Majorana Fermion Surface Code for Universal Quantum Computation

Sagar Vijay, Timothy H. Hsieh, and Liang FuDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 8 April 2015; published 10 December 2015)

We introduce an exactly solvable model of interacting Majorana fermions realizing Z2 topological orderwith a Z2 fermion parity grading and lattice symmetries permuting the three fundamental anyon types. Wepropose a concrete physical realization by utilizing quantum phase slips in an array of Josephson-coupledmesoscopic topological superconductors, which can be implemented in a wide range of solid-state systems,including topological insulators, nanowires, or two-dimensional electron gases, proximitized by s-wavesuperconductors. Our model finds a natural application as a Majorana fermion surface code for universalquantum computation, with a single-step stabilizer measurement requiring no physical ancilla qubits,increased error tolerance, and simpler logical gates than a surface code with bosonic physical qubits.We thoroughly discuss protocols for stabilizer measurements, encoding and manipulating logicalqubits, and gate implementations.

DOI: 10.1103/PhysRevX.5.041038 Subject Areas: Condensed Matter Physics,Quantum Information,Superconductivity

I. INTRODUCTION

As originally proposed by Ettore Majorana, theMajorana fermion is a particle that is its own antiparticle[1]. In the condensed-matter setting, Majorana fermionscan emerge in topological superconductors as a special typeof zero-energy quasiparticle that is bound to vortices ordefects and formed by an equal superposition of electronand hole excitations [2,3]. Theory predicts that Majoranafermions can be created in a wide array of spin-orbit-coupled materials in proximity to conventional super-conductors [4–8]. Recent observations of zero-energyconductance peaks in such systems [9–12] provide encour-aging hints of Majorana fermions [13,14].Majorana fermions in topological superconductors are of

great interest as they are predicted to produce exoticquantum phenomena such as the fractional Josephson effect[15–17] and electron teleportation [18]. Most remarkably,vortices or defects that carry Majorana fermions are pre-dicted to exhibit non-Abelian statistics [19–21], which haveyet to be observed in nature. In addition to its theoreticalsignificance, non-Abelian statistics provides the foundationfor topological quantum computation, in which logicalqubits are encoded in the topologically degenerate statesof non-Abelian anyons, and qubit operations are performedby braiding [22]. Topological quantum computation has thetheoretical advantage of being immune to errors caused bylocal perturbations [23]. Demonstrating the non-Abelian

statistics of Majorana fermions, however, requires braiding,fusing, and measuring the fusion outcome. This is achallenging task, as each of the above operations is yet tobe experimentally achieved. Furthermore, braidingMajoranafermions alone is insufficient to perform the necessary gateoperations for universal quantum computation.The “surface code” [24,25] provides an alternative

approach to universal quantum computation that usesmeasurements in an Abelian topological phase for gateoperations and error correction. In the surface code,measurements of nontrivial commuting operators (stabiliz-ers) are used to project onto a “code state,” and logicalqubits are effectively encoded in the anyon charge of aregion by ceasing certain stabilizer measurements [26–29].The logical gates necessary for universal quantum compu-tation are realized through sequences of measurements usedto move and braid the logical qubits. An advantage of thesurface code architecture is its remarkable ability for errordetection and subsequent correction during qubit readout,as the nucleation of anyons through the action of a randomoperator can be reliably tracked through stabilizer mea-surements. For a sufficiently low error rate per physicalqubit measurement, scaling the size of the surface codeproduces an exponential suppression in propagated errors[30]. Remarkably, recent experiments with superconduct-ing quantum circuits have demonstrated the ability toperform high-fidelity physical gate operations and reliableerror correction for a surface code of small size [31–33].In this work, we introduce a new scheme for surface code

quantum computation that uses Majorana fermions as thefundamental physical degrees of freedom and exploits theirunique properties for encoding and manipulating logicalqubits. Our surface code is based on a novel Z2 topological

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order with fermion parity grading (defined below), whichwe demonstrate in a class of exactly solvable Hamiltoniansof interacting Majorana fermions, termed Majorana pla-quette models. We demonstrate that charging energy-induced quantum phase slips in superconducting arrayswith Majorana fermions generates the required multifer-mion plaquette interactions, providing a physical realiza-tion of our model. We then describe a detailed physicalimplementation of the “Majorana fermion surface code,”including physical qubit and stabilizer measurements, thecreation of logical qubits, error correction, and logical gateoperations required for universal quantum computation.The Majorana fermion surface code poses significant

benefits over a surface code with bosonic physical qubits.First, stabilizer measurements in the Majorana surface codecan be performed in a single step, whereas this requiresseveral physical gate operations in the conventional surfacecode [28,29]. As a result, we anticipate that the Majoranasurface code has a significantly higher error tolerance.Furthermore, our Majorana surface code operates withsubstantially less overhead, as it requires fewer physicalqubits per encoded logical qubit and uses no physicalancilla qubits. Second, we may tune the energy gap foranyon excitations in our physical realization of theMajorana plaquette Hamiltonian, increasing error suppres-sion in the Majorana fermion surface code. Finally, thelattice symmetries in the Majorana plaquette model per-mute the three fundamental anyon types, allowing for amuch simpler implementation of the logical Hadamardgate. As we will show, these advantages arise from theunique approach taken by our Majorana fermion surfacecode and the use of Majorana fermions as the fundamentaldegrees of freedom. The fact that a Majorana fermion is“half” of an ordinary fermion—so that a pair is required toencode a qubit of information—is crucial to the Majoranafermion surface code. We emphasize that the non-Abelianstatistics of Majorana-carrying vortices or defects is of norelevance to our proposal, as our code does not involvebraiding them.This paper is organized as follows. First, we introduce a

solvable model of interacting Majorana fermions on thehoneycomb lattice realizing a novel Z2 topological orderwith a Z2 fermion parity grading and an exact S3 anyonsymmetry. We propose a physical realization of this modelthat uses the charging energy in an array of mesoscopicsuperconductors [18] to implement the required nonlocalinteractions between multiple Majorana fermions. Next, wedemonstrate that our model provides a natural setting forthe Majorana fermion surface code, in which a logical qubitis encoded in a set of physical qubits formed fromMajorana fermions. We present a physical implementationof the Majorana surface code and propose detailed proto-cols for performing gate operations for universal quantumcomputation.

II. MAJORANA PLAQUETTE MODEL

We begin by considering a honeycomb lattice with oneMajorana fermion (γ) on each lattice site; the Majoranafermions satisfy canonical anticommutation relationsfγn; γmg ¼ 2δnm. The Hamiltonian is defined as the sumof operators acting on each hexagonal plaquette:

H ¼ −uXp

Op; Op ≡ iY

n∈vertexðpÞγn: ð1Þ

We note that this model was mentioned in a work byBravyi, Terhal, and Leemhuis [34], although its noveltopological order and anyon excitations were not studiedthere; a closely related model in the same topological phasewas introduced and studied by Xu and Fu [35].It suffices to consider u > 0 in the Hamiltonian (1), as

the case with u < 0 can be mapped to u > 0 by changingthe sign of the Majorana fermions on one sublattice.The operator Op is the product of the six Majoranafermions on the vertices of plaquette p as shown inFig. 1(a). Since any two plaquettes on the honeycomblattice share an even number of vertices, all of theplaquette operators commute, and the ground-state jΨ0iis defined by the condition

OpjΨ0i ¼ jΨ0i; ð2Þ

for all plaquettes p. We note that, quite generally,Hamiltonians of interacting Majorana fermions withcommuting terms may be realized on any lattice, so longas any pair of operators in the Hamiltonian only hasoverlapping support over an even number of Majoranafermions.We demonstrate that the above Majorana plaquette

model (1) realizes a Z2 topological order of Fermi systemsby considering the ground-state degeneracy and elementaryexcitations. First, we place the system on a torus by

FIG. 1. We consider a honeycomb lattice with (a) a singleMajorana fermion on each lattice site, so the Op operator is theproduct of the six Majorana fermions on the vertices of ahexagonal plaquette. The colored plaquettes in (b) correspondto the three distinct bosonic excitations that may be obtained byviolating a plaquette constraint.

