PHYSICAL REVIEW A 90, 012323 (2014)

Majorana zero modes on a one-dimensional chain for quantum computation

Lei Chen, W. LiMing,* Jia-Hui Huang, Chengping Yin, and Zheng-Yuan XueDepartment of Physics, Laboratory of Quantum Engineering and Quantum Materials, School of Physics

and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China(Received 2 May 2014; published 17 July 2014)

Numerical calculations for Majorana zero modes on finite chains are performed using the technique of blockdiagonalization. It is found that Majorana zero modes occur near the ends of the chain and decay exponentiallyaway from the ends. The phase diagrams show that Majorana zero modes of a sufficiently long chain indeed havea parameter domain of 2t > |μ| as predicted from the bulk property of an infinite chain, but a short chain hasa much smaller parameter domain than the prediction. Through a numerical simulation Majorana zero modesare found to be robust under the disturbance of noise. Finally the reversion of the parity of the ground states isstudied by applying a bias voltage on the quantum dot at an end of the chain. It is found that for a weak couplingbetween a chain and a quantum dot the parity of the ground states can be reversed through adiabatically tuningthe bias voltage.

DOI: 10.1103/PhysRevA.90.012323 PACS number(s): 03.67.Lx, 71.10.Pm, 74.20.Rp, 74.78.−w

I. INTRODUCTION

In particle physics, Majorana fermions are their own an-tiparticles [1–4]. It is still unclear whether Majorana fermionsexist in nature as elementary building blocks, but they canbe constructed in pairs in condensed-matter systems [5–7].Dirac fermion operators a and a† can be decomposed into apair of Majorana operators γ1 and γ2, e.g., a = (γ1 + iγ2)/2.They satisfy γi = γ

†i and an unusual anticommutation relations

{γi,γj } = 2δij .One decade ago Kitaev proposed a basic theoretical model

to exhibit Majorana fermions [8]. On a quantum wire witha superconducting coupling two unpaired Majorana zeromodes (i.e., Majorana fermions) appear at the two ends ofthe wire. This model has been extensively studied in theliterature recently [9,10]. The Majorana zero modes givedoubly degenerate ground states of the system, |0〉 and|1〉, with parities ±1, respectively. These two degenerateground states can be used to encode qubit states in quantuminformation.

Majorana zero modes satisfy the braiding statistics. Aninterchange of two adjacent Majorana fermions turns γ1

and γ2 to γ2 and −γ1. Alicea et al. proposed a T-junctionstructure to interchange two Majorana fermions throughsequent switches of bias voltages on different parts of thejunction [10]. The interchange, however, can only rotate theinitial state by an angle of π/2 around the z axis of theBloch sphere. This indicates that a universal quantum gatecannot be performed through an interchange of Majoranafermions.

In order to obtain the universal set of quantum gates forquantum computation, one may consult hybrid topological andconventional qubit systems and supplement the topologicallyprotected operations with unprotected operation [11–13].Moreover, errors from these hybrid systems can be correctedfor a high threshold [14]. In particular, Flensberg recentlyconsidered a hybrid system consisting of coupled quantum

*wliming@scnu.edu.cn

dots and Majorana fermions and proposed a method forarbitrary rotations to the initial state using a coupling betweenMajorana zero modes and some quantum dots [15]. Bytuning the bias voltage on the quantum dot the parity ofthe initial state is reversed. Finally universal quantum gatesare realized using four Majorana fermions and three quantumdots with appropriate magnetic flux. Therefore, the buildingblock of quantum computation with this method is the systemconsisting of a quantum dot coupled to a Majorana fermion ofa topological qubit.

Kitaev proposed only a theoretical analysis for the one-dimensional (1D) chain model [8]. A numerical result ofthis model to show the distributions of Majorana zero modesis intriguing and necessary for real applications. AlthoughKitaev pointed out that Majorana zero modes exist in a certainparameter region for an infinitely long chain the phase diagramis never given for finite chains. Due to the nonlocal distributionof a Majorana fermion pair it is thought that Majoran zeromodes are free of decoherence in quantum information. Therobustness of the Majorana zero modes under noises, however,has not been given yet in the literature. The main goal of thepresent work is to confirm the phase diagram and to show therobustness of the Majorana zero modes using random voltagedistributions as a noise.

