedby {|0〉F , |1〉F }, then we can express the Majorana opera-tors in terms of Pauli matrices {σx, σy , σz}, i.e., γ1 = σx,γ2 = σy , γ1γ2 = iσz. By successively executing the op-eration P1 twice with different relative tunnel couplingstrengthes between modes z1 and z2, the total operationbecomes P = γ12γ

′12 = (z1γ1 + z2γ2)(z

′1γ1 + z′2γ2) =

(z1z′1 + z2z

′2) + i(z1z

′2 − z′1z2)σz , which is exactly an ar-

bitrary rotation around the z-axis.Due to the conservation of the total parity, a qubit

shall be encoded by four Majorana modes [62]. TheMajorana-based qubit can be realized by the general-ization model in Fig. 1(b), which describes a systemconsisting of three quantum dots coupling to four Ma-jorana modes (γ1, γ2, γ3, γ4) in the topological super-conductor with comb structure. In the even-parity sub-space spanned by {|0〉F1

|0〉F2, |1〉F1

|1〉F2}, the operation

P = γ12γ′12 is in fact the rotation around the z-axis, and

the operation P = γ23γ′23 is the rotation around the x-

axis (γ23 = z2γ2 + z3γ3), where the ordinary fermion F1

(F2) is formed by the Majorana modes γ1 and γ2 (γ3 andγ4).Generally speaking, the tunnel coupling between the

quantum dot and the Majorana mode depends on boththe differences among the on-site energies and the tun-nel barriers. By making use of the periodic driving onthe on-site energy of the quantum dot, the tunnel cou-pling would change consequently. Reminding that we canemploy additional electrostatic gates to manipulate thetunnel barriers, the tunnel coupling can keep a constantin practise, even the on-site energies change. Indeed, the

8

possibility of controlling the tunnel coupling in semicon-ductor nanowire has been experimentally shown recently[63]. So, we believe that in the near future it is also possi-ble to manipulate the gates such that the tunnelling rateremains unchanged in our case, especially with the peri-odic square pulses (since it has only two distinct valuesof on-site energy).In conclusion, we have proposed a method to regu-

late the total operation time of quantum computation,which can be achieved by periodic driving. By solvingthe time-dependent Schrodinger equation with perturba-tion expansion, we have given an expression of the quasi-energies and elucidated the relationship between the totaloperation time and quasi-energies in the high-frequencylimit. As a result, the operation time can be manipu-lated by designing the amplitude and frequency of thedriving field. For the case of low-frequency limit, due tothe invalidity of the perturbation theory, we study thedynamical behaviors by numerical simulations. We findthe results approach those given by adiabatic evolution.Different from the adiabatic evolution, the system in thelow-frequency limit manifests more intricate behaviorsand the operation time can also be regulated by the driv-

ing field. In particular, the total time that the high fi-delity lasts are closely related to the frequency and offsetenergy of driving field. For the case of periodic squarewave driving, we have derived an analytical expressionfor the evolution operator without any approximations.By this expression, we can calculate the amplitude ofsquare wave driving fields with fixed operation time andperiod of driving fields. We have also discussed the real-ization of quantum operations by the δ-kick, which canbe treated as a deformed square wave driving. The pe-riodic driving can also be applied to the other quantumsystem—it opens up a new avenue in manipulations ofoperation time in topological quantum computations.

ACKNOWLEDGMENTS

We thank S. M. Frolov for helpful discussions. Thiswork is supported by the National Natural ScienceFoundation of China (Grants No. 11175032 and No.61475033).

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