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imposing periodic boundary conditions and find a four-folddegenerate ground state by counting the number of degreesof freedom and constraints on the full Hilbert space. For anN-site honeycomb lattice, the 2N=2-dimensional Hilbertspace of Majorana fermions is constrained by the fixed totalfermion parity:

Γ≡ iN=2Yn

γn: ð3Þ

For convenience, we choose a unit cell for the honeycomblattice consisting of three plaquettes labeled A, B, and C,as shown in Fig. 1(b). We observe that on the torus, theproduct of plaquette operators on each of the A-, B-, andC-type plaquettes is equal to the total fermion parity:

Γ ¼Yp∈A

Op ¼Yp∈B

Op ¼Yp∈C

Op: ð4Þ

The operators fOpg on any one type of plaquette fixone-third of the plaquette eigenvalues via the condition (2)and impose 2N=6−1 constraints on the Hilbert space. Thenumber of unconstrained degrees of freedom is thereforegiven by

D ¼ 2ðN=2Þ−1=ð2ðN=6Þ−1Þ3 ¼ 4; ð5Þ

which yields a four-fold ground-state degeneracy for theMajorana plaquette model on the torus.The ground-state degeneracy is of a topological nature,

as the four ground states are distinguished only by nonlocaloperators. To see this, we construct a Wilson loop operatorWl, defined as a product of Majorana bilinears on anoncontractible loop l on the torus:

Wl ≡Y

n;m∈lðiγnγmÞ; ð6Þ

such that W2l ¼ 1 so that the Wilson loop has eigenvalues

�1. Consider the operators Wx and Wy on the twonontrivial cycles of the torus lx and ly, as shown inFig. 2. Since lx and ly traverse an even number ofvertices over any plaquette and do not contain any commonlattice sites, we have ½Wx;Wy� ¼ ½Wx;H� ¼ ½Wy;H� ¼ 0.Furthermore, we may construct Wilson loop operators W ~x

andW ~y on loops ~lx and ~ly, where ~lx is shifted from lx by a

basis vector parallel to ly and likewise for ~ly, such thatfW ~x;Wyg ¼ fW ~y;Wxg ¼ 0. As before, W ~x and W ~y com-mute with each other and with the Hamiltonian. Therefore,the four degenerate ground states may be distinguished bytheir eigenvalues under Wx and Wy, with W ~x and W ~y

transforming the ground states between distinct sectors. Inanalogy with conventional Z2 gauge theory, we mayidentify the Wilson loop operators Wx;y with electric

charges traversing the torus in two different directions,and W ~x;~y as magnetic fluxes on a dual lattice.Gapped excitations above the ground state are obtained

by flipping the eigenvalue of Op from þ1 to −1 on one ormore plaquettes. Since the total fermion parity is fixedand equal to the product of all plaquette operators of eachtype, plaquette eigenvalues can only be flipped on pairs ofplaquettes of the same type. This is achieved by stringoperators of the form (6), now acting on open pathsand anticommuting with the plaquette operators at thetwo ends of the path, thereby creating a pair of anyonexcitations.An important feature of our Majorana plaquette model,

the conservation of total fermion parity—a universalproperty of Fermi systems—makes it impossible to createor annihilate two excitations living on different types ofplaquettes or to change one type of plaquette excitation intoanother. As a result, there are three distinct elementaryplaquette excitations, labeled A, B, and C, by plaquettetype. To determine their statistics, we braid these excita-tions by acting with Majorana hopping operators iγnγm onlattice bonds [36]. We find that all three types of plaquetteexcitations have boson self-statistics and mutual semionstatistics, i.e., braiding two distinct plaquette excitationsgenerates a quantized Berry phase of π. From the elemen-tary plaquette excitations, we may build composite exci-tations AB, BC, AC, and ABC by flipping the eigenvaluesof the Op’s on two or three adjacent plaquettes. Amongthese, the composite excitation ABC is simply a physicalMajorana fermion since the Majorana operator γn acting ona lattice site flips the eigenvalues of the Op’s on the threesurrounding A, B, and C plaquettes. In contrast, thecomposite excitations AB, BC, and AC are anyons, withfermion self-statistics and mutual semion statistics with theelementary excitations. We call these excitations compositeMajorana fermions, as they are created by a string ofphysical Majorana fermions. A summary of the braidingstatistics for all anyons in our Majorana plaquette model isgiven in the following table:

FIG. 2. The action of the commuting Wilson loop operatorsWxand Wy is shown above as the product of the Majorana fermionson the lattice sites intersected by the appropriate colored lines.The operator W ~y anticommutes with Wx and takes the groundstate between two topological sectors.

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1 A B C AB BC AC ABC

1 þ1 þ1 þ1 þ1 þ1 þ1 þ1 þ1A þ1 þ1 −1 −1 −1 þ1 −1 þ1B þ1 −1 þ1 −1 −1 −1 þ1 þ1C þ1 −1 −1 þ1 þ1 −1 −1 þ1AB þ1 −1 −1 þ1 −1 −1 −1 þ1BC þ1 þ1 −1 −1 −1 −1 −1 þ1AC þ1 −1 þ1 −1 −1 −1 −1 þ1ABC þ1 þ1 þ1 þ1 þ1 þ1 þ1 −1Strange as it may seem, the existence of eight types of

quasiparticle excitations is a generic property of Z2 topo-logically ordered phases in Fermi systems, due to theconservation of fermion parity. Consider artificially dividingthe above quasiparticles into two groups: ð1; A; B; ABÞ andðABC;BC; AC;CÞ ¼ ABC × ð1; A; B; ABÞ. The former isequivalent to the four quasiparticles in Z2 gauge theorycoupled to a bosonic Ising matter field, as realized inKitaev’s toric code [22] or Wen’s plaquette model [37].The latter group of quasiparticles is obtained by attaching aphysical Majorana fermion to the former. The conservationof total fermion parity guarantees that the two groups ofquasiparticles cannot transform into each other in a closedsystem and thus have separate identities. We refer to thepresence of two groups of excitations with different fermionparity as a Z2 fermion parity grading.A remarkable property of the Majorana plaquette model

is that crystal symmetries of the honeycomb lattice permutethe three fundamental anyon excitations, A, B, and C, byinterchanging the three types of plaquettes. Examples ofsuch lattice symmetries include π=3 rotations about thecenter of a plaquette and translation by any primitive latticevector. These symmetries of the honeycomb lattice providea microscopic realization of the S3 anyon symmetry thatpermutes quasiparticle sectors, as recently studied in theformalism of topological field theory by considering thesymmetries of the K matrices of Abelian topologicalstates [38,39].