In this paper, we first study numerically the Majoranazero modes in Kitaev’s model including the wave functionsand the phase diagram of Majorana zero modes. Then theirrobustness and the parity reversion of the ground states forquantum computation are investigated. We use a technique ofSchur’s decomposition provided by the MATLAB software fordiagonalization. A brief introduction to Kitaev’s model and thediagonalization method is given in Sec. II and then numericalresults of Majorana zero modes, robustness, and the parityreversion of the ground states are reported in the followingthree sections. The last section gives a summary.

II. KITAEV’S MODEL

We begin from a finite chain of N sites with a superconduct-ing pairing potential. Each site can be either empty or occupied

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CHEN, LIMING, HUANG, YIN, AND XUE PHYSICAL REVIEW A 90, 012323 (2014)

by a spinless electron . The Hamiltonian reads [8,16]

H0 =N−1∑j=1

[−ta†j aj+1 + �ajaj+1 + H.c.]

−μ

N∑j=1

(a†j aj − 1

2

)(1)

where a†j is the electron creation operator localized at site j,

t is the hopping amplitude, μ is the chemical potential, and� = |�|eiθ is the induced superconducting gap with a phaseθ . One may try to diagonalize the above Hamiltonian using theBogoliubov de Gennes (BdG) equation after a transformationfj = ∑

i(ujiai + vjia†i ). This method, however, is not valid

because the new operators fj do not hold the fermioncommutation relations, e.g., {fj ,fk} �= 0. Kitaev provideda method by introducing the following Majorana fermionoperators [8,16]:

γ2j−1 = aj eiθ/2 + a

†j e

−iθ/2, (2)

γ2j = −i(aj eiθ/2 − a

†j e

−iθ/2). (3)

They satisfy the commutation relation of Majorana fermions{γi,γj } = 2δij and γ

†i = γi . Using these Majorana operators

Hamiltonian (1) is transformed into

H0 = i

4

2N∑i,j=1

γiAijγj , (4)

where A is symmetrized to be a real antisymmetric matrix(A = −AT )

A =

⎛⎜⎜⎜⎜⎜⎜⎝

0 −μ 0 −t + |�| ...

μ 0 t + |�| 0 ...

0 −t − |�| 0 −μ ...

t − |�| 0 μ 0 ...

......

......

. . .

⎞⎟⎟⎟⎟⎟⎟⎠

.

In general A has eigenvalues of purely imaginary numbersin pairs ±iεj (εj � 0) [8,16,17]. It allows one to bring Ato a block diagonal form through a real orthogonal matrixtransformation as follows:

WAWT =

⎛⎜⎜⎜⎜⎜⎜⎝

0 ε1

−ε1 0

. . .

0 εN

−εN 0

⎞⎟⎟⎟⎟⎟⎟⎠

. (5)

The matrix W is worked out through the Schur routine in thesoftware MATLAB. In terms of Eq. (5), the Hamiltonian (4) is

reduced into a canonical form

H0 = i

2

N∑m=1

εmγ2m−1γ2m (6)

=N∑

m=1

εm

(a†

mam − 1

2

), εm � 0, (7)

where γm = ∑j Wmjγj and am = (γ2m−1 + iγ2m)/2,a

†m =

(γ2m−1 − iγ2m)/2. Note that γm = γ†m are Majorana fermion

operators and a†m,am are Dirac fermion operators representing

quasiparticles of the present model. The single-particle ener-gies εm have been arranged in an increasing order. Majoranazero modes appear when ε1 vanishes. They in general distributeat boundaries of a system. A pair of Majorana zero modescomposes a nonlocal Dirac fermion.

