Eur. Phys. J. D (2013) 67: 89DOI: 10.1140/epjd/e2013-30582-y

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Measurement based controlled not gate for topological qubitsin a Majorana fermion and quantum-dot hybrid system

Zheng-Yuan Xuea

Laboratory of Quantum Information Technology, and School of Physics and Telecommunication Engineering,South China Normal University, Guangzhou 510006, P.R. China

Received 21 September 2012 / Received in final form 15 February 2013Published online 30 April 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. We propose a scheme to implement controlled not gate for topological qubits in a quantum-dotand Majorana fermion hybrid system. Quantum information is encoded on pairs of Majorana fermions,which live on the the interface between topologically trivial and nontrivial sections of a quantum nanowiredeposited on an s-wave superconductor. A measurement based two-qubit controlled not gate is producedwith the help of parity measurements assisted by the quantum-dot and followed by prescribed single-qubit gates. The parity measurement, on the quantum-dot and a topological qubit, is achieved by theAharonov-Casher effect.

Recently, physical implementation of quantum computershas attracted much attention. One of the main difficul-ties of scalable quantum computation is decoherence ofquantum information. A promising strategy against deco-henrence is based on the topological idea [1] where gateoperations depend only on global features of the controlprocess, and thus largely insensitive to local noises. Topo-logical ordered states emerge as a new kind of states ofquantum matter beyond the description of conventionalLandaus theory [2]. A paradigmatic system for the exis-tence of anyons is a kind of so-called fractional quantumHall states [2]. Alternatively, artificial spin lattice modelsare also promising for observing these exotic excitations,e.g., Kitaev models [3,4] are most famous for demonstrat-ing anyonic statistics [5–7] and braiding operations fortopological quantum computation.

For universal quantum computation, one needs non-Abelian anyon to serve as qubit. With the poten-tial applications in topological quantum computation,Majorana fermions (MF) with non-Abelian statistics haveattracted strong renewed interests. MF are a kind of self-conjugate quasi-particles induced from a vortex excitationin px + ipy superconductor [1]. However, due to the in-stability of the p-wave superconducting states, its imple-mentation remains an experimental challenge. Recently,it is recognized that topologically protected states may bemost easily engineered in 1D semiconducting nanowiresdeposited on an s-wave superconductor [8–11], which pro-vides the first realistic experimental setting for Kitaev’s1D topological superconducting state [12]. Note that thequench dynamics of this model across a quantum criticalpoint is also presented [13,14].

a e-mail: zyxue@scnu.edu.cn

Although not universal for quantum computation, par-ity protected quantum logical operations schemes with twopairs of MF as a topological qubit are proposed [15–21],which mainly focus on braiding operations of singlequbit [15–18] and quantum information tranfer betweena topological qubit and a quantum bus [19–21]. As theabsence of topologically ordered system that is univer-sal for quantum computation and extremely difficult toperform certain topology changing operations, present re-search of universal quantum computation with MF, aswell as other exotic topological qubits, need to supple-ment braiding operations with topologically unprotectedoperations. These operations can be error-corrected for ahigh error-rate threshold of approximately 0.14 [22]. How-ever, such a high error threshold may still prove difficultusing unprotected operations within a topological system.

One of the main difficulties in implementing topologi-cal quantum computation lies in the difficulty of braidingof MF from different topological qubits for entangling op-eration. Therefore, a topological quantum bus, usually ina topological and conventional qubit hybrid architecture,would be of great help, where error rates below 0.14 havealready been achieved [23]. A topological quantum busmakes the implementation much easier as the quantumbus can transfer the quantum information between dif-ferent topological qubits, and thus only braiding withina topological qubit, to realize single-qubit operation, isneeded for quantum computation.