III. PHYSICAL REALIZATION

A. Physical platforms

In this section, we show that the Majorana plaquettemodel can be physically realized in an array of mesoscopictopological superconductors that are Josephson coupled. Awide range of material platforms for engineering a topo-logical superconductor have been proposed and are beingexperimentally studied [2,3]. As will be clear in thefollowing, the scheme we propose for realizing theMajorana plaquette model is independent of which plat-form is used. For the sake of concreteness, we use aplatform based on topological insulators in describing thegeneral scheme below and discuss other platforms based onnanowires and a two-dimensional electron gas with spin-orbit coupling in Sec. III C.

We place an array of hexagon-shaped s-wave super-conducting islands on a topological insulator (TI) to inducea superconducting proximity effect on the TI surface states.The Hamiltonian for this superconductor-TI hybrid systemis given by

H0 ¼Z

drð−ivÞψ†ðrÞð∂xsy − ∂ysx − μÞψðrÞ

þXj

Zdrj½Δeiφjψ†

↑ðrjÞψ†↓ðrjÞ þ H:c:�; ð7Þ

where ψ ¼ ðψ↑;ψ↓ÞT is a two-component fermion fieldand sx;y are spin Pauli matrices. The first term describes thepristine TI surface states, with a single spin-nondegenerateFermi surface and helical spin texture in momentum space.The second term describes the superconducting proximityeffect: rj belongs to the region underneath the jth super-conducting island, whose phase is denoted by φj.As found by Fu and Kane [4], a vortex or antivortex

trapped at a trijunction, where three islands meet, hosts asingle Majorana fermion zero mode. Let us consider settingup the phases of superconducting islands to realize an arrayof vortices and antivortices at trijunctions. For example, thephases can be set to φj ¼ 0, 2π=3, and −2π=3 on the A-,B-, and C-type islands, respectively, as shown in Fig. 3.This yields a two-dimensional (2D) array of Majoranafermions on a honeycomb lattice. In practice, the desiredphase configuration can be engineered by external elec-trical circuits [40] and/or magnetic flux. Alternatively,applying a perpendicular magnetic field generates a vortexlattice. These vortices may naturally sit at these trijunctionswhere the induced superconductivity is weak, leading to thedesired lattice of Majorana fermions.We take the size of the islands to be larger than the

coherence length of the superconducting TI surface states.Under this condition, Majorana fermions at different siteshave negligible wave-function overlap, preventing any

FIG. 3. Array of hexagonal s-wave superconducting islandsplaced on a TI surface. Each arrow points in the direction of therelative phase of the associated island, with φ ¼ 0, �2π=3. Thisproduces a honeycomb lattice of vortices (blue) and antivortices(red) at trijunctions, hosting Majorana fermions.

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unwanted direct coupling between them. (We note that evenweak couplings from wave-function overlap will not affectthe Z2 topological order of the Majorana plaquette modelbecause of its finite energy gap.) Nonetheless, as we showbelow, the charging energy of superconductors induces anonlocal interaction between the six Majorana fermions oneach island, providing the key ingredient of the Majoranaplaquette model.

B. Phase-slip-induced multifermion interactions

The important but subtle interplay between Majoranafermions and charging energy was first recognized by Fuand formulated for superconductors with a fixed number ofelectrons [18]. Later works have extended this to multiplesuperconductors connected by Josephson coupling andsingle-electron tunneling [35,41–43]. In all of these cases,the charging energy of a given superconductor inducesquantum phase slips φ → φ� 2π, from which theMajorana fermions in the superconductor acquire a minussign: γi → −γi. This property is due to the inherentlydouble-valued dependence of Majorana operators on thesuperconducting phase [18].In our setup for the Majorana plaquette model, the

charging energy of the superconducting islands exertseven more dramatic and interesting effects on theMajorana fermions at trijunctions, which have not beenpreviously studied. In the presence of a charging energy, thephase of each island becomes a quantum rotor. The kineticenergy of the rotor is provided by the charging energy Ec,which depends on the capacitance between an island andthe rest of the array and is described by the followingHamiltonian:

Hc ¼ 4Ec

Xj

ðnj − ngÞ2; ð8Þ

where nj ≡ ð−iÞ∂=∂φj is the Cooper-pair number operatorfor the jth island and ng is the offset charge, which can betuned by an externally applied electric field. The potentialenergy of the rotor is provided by the Josephson couplingEJ between adjacent superconducting islands, given by

HJ ¼ −EJ

Xhj;j0i

cosðφj − φj0 − ajj0 Þ; ð9Þ

where ajj0 ¼ φ0;j − φ0;j0 is externally set up such that theminimum of the Josephson energy corresponds to φj ¼φ0;j mod 2π, with φ0;j ¼ 0, 2π=3, and −2π=3 for the A, B,and C-type islands, respectively.Combining Eqs. (7), (8), and (9), the full Hamiltonian for

our setup, i.e., an array of superconducting islands on a TIsurface, is given by

H ¼ H0 þHc þHJ: ð10Þ

Wework in the regimeEJ ≫ Ec. Under this condition, low-energy states of the quantum rotor on a given island φjconsist of small-amplitude fluctuations around each poten-tial minimum φ0;j þ 2πm. Moreover, different minima areconnected by quantum phase slips, in which the phase φtunnels through a high-energy barrier to wind by 2πn, withn an integer. The small-amplitude phase fluctuationsaround a potential minimum correspond to a quantumharmonic oscillator and thus generate a set of energy levelsgiven by

ϵ0α ≈ ðαþ 1=2Þffiffiffiffiffiffiffiffiffiffiffiffiffi8EJEc

p; ð11Þ

with α ∈ N.On the other hand, quantum phase slips on a super-

conducting island strongly couple to the Majorana fermionsthat reside on the border with its neighbors, previouslyobtained by holding the phase fixed at φ0;j. In other words,Majorana fermions enter the low-energy effective theory ofEq. (10) via quantum phase slips induced by the smallcharging energy on each superconducting island. This newphysics makes our system different from a conventionalCooper-pair box. Remarkably, the action of a quantumphase slip involves Majorana fermions in a way thatdepends periodically on the phase winding number nmod 6. Consider, for example, phase slips at the centralsuperconducting island in Fig. 3. For n ¼ 1, a 2π phase slipφ ¼ 0 → 2π cyclically permutes the three Majorana fer-mions bound to vortices in the counterclockwise directionand the three Majorana fermions bound to antivortices inthe clockwise direction, i.e.,

φ¼ 0→ 2π∶ γ1 → γ3; γ3 → γ5; γ5 → −γ1γ2 → −γ6; γ4 → γ2; γ6 → γ4; ð12Þ

as shown in Fig. 4, where i ¼ 1;…; 6 labels the sixMajorana fermions at vertices of this island in clockwiseorder. The physical movement of Majorana fermionsinduced by phase slips is a unique and attractive advantageof our setup, compared to other setups in which thepositions of Majorana fermions are fixed [35,41–43]. Onthe other hand, for n ¼ 3, a 6π phase slip takes eachMajorana fermion over a full circle and back to its originalposition, from which it acquires a minus sign [4], i.e.,

φ ¼ 0 → 6π∶ γi → −γi: ð13Þ

Only for n ¼ 6 does each Majorana fermion come back toits original position unchanged.We now add up the contributions of various phase slips

to derive an effective Hamiltonian for Majorana fermions asa function of the offset charge ng for each state of theharmonic oscillator:

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HαðngÞ ¼ ϵ0α þX6n¼1

ðtα;nUnei2πnng þ H:c:Þ: ð14Þ

Here, ϵ0α is the quantized energy of the harmonic oscillatorgiven by Eq. (11), which is the same for all internal states ofthe Majorana fermions. The second term describes quan-tum phase slips: tα;n denotes the amplitude of the αthenergy level of the harmonic oscillator tunneling betweentwo potential minima that differ by 2πn, while Un is theunitary operator acting on the Majorana fermions due to a2πn phase slip. The coupling tα;n depends on the energybarrier in the phase-slip event and can be modulated by

tuning Ec=EJ; for example, tα;1 ∝ e−ffiffiffiffiffiffiffiffiffiffiffi8EJ=Ec

p[44]. The

offset charge ng provides an Aharonov-Bohm flux propor-tional to the winding number n.The Hamiltonian (14) is analogous to the Bloch

Hamiltonian that describes the band structure of a particlehopping in a one-dimensional periodic potential, with theoffset charge ng playing the role of crystal momentum.Importantly, the phase particle carries internal degrees offreedom resulting from Majorana fermions γ1;…; γ6 thatare unique to our system. A phase slip that moves the phaseparticle to a different potential minimum also permutes theMajorana fermions as shown in Eqs. (12) and (13), similarto a spinful particle hopping in the presence of a non-Abelian gauge field. These permutations are represented bythe unitary operators Un in the effective Hamiltonian (14)acting on Majorana fermions. For example, the operator U1

that generates the transformation (12) is given by

U1 ¼1þ γ2γ3ffiffiffi

2p 1þ γ4γ5ffiffiffi

2p 1 − γ6γ1ffiffiffi

2p

×1þ γ1γ2ffiffiffi

2p 1þ γ3γ4ffiffiffi

2p 1þ γ5γ6ffiffiffi

2p : ð15Þ

It follows from the addition of phase slips that Un ¼ ðU1Þn.In particular, the unitary operator U3, which takes γi to −γias shown in Eq. (13), has a simple form

U3 ¼ −Y6i¼1

γi ¼ iO; ð16Þ

where O is the plaquette operator defined in the Majoranaplaquette model (2). On the other hand, for n ¼ 1, 2, 4, or 5,Un is a sum of operators γiγj, γiγjγkγl, and iO.Substituting the expressions for the Un’s into Eq. (14),

we find that the effective Hamiltonian induced by the smallcharging energy of a single island takes the following form:

HαðngÞ ¼ ϵ0α þ ΔαðngÞOþ VαðngÞ; ð17Þ

with

ΔαðngÞ ¼X5m¼1

tα;m sinð2πmngÞ: ð18Þ

VαðngÞ includes a constant tα;6 cosð12πngÞ, as well asMajorana bilinear and quartic operators generated by phaseslips with winding number n ≠ 0 mod 3. Unlike O, theseoperators on neighboring islands do not commute.However, by appropriately tuning ng, the contribution ofquartic operators to the effective Hamiltonian may vanish,so the only remaining terms in the Hamiltonian will be thesix-Majorana interaction and Majorana bilinear terms. Thebilinear term receives no contribution from any �6πmphase slip, while ΔαðngÞ receives contributions from every�2πm phase-slip process. Therefore, for the remainder ofthis work, we assume that V can be treated as a perturbationto the Majorana plaquette model that does not destroy theZ2 topological order of the gapped phase. We note that an

FIG. 4. Schematic of a 2π phase slip on the central superconducting island in a hexagonal superconducting array on a TI surface, withthe phase of the central island indicated in each panel. When the phase difference between neighboring islands is π, the pair of Majoranafermions on the shared edges couple [4] as indicated. The 2π phase slip permutes the Majorana fermions as shown, leading to thetransformation in Eq. (12).

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alternative setup without the presence of V has beenpresented in a recent work [45].

C. Discussion

In deriving the effective Hamiltonian (14), we haveimplicitly assumed that Majorana fermions are the onlylow-lying excitations involved in phase-slip events, sepa-rated by an energy gap from other Andreev bound states inthe junctions between islands. This assumption is validbecause of the finite size of the islands, which leads to adiscrete Andreev bound-state spectrum with a finite gap forall values of the phase. The presence of this gap justifies ourderivation of the effective Hamiltonian (14) in a controlledmanner.Over the last few years, considerable experimental

progress has been made in hybrid TI-superconductingsystems. Proximity-induced superconductivity and super-currents have been observed in a number of TI materials[46–52]. Low-temperature scanning tunneling microscopy(STM) experiments have found a proximity-induced super-conducting gap on TI surface states, and the tunnelingspectrum of Abrikosov vortices shows a zero-bias con-ductance peak, which is robust in a range of a magneticfield and splits at higher fields [12]. This peak has beenattributed to the predicted Majorana fermion zero modes inthe vortex cores of superconducting TI surface states. Inview of these rapid, unabated advances, we regard thehybrid TI-superconductor system as a very promisingmaterial platform for realizing the Majorana plaquettemodel and studying the exciting physics of Majoranafermions enabled by quantum phase slips.Besides TIs, a two-dimensional electron gas (2DEG)

with spin-orbit coupling (such as InAs) can be driven into ahelical state with an odd number of spin-polarized Fermisurfaces by an external Zeeman field, which providesanother promising platform for realizing topological super-conductivity via the proximity effect [53,54]. In thistopological regime, vortices and trijunctions of a super-conducting 2DEG host a single Majorana fermion, similarto the TI surface. Thus, our proposed setup for theMajorana plaquette model in Sec. III A directly appliesto this system as well.In addition to TIs and 2DEG, (quasi-)one-dimensional

semiconductors and metals with strong spin-orbit couplinghave become a hotly pursued system to search for Majoranafermions [5–7]. Signatures of Majorana fermions werereported in 2012, based on the observation of a zero-biasconductance peak in hybrid nanowire-superconductor sys-tems [9,10]. One can envision a network of nanowires inproximity to Cooper-pair boxes to realize our Majoranaplaquette model. In this direction, it is worth noting that anew physical system—a nanowire with an epitaxiallygrown superconductor layer—has recently been introducedto study Andreev bound states in the presence of chargingenergy [55].

Many other physical systems for Majorana fermionshave been theoretically proposed and experimentally pur-sued; they are too numerous to list. Regardless of theparticular system, nonlocal interactions between multipleMajorana fermions emerge from the charging energy ofsuperconductors via quantum phase slips, and in theuniversal regime, such interactions are determined by thetransformation of Majorana fermions under phase slips, aswe have shown in Sec. III B.Finally, we note several previous works related to our

Majorana plaquette model and its physical realization. InRef. [35], Xu and Fu first introduced a model of interactingMajorana fermions that realizes Z2 topological order. Thismodel involves four-body and eight-body plaquette inter-actions on square and octagonal plaquettes in a two-dimensional lattice. Physical realizations of this modelwere proposed using an array of superconductor islands inproximity to either a 2D TI [35] or semiconductor nano-wires [56,57]. The four-body nonlocal interaction betweenMajorana fermions comes directly from the chargingenergy, whereas the eight-body interaction comes from ahigh-order ring-exchange process generated by single-electron tunneling between islands. In comparison, ourMajorana plaquette model on the honeycomb lattice has thetheoretical novelty of possessing an exact anyon permuta-tion symmetry and can be realized in a much simplermanner using an array of superconductors on a 3D TI withglobal phase coherence, with all the required interactionscoming directly from the charging energy. We also note arecent work on lattice models of Majorana fermions inAbrikosov vortices on a superconducting TI surface [58],which use different interactions and do not exhibittopological order.