III. MAJORANA ZERO MODES

Through a Fourier transformation ak = N−1/2 ∑i e

−ikRi ai

to the Hamiltonian (1) an infinite chain’s bulk energy can beobtained and is given by

ε±(k) = ±√

(2t cos k + μ)2 + 4�2 sin2 k, |k| � π. (8)

The gap between these two energies closes at 2t = |μ|.Kitaev found Majorana zero modes at a special case |�| = t >

0,μ = 0 [8,16]. Two unpaired Majorana zero modes with zeroenergy appear sharply at the two ends of the chain. In addition,Kitaev made a conjecture and a mathematical analysis thatMajorana zero modes exist in the domain |μ| < 2t . It isbelieved that Majorana zero modes in a general case distributenear the ends and decay exponentially away from the ends. Wefirst verify this conjecture through numerical computations.Throughout the whole paper the parameter t in the Hamiltonianis set to unity.

First the parameters are set in the domain |μ| < 2t for a50-site chain. Two zero modes indeed appear with approximatezero energy (<0.002). The components W1i and W2i onMajorana operators γi of the zero modes are plotted in Fig. 1.As |�| deviates from t the zero modes disperse gradually awayfrom the two ends but decay exponentially as expected. Whenμ approaches to 2t , the phase transition point, the two zeromodes overlap more and disappear finally.

To see the phase transition we plot the phase diagram ofMajorana zero modes with respect to the superconductor gap� and the chemical potential μ in Fig. 2. It is interesting to seethat the Majorana zero modes exist indeed below μ < 2t butthe boundary of the phase inclines to smaller μ for larger �.The Majorana zero modes of a short chain have only a smallparameter domain, deviating significantly from the predictedphase transition point |μ| = 2t . In addition, the Majorana zeromodes have a threshold value for the superconducting gap.This indicates that a superconducting gap is necessary for theMajorana zero modes.

IV. PARITIES AND ROBUSTNESS

How the ground state looks like on the chain is an interestingissue. It is seen from the canonical form (7) of the Hamiltonian

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MAJORANA ZERO MODES ON A ONE-DIMENSIONAL . . . PHYSICAL REVIEW A 90, 012323 (2014)

0 20 40 60 80 100−1

0

1

Com

pone

nts

Site× 2

Δ=0.8μ=0.4

(a)

0 20 40 60 80 100−1

0

0.5

Com

pone

nts

Site× 2

Δ=0.2μ=0.4

(b)

0 20 40 60 80 100−0.4

−0.2

0

0.2

0.4

Com

pone

nts

Site× 2

Δ=0.2μ=1.97

(c)

FIG. 1. (Color online) Components W1i and W2i on γi of zero modes of a 50-site chain with different parameters (a) t = 1, � = 0.8,μ = 0.4, (b) t = 1, � = 0.2, μ = 0.4, and (c) t = 1, � = 0.2, μ = 1.97. The solid (blue online) and dashed (red online) lines correspond tothe two Majorana zero modes γ1 and γ2, respectively.

that the ground state |ψ〉 satisfies 〈ψ |a†mam|ψ〉 = 0 for m � 2

as εm > 0 but 〈ψ |a†1a1|ψ〉 = 0 or 1 as ε1 = 0 for the Majorana

zero modes. These two cases lead to twofold degenerateground states. In the occupation number representation basisvectors can be expressed as

|nN nN−1 . . . n1〉 =N∏j

(a†j )nj |0〉, (9)

where |0〉 is one of the two degenerate ground states withoutany quasiparticles, i.e., aj |0〉 = 0,j = 1,2, . . . ,N . Obviously,|0〉 can be given by

|0〉 = Ca1a2...aN |vac〉, (10)

where C is a normalization factor and |vac〉 is the vacuum state.The other ground state is |1〉 = a

†1|0〉, which has a quasiparticle

with zero energy.One defines a parity operator in the following form:

P =N∏

j=1

(−iγ2j−1γ2j ) =N∏

j=1

(−2a†j aj + 1). (11)

Since P 2 = 1, P has two eigenvalues ±1, called even and oddparities, respectively. It can be easily verified that [P ,H ] = 0.This indicates that a nondegenerate eigenstate of H musthave a determinant parity, but degenerate eigenstates can berecomposed to have determinant parities. Most important isthat an even (odd) parity corresponds to an superposition ofstates with even (odd) numbers of electrons.