Here, we propose a scheme to implement controlled notgate for topological qubits in a quantum-dot (QD) andMF hybrid system. The architecture consists of 1D semi-conducting wires deposited on an s-wave superconductor,under certain conditions, the endpoints of such wires sup-port localized zero-energy MF. Qubit is encoded on two

Page 2 of 4 Eur. Phys. J. D (2013) 67: 89

x

L R

J1 J2 J3

124 3

Fig. 1. Superconducting flux qubit with three Josephson junc-tions (pink) and an enclosing magnetic flux Φx. MF (greendots) are induced at the interface between a topologically triv-ial (blue) and a topologically nontrivial (red) section of anquantum nanowire. The right superconducting island is locatedby a semiconductor double quantum-dot qubit. Gate electrodes(not shown) can be used to move the MF along the wire.

pairs of MF, the realization of parity measurements ona QD and a topological qubit, together with control andmeasurement on the QD state and single-qubit gates onthe target topological qubit, is able to generate a two-qubit controlled not (CNOT) gate [24]. In this sense, thisscheme is a measurement based one. Supplemented witharbitrary single qubit rotation, which is already proposedin such MF quanutm dot hybrid system [10,16], universalquantum computation can be realized.

The setup we consider is shown in Figure 1, which isa spin-orbit coupled semiconducting wire deposited on ans-wave superconductor, which, together with Josephsonjunctions, form a superconducting flux qubit configura-tion. Applying a magnetic field perpendicular to the su-perconductor surface, the Hamiltonian describing such awire is [8]

H =∫ [

ψ†x

(− �

2∂2x

2m− μ− i�ueσ∂x + VBσz

)ψx

+(|Δ|eiϕψ↓xψ↑x + h.c.)]dx, (1)

where ψαx corresponds to electrons with spin α, effec-tive mass m, and chemical potential μ; the third termdenotes spin-orbit coupling with u being the strength, andσ = (σx, σy, σz) is the vector of Pauli matrices; the fourthterm represents the energy shift due to the magnetic field;and the terms in the second line are the spin-singlet pair-ing from the s-wave superconductor via proximity effect.The interplay of Zeeman effect, spin-orbit coupling, andthe proximity to an s-wave superconductor drive the wireinto a chiral p-wave superconducting state [8,9], providingthat the wire is long compared to the superconducting co-herence length (ξ � 40 nm for the superconducting sub-strate being Nb). For |μ| < μc =

√V 2

B − |Δ|2 the topolog-ical phase with end MF emerges, or a topologically trivialphase. Thus, applying a gate voltage uniformly allows oneto create or remove the MF. To avoid gap closure [10], a“keyboard” of local tunable gate electrodes to the wire isused to control whether a region of the wire is topological

or not. For InAs quantum narowire, assuming |VB | ∼ 2|Δ|and �u ∼ 0.1 eV A, the gap for a 0.1 μm wide gate is oforder 1 K [10]. The QD used here is semiconductor doublequantum dot molecule with one dot deposit on the super-conductor of the flux qubit circuit while the other is not.We assume here that there is a galvanic isolation betweenthe superconductor and semiconductor, so that there is nocharge transfer between them [20]. Remarkably, one canalso realize this QD using InAs nanowires [25], which ispreviously used for supporting MF, and thus reduce theexperimental difficulties of implementation.

A pair of MF can be combined into a complex fermion.The fermion parity operator np has eigenvalues −1 and +1for states |0〉 and |1〉, respectively. But, the two states dif-fer by fermion parity, which prevents the coherent super-position of the two. Therefore, for the purpose of quantumcomputation, where coherent superposition is inevitable,we combine four MF to form a topological qubit. In thisway, coherent superposition is permitted for the two en-coded qubit subspaces with same fermion parity. Withoutloss of generality, we can use the subspace with fermionparity is even as the encoded qubit states, i.e., the twostates of the topological qubit are encoded as |00〉 and |11〉of the four MF.

We now turn to the problem of reading out the twoqubit states, which is one of the preliminary requirementsfor quantum computation purpose. As the states of thetwo pairs of MF in a topological qubit are always the same,detecting one of them can then fulfil the purpose of distin-guish them. As noted above, the two states |0〉 and |1〉 aredifferent in fermion parity, so they can be distinguishedby np. Without loss of generality, we choose to detect thepair of γ1 and γ2 while move γ3 and γ4 out of the fluxqubit circuit, also our measurement circuit, as shown inFigure 1. Note that the topological property of the wirewill not be interrupted by the junctions if the Josephsonjunctions’ thickness is much smaller than the supercon-ducting coherence length ξ. To measure the parity of np,we use the suppression of macroscopic quantum tunnel-ing by the Aharonov-Casher effect [26]: a Josephson vor-tex encircling a superconducting island picks up a phaseφ = πq/e determined by the total charge q coupled ca-pacitively to the superconductor, which includes both thecharge on the superconducting island and on a nearby gateelectrode.