IV. MAJORANA SURFACE CODE

In the rest of this work, we demonstrate that theMajorana plaquette model finds a natural application asa Majorana fermion surface code, on which universalquantum computation and error correction may be per-formed. The main idea of the surface code is to (i) useanyons of the Majorana plaquette model to encode logicalqubits, (ii) manipulate anyons to perform gate operationson logical qubits, and (iii) use commuting measurements ofthe Majorana plaquette operators for error correction. Wedescribe the detailed implementation of the Majoranasurface code, including the creation of logical qubits, errorcorrection, and protocols for logical gate operationsrequired for universal quantum computation.The surface code architecture [24,25,28] is a measure-

ment-based scheme for quantum computation. It usesprojective measurements of commuting operators—called“stabilizers”—acting on a 2D array of physical qubits toproduce a highly entangled “code state” jψi. Logical qubitsare created by stopping the measurement of certain com-muting operators to create “holes.” The different possible

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anyon charges at a hole are the degrees of freedom thatdefine a logical qubit. Logical gates are realized bymanipulating and braiding holes via a sequence ofmeasurements.A key advantage of the surface code is its remarkable

capability for error detection. The random measurement ofan operator in the surface code corresponds to nucleatingpairs of anyons, a process that can be reliably measured bytracking the eigenvalues of the commuting stabilizers.Reliable error detection hinges on having (i) a large numberof physical qubits for a given encoded logical qubit and(ii) a sufficiently low error rate for stabilizer measurements[28]. For the previously studied surface code with bosonicphysical qubits, it has been estimated [59,60] that below athreshold as high as 1% error rate per physical qubitoperation, scaling the size of the surface code permits anexponential suppression of errors propagated. This errortolerance makes the surface code architecture one of themost realistic approaches to practical, large-scale quantumcomputation.Recent practical realizations of the surface code have

used superconducting qubits coupled to a microwavetransmission line resonator to perform qubit manipulationsand measurements [31–33]. Here, a physical qubit isdefined by two energy levels arising from quantizationof phase fluctuations in a conventional Cooper-pair box.The surface code is implemented on a 2D array of physicalqubits with the four-qubit interactions of Kitaev’s toric codeHamiltonian [22] as the set of commuting stabilizers. Thefour-qubit stabilizer is measured by performing a sequenceof single- and two-qubit gates between the four physicalqubits and additional ancilla qubits [28]. Experiments havedemonstrated the remarkable ability to operate thesephysical gates with fidelity above the threshold requiredfor surface code error correction [31]. Recent experimentshave also used error detection to preserve entangled codestates on a surface code with a 9 × 1 [32] and a 2 × 2 [33]array of stabilizers. It remains to be shown that logicalqubits can be successfully encoded and manipulated vialogical gates in these surface code arrays.

A. Implementation

We implement the Majorana surface code on a 2D arrayof Majorana fermions by performing projective measure-ments of the Majorana plaquette operators fOpg, whichform a complete set of commuting stabilizers. For theremainder of this paper, we will use “plaquette operators”and “stabilizers” interchangeably to refer to fOpg. Apractical physical system for implementing the Majoranasurface code is the superconductor-TI hybrid systemintroduced in the previous section. We place an array ofsuperconducting islands on the TI surface, which arestrongly Josephson coupled. By introducing external cir-cuits or applying fluxes, we engineer the Josephsoncoupling between islands to achieve the phase

configuration in Fig. 3, leading to a honeycomb latticeof Majorana fermions at trijunctions.To perform a projective measurement of the Majorana

plaquette operator on a given island, i.e., a single stabilizer,we decrease the Josephson coupling of the island with therest of the array to activate quantum phase slips from thesmall but nonzero charging energy on this island. As shownby the effective Hamiltonian in Eq. (17), these quantumphase slips (partially) lift the degeneracy between states inthe eight-dimensional Fock space of the six Majoranafermions. In particular, for every energy level of theharmonic oscillator, there is an energy splitting ΔαðngÞbetween states of Majorana fermions with Γ ¼ þ1 (evenfermion parity) and with Γ ¼ −1 (odd fermion parity) fromEq. (18), where Γ is the stabilizer eigenvalue; this is shownschematically in Fig. 5(b). Therefore, the charging energyof the island creates an energy difference between differentstabilizer eigenstates. Furthermore, the energy gap betweenthe two lowest harmonic oscillator levels on the island is afunction of the stabilizer eigenvalue Γ ¼ �1, and in thelimit of negligible interaction V, it takes the followingform:

FIG. 5. (a) Schematic of the harmonic oscillator energy levels ofthe effective Hamiltonian (17), centered at φ ¼ 2πn, with the 2πand 4π phase-slip amplitudes for the lowest energy levels shown.In (b), we show a schematic plot of the two lowest harmonicoscillator levels as a function of the gate charge. The energysplittings Δ1 and Δ2 are between states with even (Γ ¼ þ1) andodd fermion parity (Γ ¼ −1) within the first and second harmonicoscillator levels, respectively. Each level within a fixed fermionparity sector is nearly four-fold degenerate.

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ΔEΓðngÞ ¼ ϵ0 þ ½Δ2ðngÞ − Δ1ðngÞ�Γþ � � � ; ð19Þ

where ϵ0 ≡ ϵ02 − ϵ01 ≈ffiffiffiffiffiffiffiffiffiffiffiffiffi8EJEc

p. The sensitivity of the

energy gap to the stabilizer eigenvalue now permits astabilizer measurement by simply measuring the energygap. By shining a probe microwave beam on this island, wemay measure the phase shift of the transmitted photons todetermine the gap between the two harmonic oscillatorlevels [61,62].We now perform these stabilizer measurements on all of

the superconducting islands to project onto an eigenstate ofthe Majorana plaquette Hamiltonian (1); this will be ourreference “code state.” We continue to perform measure-ments on all hexagonal islands in each cycle of the surfacecode in order to maintain the state. In subsequent cycles, wemay encode logical qubits into the code state and manipu-late the qubits via measurement. While projection onto thecode state and error correction in the surface code relyexclusively on measuring the six-Majorana plaquette inter-action, manipulation of logical qubits also requires meas-uring nearest-neighbor Majorana bilinears on thehexagonal lattice. This may be done by tuning the phaseof neighboring superconducting islands to bring the pair ofMajorana fermions on the shared edge sufficiently closetogether [4] so that the resulting wave-function overlapfurther splits the nearly four-fold degeneracy within asingle fermion parity sector, shown in Fig. 5(b). Again,the Majorana bilinear may be measured by shining a probebeam to measure the energy gap to the next harmonicoscillator level.Using the commuting six-Majorana operators in our

plaquette model to realize a surface code provides uniqueadvantages over the more conventional surface code withbosonic physical qubits. First, while a four-spin stabilizermeasurement in the usual surface code requires performing6–8 gates or measurements between a set of physical andancilla qubits [28,59], stabilizer eigenvalues in theMajorana surface code are obtained via a single-stepmeasurement by shining a probe beam. We emphasizethat even when measurement is not being performed, theintrinsic charging energy of the islands generates a finitegap Δ1ðngÞ to creating anyon excitations and naturallysuppresses errors at temperatures kBT < Δ1ðngÞ. We antici-pate that the corresponding error tolerance for scalablequantum computation is substantially improved for theMajorana surface code. Second, the Majorana surface codeoperates with lower overhead than its bosonic counterpart,using three-qubit stabilizers and requiring no ancilla qubits.Finally, the anyon transmutation required to perform alogical Hadamard gate in the conventional surface codecorresponds to a duality transformation that exchanges thestar and plaquette toric code operators. This operation isquite difficult to perform on a single logical qubit as it alsorequires lattice surgery to patch the transformed logicalqubit back into the remaining surface code [28,63].