The parity of the ground state |0〉 can be deduced in thefollowing way:

P |0〉 = (−1)NCa1a2...aN P |vac〉= (−1)N |0〉, (12)

where P |vac〉 = |vac〉. Similarly, the other ground state has

P |1〉 = (−1)N+1|1〉. (13)

Therefore, the two degenerate ground states, |0〉 and |1〉, haveopposite parities, (−1)N and (−1)N+1, and contain even (odd)and odd (even) numbers of electrons, respectively, for a chainwith an even (odd) number of sites. These two degenerateground states |0〉 and |1〉 can be used to encode qubit states inquantum information.

How do the electrons in the ground states distribute on thechain? The electron density is given by the expectation value〈a†

j aj 〉 in the ground states. As shown in Fig. 3(a) the electrondensity drops sharply about 10% at both ends at � = 0.8,μ =0.4 but oscillates around the ends at � = 0.2,μ = 0.4 on a50-site chain, in accordance with the Majorana zero modesshown in Figs. 1(a) and 1(b). This provides a possibility ofmeasuring the Majorana zero modes. Electron densities inboth ground states, however, are indistinguishable due to thedegeneracy.

The single-particle energies are shown in Fig. 3(b). Thereis a big gap between the zero mode energy and the not-zeromodes. This gap allows the Majorana zero modes to be robustfor weak disturbance.

How robust the Majorana zero modes are under the distur-bance of noise is an essential issue for quantum computing. Tosimulate the disturbance of noise we add to Hamiltonian (1)

0 1 2 30

1

2

μ

Δ N=8

(a)

0 1 2 30

1

2

μ

N=31

(b)

0 1 2 30

1

2

μ

N=60

(c)

FIG. 2. (Color online) Phase diagrams of Majorana zero modes on chains with (a) 8 sites, (b) 31 sites, and (c) 60 sites. Shadow regionsstand for the parameter domains of Majorana zero modes.

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CHEN, LIMING, HUANG, YIN, AND XUE PHYSICAL REVIEW A 90, 012323 (2014)

0 10 20 30 40 500.52

0.54

0.56

0.58

0.6

0.62

0.64

Sites

Ele

ctro

n de

nsiti

es Δ=0.8,μ=0.4Δ=0.2,μ=0.4

(a)

0 10 20 30 40 500

0.5

1

1.5

2

2.5

Ene

rgie

s

States

Δ=0.8,μ=0.2Δ=0.2,μ=0.2

(b)

FIG. 3. (Color online) (a) Electron densities in the ground statesfor different parameter settings as shown in the labeling and (b)single-particle energies εm on a 50-site chain with parameters asshown.

a local noise Hn = ∑j Vja

†j aj , where the noise energies Vj

are simulated by Vj = 2V0(Rj − 1/2) with a series of randomnumbers Rj ∈ (0,1). The phase diagrams after adding thisnoise are shown in Fig. 4. It is seen that the boundary of theMajorana zero modes becomes more indistinct and diffusesinto the forbidden region as increasing the intensity V0, but theregion with smaller chemical potentials is still robust underthe disturbance.

V. QUANTUM COMPUTATION

An arbitrary operation of a quantum bit can be realizedthrough the coupling between quantum dots and Majoranafermions [15]. Here we consider the ending site of the chainas a quantum dot. The Hamiltonian takes the following form:

H = H0 + (V − μ)

(D†D − 1

2

)+ H1, (14)

H1 = i

2(t + |�|)γ2N1 + i

2(−t + |�|)γ2N−12, (15)

0 1 2 30

1

2

μ

Δ V =1.0

0 1 2 30

1

2

μ

V =2.0

(a) (b)

FIG. 4. (Color online) Majorana zero modes (blue [gray]crosses) of a 31-site chain with noise (a) V0 = 1 and (b) V0 = 2.