Following references [16,26], we consider a supercon-ducting flux qubit with three Josephson junctions, asshown in Figure 1, where junctions 1 and 3 have thesame Josephson coupling energy EJ while that of junc-tion 2 is αEJ . The charging energy of the islands is muchlarger than EJ so that the considered qubit works inthe flux regime. The gauge-invariant phase drops of thethree junctions 1, 2, 3 are related to the total magneticflux Φx through the flux qubit loop by the constraintφ1 + φ2 + φ3 = 2πΦx/Φ0 with Φ0 = h/2e being the fluxquanta. On the condition that the size of the qubit is suffi-ciently small, the flux generated by the circular supercur-rent along the loop can be neglected. Then, the enclosedflux of the qubit contour comes solely from the external

Eur. Phys. J. D (2013) 67: 89 Page 3 of 4

magnetic filed. Then, the superconducting energy of theflux qubit reads

U = −EJ

[cosφ1 + cosφ3 + α cos

(2πΦx

Φ0− φ1 − φ3

)],

(2)which exists two lowest degenerate states, the supercur-rent of which flows clockwise and counterclockwise, re-spectively. These states are usually defined as the super-conducting flux qubit states. The qubit states correspondto the potential energy minima in equation (2), and thusthe tunneling between the two states requires quantumphase slips.

When α > 1, the two energy minima are connectedby two different tunneling paths, differ by 2π in φ1

and −2π in φ3. The interference between the two tunnel-ing paths constitutes the circulation of a Josephson vortexaround both superconducting islands L and R. Accodingto Aharonov-Casher effect, the acquired phase is ψ = πq/ewith q =

∑i=L,R(eni

p + qi) being the total charge on thetwo islands eni

p and gate capacitors qi to the qubit loop.As we chose junctions 1 and 3 have the same Josephsoncoupling energy, the two tunneling paths have the sameamplitude. The interference between the two tunnelingpaths produces an oscillating tunnel splitting of the twolevels of the flux qubit

Δ = Δ0

∣∣∣∣cos(ψ

2

)∣∣∣∣, (3)

where Δ0 is the tunnel splitting associated with one path.Therefore, if q is an odd (even) multiple of the electroncharge e, the two tunneling paths interfere destructively(constructive), and thus the tunnel splitting is minimum(maximal).

As we only need to distinguish maximal from minimaltunnel splitting, the flux qubit does not need to have ahigh quality factor. In addition, Δ0 � 100 μeV � 1 Kfor parameters in typical experiments of flux qubits [26],which should be readily observable by microwave absorp-tion. To make sure the total charge is solely comes fromthe two superconducting islands, one would first calibratethe charge on the gate capacitor to zero, i.e., qi = 0, bymaximizing the tunnel splitting in the absence of vorticesin the island. The read-out is nondestructive, which is nec-essary for our proposed way of implementation the two-qubit CNOT gate. Meanwhile, it is insensitive to sub-gapexcitations in the superconductor as they do not changethe fermion parity.

Following reference [20], as shown in Figure 1, q canalso have a quantum component: charge comes fromthe QD. Logical basis of QD can be defined as semiconduc-tor charge qubit |0〉 = |0〉⊗|1〉 and |1〉 = |1〉⊗|0〉 with theelectron occupies the lower and upper dot, respectively.Therefore, the qubit basis states correspond to the elec-tron parity on the upper dot enclosed by the Josephsonvortex circulation. To read-out the MF state, one can sim-ply set the QD to |0〉 so that its charge do not interruptthe measurement. For our purpose, we are also interestin the joint parity of QD and a pair of MF. Indeed, the

P

PH

C

MH

H H

AT

H

Fig. 2. Measurement-based CNOT gate for two topologicalqubits. Capital letters “A” represents ancillia of QD, “H” isthe Hadamard gate, “C” and “T” represent the control andtarget qubit, respectively. The measurement “M” results of“A” together with the outcomes of the two joint parity mea-surements “P” determine which operation one has to apply onthe “C” and “T” qubit in order to complete the CNOT gate.The arrowed line in the bottom represents the sequence of theprocess.

flux qubit splitting energy Δ is the same for combinedtopological-QD qubit states with equal joint parity [20].Thus, measurement of the flux qubit splitting energy isequivalent to a joint parity measurement to the states ofQD and a pair of MF. In reference [20], quantum statetransfer between topological and QD qubits is achieved bystandard quantum teleportation: the proposed parity mea-surements supplied with Hadmard gates are equivalent toBell state measurements. Here, our scheme for CNOT gateis a measurement based one.