As lattice symmetries permute anyon sectors in theMajorana plaquette model, anyon transmutation in theMajorana surface code corresponds to a lattice translationof the logical qubit, substantially simplifying the Hadamardgate implementation.

B. Logical qubits and error correction

Logical qubits may be encoded in the surface code byceasing the measurement of the plaquette operator on ahexagonal superconducting island in a surface code cycle,while continuing measurements on all other plaquettes. Intheory, we could stop measuring a single plaquette anddefine a two-level system, with the Z and X operators of thelogical qubit defined by the plaquette operator and aWilsonline connecting the plaquette to the boundary, respectively.A pair of such qubits on the A-type plaquettes is shown inFig. 6(a), where the solid and dashed lines correspond toproducts of Majorana fermions that define the indicatedlogical operators. The two qubits shown may also becoherently manipulated by acting with the Wilson lineoperator connecting the two plaquettes, denoted X12.In practice, however, it is difficult to manipulate qubits

with an operator that connects to a distant boundary, so it issimpler to encode a logical qubit by stopping the stabilizermeasurement on two well-separated plaquettes of the sametype. We choose to only manipulate 2 of the 4 resultingdegrees of freedom by defining Z≡Op and X ≡Wpq, theWilson line operator connecting the two plaquettes. We usethe opposite convention to define the logical Z and Xoperators for a qubit on the adjacent B plaquettes; anexample of such logical qubits is shown in Fig. 6(b). Wenote that when such a qubit is created, it is automaticallyinitialized to an eigenstate of the plaquette operator, withthe eigenvalue given by the measurement performed in theprevious surface code cycle. As a result, logical qubits oftype A (B) are initialized to an eigenstate of the Z (X)logical operator.To reduce errors during qubit manipulation, we may

define a qubit by ceasing measurement of multiple adjacentplaquettes as shown in Fig. 6(b). In this particular case, thelogical operator X is still a Wilson line connecting toanother set of distant holes. However, the logical Z isdefined as Z≡Op ⊗ Oq ⊗ Or. For the remainder of ourdiscussion, we consider logical qubits with only a singleplaquette operator used to define the logical Z; thegeneralization to larger qubits is straightforward.Errors may occur during qubit manipulation, including

(1) single-qubit errors due to the unintended measurementof a local operator involving an even number of Majoranafermions and (2) measurement errors. Single-qubit errorcorrection may be performed on logical qubits by con-stantly measuring the remaining plaquette eigenvaluesduring surface code cycles. Since only pairs of plaquettesmay be flipped simultaneously by a random measurement,corresponding to the nucleation of a pair of anyons of a

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single type, detecting the change of an odd number ofplaquette eigenvalues in a single surface code cycle willgenerally signal the presence of a random measurementperformed on a nearby logical qubit. More precisely, whena stabilizer eigenvalue changes in a surface code cycle, it isefficient to store the location of that stabilizer and waitseveral code cycles, accumulating a spacetime diagram ofstabilizer errors as additional errors occur [30,59,60]. Aftersufficiently many code cycles, the spacetime diagram maybe used to determine the most likely configuration ofWilson lines that could have generated those errors [28,29]using a minimum-weight perfect matching algorithm[64,65]. Errors may be subsequently corrected by softwarewhen performing logical qubit manipulations and readouts[28]. Random measurement errors involve incorrectly

registering the eigenvalue of a plaquette operator; theseare naturally corrected by performing multiple surface codecycles to verify the accuracy of a measurement.

C. Logical gate implementations

The Majorana surface code may be used for universalquantum computation by implementing CNOT, T, andHadamard gates on logical qubits; this has been extensivelystudied in the context of the surface code architecture withunderlying bosonic degrees of freedom [28,63]. Here, wedescribe the implementations of these gates in our reali-zation of quantum computation with a Majorana surfacecode. Our gate implementations follow the spirit of theimplementations presented in Ref. [28].All gates in the Majorana surface code are implemented

on logical qubits via a sequence of measurements. Let U bethe desired unitary we wish to perform on the quantumstate of several logical qubits, defined by the logicaloperators fXig and fZig. It is convenient to keep trackof the transformation of the logical state by monitoringthe transformation of logical operators Xi → UXiU

†,Zi → UZiU

†. In practice, performing the appropriatesequence of measurements will yield the transformationW, such that

UXiU† ¼ �WXiW

†; ð20Þ

UZiU† ¼ �WZiW

†; ð21Þ

where the signs depend on the outcomes of the specificmeasurements performed. These measurement outcomesare stored in software and used to correctly interpret thereadout of a logical qubit.In what follows, we often demonstrate our gate imple-

mentations in an “operator picture,” where a set ofoperators in the surface code o1;…; on and p1;…; pmwith eigenvalues �1 are measured in an appropriatesequence. This implements a logical gate via the desiredtransformations:

Z → Z ⊗Ynj¼1

oj ¼ U Z U†; ð22Þ

X → X ⊗Ymj¼1

pj ¼ U X U†: ð23Þ

In practice, the measured outcomes for the foig and fpjgoperators will be stored by software and used to obtain theabove transformations during logical qubit readout.CNOT gate.—A CNOT gate takes two qubits, a

“control” and a “target,” and flips the value of the targetqubit based on the value of the control; it then returns the

FIG. 6. Logical qubits in the Majorana surface code. In (a) westop the measurement of two plaquette operators in subsequentsurface code cycles, increasing the ground-state degeneracy by afactor of 4. If we take Z1 and Z2 to be the logical Z operators forthe two encoded qubits, the corresponding X1 and X2 operatorsare given by Wilson lines connecting to the boundary. The twoqubits may be coherently manipulated by applying the operatorX12 as shown. In practice, it is simpler to define logical qubits bystopping the measurement of pairs of plaquettes of a single type,with the logical X and Z defined as shown in (b). We may alsoconsider a logical qubit made of several “holes,” as in (c), tominimize errors during qubit manipulation.