−2 −1 0 1 2 3−2

−1

0

1

2

Bias voltage V

Ene

rgie

s

(a)

−2 −1 0 1 2 30

0.5

1

Bias voltage V

C

C C

(b)

−2 −1 0 1 2 30

0.5

1

Bias voltage V

C

C C

(c)

FIG. 5. (Color online) (a) Energies of a quantum dot coupled toa 50-site chain with parameters � = 0.8,μ = 0.4; (b) amplitudes Ci

in Eq. (17) with the original coupling constant S; (c) amplitudes Ci

with a 10-time reduced coupling constant 0.1S.

where 1 and 2 denote Majorana fermion operators of thequantum dot at site N + 1, D† = (1 − i2)/2 denotes thecreation of electrons on the quantum dot, and V is the biasvoltage on the quantum dot. The operators γ2N−1 and γ2N inthe above Hamiltonian change the electron number and thusthe parity of the ground state. At high bias voltage this couplingresults in nonzero excitations but it approaches Karsten’s resultat small bias voltages near the ground states [15].

We consider this coupling as a first-order perturbation onfour near states |nm〉 = (a†)n(D†)m|00〉,n,m = 0,1. They haveunperturbed energies ±(V − μ)/2 with double degeneracy.Using the theory of the first-order perturbation we obtain twoenergies,

E = ±√

(V − μ)2/4 + |S|2, (16)

where S = − 12 (−t + |�|)T2N−1,1 + i

2 (t + |�|)T2N,1 withTJ,j = WT

J,2j−1 + iWTJ,2j is the coupling constant. A superpos-

tion of the two degenerate states corresponding to the lowerenergy gives

|ψ〉 = C1(α|00〉 + β|10〉) + C2(α|11〉 + β|01〉), (17)

where amplitudes C1 = ρ/√

1 + |ρ|2,C2 = 1/√

1 + |ρ|2 withρ = −[(V − μ)/2 +

√(V − μ)2/4 + |S|2]/S. This result is

in coincidence with Karsten’s result [15]. Through adiabat-ically tuning the bias voltage V the above state can bedriven from |i〉 = α|00〉 + β|10〉 to γ1|i〉 = α|11〉 + β|01〉continuously, where the parity of the chain has been reversed(|0〉 ↔ |1〉). This process, however, is limited by the gapabove the ground states as shown in Fig. 3(b). Since nonzeroexcitations may occur above the gap the initial state may bemissing in the tuning process. The energies of a 50-site chaincoupled to a quantum dot and its amplitudes Ci are plotted inFig. 5 for different bias voltages. The bias voltage has beenlimited in the region so that the energy does not exceed thegap of the chain. It is seen from Fig. 5(b) that a reversal ofC1 and C2 is truly taking place but they do not have a clear 0to 1 reversal. Reducing the coupling constant there will be aclearer reversal as shown in Fig. 5(c) for a coupling constant0.1S. Therefore, a parity reversion can be realized in a weakcoupling between a chain and quantum dot.

VI. SUMMARY AND CONCLUSION

In this work numerical calculations for Majorana zeromodes on a finite superconducting chain are performed usingthe technique of block diagonalization in Schur’s decompo-sition. It is found that Majorana zero modes occur near the

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MAJORANA ZERO MODES ON A ONE-DIMENSIONAL . . . PHYSICAL REVIEW A 90, 012323 (2014)

ends of the chain and decay exponentially away from theends. The phase diagrams show that Majorana zero modesof a sufficiently long chain indeed have a parameter domainof 2t > |μ| as predicted from the bulk property of the chain,but a short chain has a much smaller parameter domain thanthe prediction. Through a simulation Majorana zero modesare found to be robust under the disturbance of noise. Finallythe reversion of the parity of the ground states is studied byapplying a bias voltage on a quantum dot at an end of thechain. For a weak coupling between a chain and a quantum

dot the parity of the ground states can be reversed throughadiabatically tuning the bias voltage.

ACKNOWLEDGMENTS

I thank Ming Gong for helpful discussions. This workwas supported by the NFRPC (No. 2013CB921804) andthe National Natural Science Foundation of China (GrantNo. 11104086).

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