We next proceed to implement a CNOT gate betweentwo topological qubits with the help of QD as an ancilla.Here, we propose a measurement-based CNOT gate op-eration [24]. The relevant operations are single-qubit ro-tations, single-QD rotations/measurements, and effectivejoint parity measurements for QD and a topological qubit.The circuit for the CNOT gate is depicted in Figure 2. Theauxiliary QD is initially prepared in the state of |0〉A. Aftera Hadamard gate on the QD, the first joint parity mea-surement P1 in Figure 2 is implemented on the QD and apair of MF from “C” qubit. After Hadamard rotation ofthe QD and the target qubit, the second parity measure-ment P2 is implemented on the QD and a pair of MF in the“T” qubit. Then, we rotate back the QD and the targetqubit state by Hadamard gate. The last step is the mea-surement of the QD in the {|0〉, |1〉} basis. The two paritymeasurement results, together with the measurement re-sult of the QD determine which single-qubit gates to beoperated on the control and target qubits to generate aCNOT gate. The relationship between the measurementresults and the gates to be operated is summarized in theTable 1. After completing the required gates on the cor-responding qubits, it is straightforward to check that theprocess is a CNOT gate operation between the two qubits.

Here, we want to emphasize that our implementa-tion is different from that of reference [16], which isalone the line presented in reference [27]. In the pro-posal [16,27], they use the parity measurement of two MFqubit, to entangle them, and braiding operations of MFfrom qubit and ancilla to implement a CNOT gate. Whilewe use joint MF-QD parity measurements to constructthe measurement-based CNOT gate, where only braidingoperations of MF within topological qubits are needed. In

Page 4 of 4 Eur. Phys. J. D (2013) 67: 89

Table 1. Correspondence between the measurement resultsand the gates operated on the control and target qubits. “0”and “1” represent odd and even parity, respectively.

“P1” “P2” Result of “M” Gate on “C” Gate on “T”1 1 |0〉A I I1 1 |1〉A I X1 0 |0〉A Z I1 0 |1〉A Z X0 1 |0〉A I X0 1 |1〉A I I0 0 |0〉A Z X0 0 |1〉A Z I

the proposal [16,27], only one parity measurement of twoMF qubit is needed, thus it is more efficient in terms ofcomplexity. But, it requires long range braiding operationsbetween qubit and the ancilla [27], which is experimen-tally challenging. As for our introducing of a QD qubit toserve as the ancilla, it breaks the topological protect ofthe proposal in [27] in some sense. But, our joint MF-QDparity measurement is based on the Aharonov-Casher ef-fect, which is also of topological nature and insensitive tolocal noises.

Before concluding, we want to further emphasize thatour scheme have the following distinct merits. (1) As inreference [20], one can host both MF and QD using a sin-gle InAs nanowires [25], which may thus reduce the tech-nical challenges of implementation. (2) Here, QD servesas ancillary qubit. Therefore, only braiding between MFwithin a topological qubit is needed, noting that braidingof two MF from different topological qubits will be ex-perimentally challenging. (3) Measurement of topologicalqubit uses the Aharonov-Casher effect [26], which is non-destructive. (4) Parity measurement on QD and a certaintopological qubit can be achieved by moving the topolog-ical qubit into the measurement circuit by local tunablegate on quantum wire [16].

In summary, a measurement based two-qubit con-trolled not gate is produced with the help of joint par-ity measurements and followed by prescribed single-qubitgates.

This work was supported by the NSFC (No. 11004065), theNFRPC (No. 2013CB921804), the PCSIRT, and the NSF ofGuangdong Province.

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