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control unchanged. The action of a CNOT takes thefollowing form in the basis of two-qubit states:

C ¼

0BBB@

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

1CCCA: ð24Þ

A CNOT gate may be implemented by braiding logicalqubits in the Majorana surface code. In the simplest case, aCNOT between two logical qubits of different types isimplemented through a single braiding operation thatproduces an overall sign if the hexagonal ends of bothqubits contain an anyon, due to the π mutual statistics,demonstrated in Sec. II. In the following section, we firstdescribe the procedure to move a logical qubit along agiven type of plaquette before discussing the braidingprocedure required to produce a CNOT gate.Consider the A-type logical qubit shown in Fig. 7. To

move the qubit one unit to the right, we perform thefollowing sequence of measurements. We begin by multi-plying the Z logical operator by the eigenvalue of theadjacent r plaquette operator to perform the transformation

Z → Z0 ≡ Z ⊗ Or: ð25Þ

As the r plaquette is being continuously measured, itseigenvalue Or ¼ �1 is known from the previous surfacecode cycle. In the next cycle, we stop measuring Or andmeasure the Majorana bilinear iγη. We then multiply the Xoperator by the measurement outcome, affecting thetransformation

X → X0 ≡ X ⊗ iγη: ð26Þ

In the final surface code cycle, we begin measuring theoriginal Z stabilizer and continue to include the measure-ment of the Z stabilizer in all subsequent surface codecycles. Furthermore, we redefine the logical operator Z0 as

Z0 → Z00 ≡ Z0 ⊗ Z: ð27Þ

The initial qubit configuration and final outcome aredepicted schematically in Fig. 7. This sequence of mea-surements has shifted the A-type qubit by moving itshexagonal end one unit to the right, and it may generallybe used to move an A- or a B-type logical qubit within the Aor B plaquettes, respectively.We may now braid pairs of logical qubits to perform a

CNOT gate in the Majorana surface code. The simplestCNOT that we may realize is between two distinct types ofqubits, taking the A qubit as the control, as shown in Fig. 8.Since the qubits are distinct, braiding the B-type qubit—with logical operators XB and ZB—along a closed path l,

enclosing the second qubit, (i) multiplies the Wilson line ofthe B-type qubit by the anyon charge enclosed by l and(ii) multiplies the Wilson line of the A qubit by the anyoncharge of the B qubit. This results in the transformation

XA → XA ⊗ XB; ZB → ZA ⊗ ZB ⊗Yp∈l

Op; ð28Þ

where fOpg are A- andC-type plaquette operators enclosedby the braiding trajectory, as shown in Fig. 8. Since theeigenvalues of the enclosed plaquette operators are knownfrom the previous surface code cycle, we may implementthe logical CNOT (ZA → ZA, ZB → ZA ⊗ ZB) by multi-plying the transformed ZB by an appropriate sign. Insummary, the simplest braiding process between an Aand a B logical qubit implements a CNOT on the B qubit,with the A qubit as the control.A CNOT between two logical qubits of the same type

may also be performed by appropriately braiding pairs ofdistinct types of logical qubits. In this case, we take onequbit as the control by convention and store the outcome ofthe CNOT gate in a third ancilla qubit. First, considerperforming a CNOT gate on two A-type qubits. To imple-ment the CNOT, we prepare two additional ancilla qubits;the first is an A qubit prepared in the state jφi≡½jþzi þ j−zi�=

ffiffiffi2

p, while the second is a B qubit prepared

in the state jþxi, with j�zi and j�xi the eigenstates of thelogical Z and X operators, respectively. Both ancilla qubitsare prepared by measuring a þ1 eigenvalue for the Wilsonline joining the pair of plaquettes of the appropriate qubit.For the A (B) qubit, this projects onto an eigenstate of the

FIG. 7. We may move a logical qubit defined by X and Zoperators along a given sublattice. We first multiply the logical Zby Or. After measuring iγη in the next code cycle, we extend thelogical X → X ⊗ iγη. Finally, we begin measuring Z in the nextsurface code cycle and restore Op and Oq to six-Majoranaoperators.

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logical X (Z) operator and produces the desired ancillastates.We now represent a complete basis of the four-qubit

states as jzB; zc; zt; zouti, referring to the eigenvalues of thelogical Z operators of the B ancilla, the control, the target,and the ancilla A qubits, respectively. We start out with aninitial state jψ initi of the form

jψ initi≡ 1ffiffiffi2

p ½jþ; zc; zt;þi þ jþ; zc; zt;−i�: ð29Þ

Next, we braid the B ancilla qubit around all threeremaining qubits as shown in Fig. 9(a). Up to an overallsign determined by the eigenvalues of plaquette operatorsenclosed by the braiding trajectory that are known fromprevious surface code cycles, this braiding implements thetransformation ZB → ZB ⊗ Zc ⊗ Zt ⊗ Zout on the logical

Z of the ancilla qubit, where Zc, Zt, and Zout are the logicalZ operators for the control, target, and ancilla A-typequbits, respectively. The final state we obtain is then ofthe form

jψ finali ¼1ffiffiffi2

p ½jzczt; zc; zt;þi þ j−zczt; zc; zt;−i�: ð30Þ

This braiding process is convenient, as a measurement ofthe state of the B qubit can determine whether the state ofthe A ancilla contains the correct outcome of the CNOToperation. If we now measure the logical Z of the B qubitand obtain ZB ¼ þ1, then we project onto a state withzczt ¼ zout. In this case, the A ancilla qubit contains thecorrect outcome of the CNOT between the other A qubits. IfZB ¼ −1, however, then zczt ¼ −zout and the A ancillacontains the opposite of the correct CNOT outcome. In thiscase, we may act with Xout on the A ancilla qubit in thesurface code software [28] to obtain the desired final state.

FIG. 9. Braiding processes that implement the transformation(a) Za → Za ⊗ Zc ⊗ Zt ⊗ Zout up to an overall sign, as deter-mined by the product of the appropriate plaquette operatorsenclosed by the path l, and (b) Zout→ Zout⊗ ZA, Zt→ Zt⊗ ZA,Zc → Zc ⊗ ZA. The two braids are used to realize CNOT gatesbetween two (a) A-type and (b) B-type logical qubits, respec-tively. By convention, we take the lowest qubit enclosed by thebraiding trajectory to be the control for the logical CNOT.

FIG. 8. CNOT gate. Braiding two logical qubits to perform alogical CNOT. In (a), a possible trajectory for braiding the firstqubit around the second is indicated by the dotted line. Sincethe two qubits live on distinct sublattices, the braiding procedureinduces the transformation XA → XA ⊗ XB and ZB →ZA ⊗ ZB ⊗ O, where O is the product of the colored plaquettesshown. As a result, this operation performs a CNOT trans-formation on the braided qubit.

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A similar process may be used to perform logicalCNOT’s between two B qubits; now, we prepare an Aancilla qubit and a B ancilla qubit in the states shown inFig. 9(b). After braiding the ancilla A qubit around thecontrol, target, and ancilla B qubits, if we measureXA ¼ þ1, then the B ancilla contains the desired outcomeof the CNOT operation. Again, by convention, we take thecontrol qubit to be the first one enclosed by the braidingtrajectory, as shown in Fig. 9(b).Hadamard gate.—The Hadamard gate is a single-qubit

gate taking the matrix form

H ¼ 1ffiffiffi2

p�1 1

1 −1�: ð31Þ

The action of a Hadamard gate is to exchange the logical Xand Z operators so that H X H† ¼ Z and H Z H† ¼ X. Asthe logical X and Z are defined oppositely on differenttypes of qubits, a Hadamard operation in the bosonicsurface code corresponds to an electric-magnetic dualitytransformation that interchanges star and plaquette oper-ators in the toric code. In the ordinary surface code, such atransformation is quite difficult to implement, requiring aseries of Hadamards on physical qubits enclosing thelogical qubit so as to interchange the X and Z stabilizers,followed by physical swap gates in order to correctly patchthe transformed logical qubit back into the remainingsurface code array [63]. As lattice symmetries permutethe anyons in the Majorana plaquette model, however, thelogical Hadamard gate may be realized in the Majoranasurface code by simply moving a logical qubit betweendistinct plaquettes.We implement the logical Hadamard by the procedure

shown in Fig. 10. Consider an A-type logical qubit. Wemultiply the logical X operator of the qubit, defined by theWilson line in Fig. 10(a), by the product of the adjacentplaquette operators fOμkg extending between the hexago-nal ends of the qubit. The eigenvalues of these plaquetteoperators are known from previous surface code cycles.This operation implements the transformation

X → Z0 ≡ X ⊗Yk

Oμk : ð32Þ

At the same time, we multiply the logical Z by the adjacentplaquette operator Op, shown in Fig. 10(a), that borders thelogical qubit above:

Z → X0 ≡ Z ⊗ Op: ð33Þ

In subsequent surface code cycles, we stop measuring theeigenvalue of Op. We implement a similar transformationon the other hexagonal end of the logical qubit by stoppingthe measurement of the plaquette operator above theother qubit hole. The end result, after performing these

operations, is shown in Fig. 10(b). The solid and dashedblue lines indicate the products of the Majorana fermionson the appropriate sites that define the X0 and Z0 operators,respectively.In the next surface code cycle, we measure the product

ðiη1η2Þðiη3η4Þ… of the Majorana fermions along the lower“string” that defines the Z0 operator; this measurementcommutes with X0 since the two operators do not overlap,as shown in Fig. 10(c). Afterwards, we measure Oq, as well

FIG. 10. Hadamard gate. A logical Hadamard is performed bytransferring a qubit between distinct sublattices so that the logicalX and Z operators are exchanged. We do this by taking the qubitin (a) and multiplying the logical X by the plaquette operatorsfOμkg and the logical Z by Op and ceasing measurement of thefermion parity of plaquette p, yielding the operators shown in (b).Next, we measure the product ðiη1η2Þðiη3η4Þ… and Oq andmultiply by Z0 and X0, respectively. The final result is shownin (d).

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as the plaquette operator Oh for the other hole of theoriginal logical qubit. Then, we may perform the followingtransformations on the logical X0 and Z0 operators:

X0 → X00 ≡ X0 ⊗ Oq; ð34Þ

Z0 → Z00 ≡ Z0 ⊗Yl

ðiη2l−1η2lÞ: ð35Þ

This yields the logical qubit shown in Fig. 10(d). Insubsequent surface code cycles, we continue measuringthe eigenvalues of Oq and Oh. Since the logical Z and Xoperators are defined differently on the A and B-typeplaquettes, our procedure for transforming our A qubitinto a B qubit implements a logical Hadamard gate. Anidentical protocol may be used to perform a Hadamard gateon a logical B qubit.S and T gates.—Finally, we implement the logical S and

T gates, described by the following single-qubit operations:

S ¼�1 0

0 i

�; T ¼

�1 0

0 eiπ=4

�: ð36Þ

As demonstrated in Ref. [28], it is possible to realize thesegates by performing a series of logical Hadamard andCNOT gates between the logical qubit and an appropriatelogical ancilla qubit. Here, we first discuss the S- andT-gate implementations, given the appropriate ancillaqubit, before outlining a procedure for creating theselogical ancillas in the surface code.To implement an S gate, we prepare a logical ancilla in

the state

jφSi≡ 1ffiffiffi2

p ½jþzi þ ij−zi�: ð37Þ

Then, if jΨi is the state of the logical qubit of interest, thefollowing sequence of logical Hadamard and CNOT gatesimplements the transformation jΨi → SjΨi [28]:

To perform a T gate, we first prepare a logical ancilla inthe state

jφTi≡ 1ffiffiffi2

p ½jþzi þ eiπ=4j−zi�: ð38Þ

The T gate is then implemented via a probabilistic circuit.We perform a CNOT between the ancilla and the logicalqubit of interest and then measure the logical Z of the qubit.

Depending on the measurement outcome, we implement anS gate as shown below:

If the measurement outcome MZ ¼ þ1, then we obtainthe correct output TjΨi; otherwise, if MZ ¼ −1, then wehave performed the transformation jΨi → XT†jΨi. In thiscase, we implement an S gate on the logical qubit andobtain the final state iX Z T jΨi. The action of the operatoriX Z may be undone in the surface code software toimplement a pure T gate on the logical qubit [28].To realize the above implementations, we may prepare

logical ancilla qubits in the states jφTi and jφSi as follows.First, we create a “short qubit” [28] by ceasing the fermionparity measurement on two adjacent plaquettes p, qbelonging to the same sublattice, as shown in Fig. 11(a).For this qubit, let X ≡ Op and Z≡ iγη. The qubit isinitialized to a state jΨ�i such that XjΨ�i ¼ �jΨi. In abasis of eigenstates of the logical Z, the qubit state takes theform jΨ�i ¼ ðjþzi � j−ziÞ=

ffiffiffi2

p. Now, we assume that the

two-level system formed by the pair of Majorana fermionsγ and η can be manipulated by performing a rotation

RðθÞ ¼�1 0

0 eiθ

�ð39Þ

that acts in the basis of j�zi states. This may be imple-mented by using the phase of the adjacent superconductingislands to tune the coupling between the Majorana zeromodes [66]. To prepare the state jφSi, we perform therotation Rð�π=2ÞjΨ�i in the next surface code cycle,while to prepare jφTi, we perform the operation

FIG. 11. S- and T-gate ancilla preparation. We create the jφSiand jφTi ancilla states, which are needed to realize logical S andT gates by preparing the short qubit [28] shown above. We ceasestabilizer measurements on two adjacent plaquettes p and q. Inthe next surface code cycle, we perform a rotation of the two-levelsystem defined by iγη. Finally, we enlarge the logical qubit byextending one end of the qubit to guarantee stability againstnoise.

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Rðð2π � πÞ=4ÞjΨ�i. Afterwards, to guarantee the stabilityof the qubit against noise generated by the environment, weincrease the length of the logical X operator by extendingone end of the logical qubit, as discussed in detailpreviously and shown schematically in Fig. 7. In practice,a high-fidelity implementation of the S and T gates requiresthat the short qubits are put through a distillation circuit, asdiscussed in Ref. [28], which may be implemented using asequence of logical CNOT gates with other ancilla logicalqubits.We have presented a two-dimensional model of interact-

ing Majorana fermions that realizes a novel type of Z2

topological order with a microscopic S3 anyon symmetry.The required multifermion interactions in the plaquettemodel are naturally generated by phase slips in an array ofphase-locked s-wave superconducting islands on a TIsurface. Based on this physical realization, we proposethe Majorana surface code and provide the necessarymeasurement protocols and gate implementations foruniversal quantum computation. The Majorana surfacecode provides substantially increased error tolerance,reduced overhead, and simpler logical gate implementa-tions over a surface code with bosonic physical qubits. Weare optimistic that the Majorana fermion surface code willbe physically implemented and may provide an advanta-geous platform for fault-tolerant quantum computation.

ACKNOWLEDGMENTS

We thank Patrick Lee for helpful comments and dis-cussion. This work was supported by the PackardFoundation (L. F.), and the DOE Office of Basic EnergySciences, Division of Materials Sciences and Engineeringunder Grant No. DE-SC0010526 (S. V. and T. H.). S. V. andL. F. conceived and developed the Majorana fermionsurface code for universal quantum computation and wrotethe manuscript. T. H. contributed to the analysis of theMajorana plaquette model.

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