Braiding without Braiding: Teleportation-Based Quantum Information Processing with Majorana Zero Modes
BBraiding without Braiding: Teleportation-BasedQuantum Information Processing with Majorana Zero Modes
Sagar Vijay
1, 2 and Liang Fu Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Kavli Institute for Theoretical Physics, Santa Barbara, CA 93106, USA
We present a new measurement-based scheme for performing braiding operations on Majoranazero modes and for detecting their non-Abelian statistics without moving or hybridizing them.In our scheme, the topological qubit encoded in any pair of well-separated Majorana zero modesin a mesoscopic superconductor island is read out from the transmission phase shift in electronteleportation through the island in the Coulomb blockade regime. We propose experimental setupsto measure the teleportation phase shift via conductance in an electron interferometer or persistentcurrent in a closed loop.
Majorana zero modes are exotic quasiparticle excita-tions in topological superconductors. Theory predictsthat the presence of spatially separated Majorana zeromodes gives rise to degenerate superconducting groundstates that are indistinguishable by local observables [1].Furthermore, braiding Majoranas is expected to performa quantized unitary evolution on these ground states, ahallmark of their non-Abelian statistics [2–4]. Due tothese remarkable properties, Majorana zero modes havebeen proposed as topological qubits for robust quantuminformation processing that is (ideally) error-free at zerotemperature [5–7].Following theoretical proposals, over the last few yearstransport and scanning tunneling microscopy experi-ments have reported the observation of zero-bias con-ductance peak as a signature of Majorana zero modesin various material platforms including nanowires [8],atomic chains [9] and topological insulators [10], proxim-itized by s -wave superconductors. These results suggestthe existence of Majorana zero modes, and encourageresearch towards demonstrating their topological prop-erties. Among these, non-Abelian statistics is widely re-garded as the “holy grail” for topological phases of matterand for topological quantum computation.Theoretical proposals for detecting the non-Abelianstatistics of Majoranas have mostly relied on braiding,i.e. moving Majoranas around each other via a sequenceof operations. For example, by changing the phase ofJosephson junctions, Majorana zero modes localized inJosephson vortices can be braided in an array of su-perconducting islands on a topological insulator [11].By tuning the gate voltage, Majoranas in proximitizednanowires can be braided in a T-junction [12, 13]. De-tecting non-Abelian statistics further requires measuringthe state of Majoranas before and after braiding. Boththe motion and measurement of Majoranas are yet to beexperimentally achieved. Furthermore, physically mov-ing Majoranas in nanowire networks suffers from danger-ous thermal errors that are very difficult to correct [14].These errors may be avoidable in other proposals thatselectively tune couplings between Majoranas to imple-ment braiding transformations [16–18].In this work, we introduce a new scheme for (i) detect- ing the non-Abelian statistics of Majorana zero modesand (ii) implementing braiding operations, without anyphysical braiding, which is entirely based on projectivemeasurement as opposed to unitary evolution. In ourscheme, a topological qubit encoded in any pair of well-separated Majoranas is read out from the transmissionphase shift in electron teleportation through the topo-logical superconductor that hosts these Majoranas [19].Electron teleportation is a remarkable mesoscopic trans-port phenomenon enabled by the fractional nature of Ma-jorana zero modes and the charging energy of the super-conductor. Here we use electron teleportation to directlymeasure and manipulate Majorana qubits without mov-ing, hybridizing or destroying Majorana zero modes.In our scheme for “braiding without braiding”, the uni-tary transformation that would be generated by physi-cally exchanging a pair of Majoranas is realized by per-forming a sequence of projective measurements of Majo-rana bilinear operators. The theoretical basis for usingprojective measurements to implement quantum gateswas provided in Ref. [20, 21]. Within the abstract set-ting of non-Abelian topological order, replacing anyonbraiding by topological charge measurements was pro-posed by Bonderson, Freedman and Nayak [22]. On theother hand, electron teleportation provides an ideal wayof measuring Majorana qubits in mesoscopic topologi-cal systems, where the charging energy required for tele-portation comes from the long-range Coulomb interac-tion. As a result, the physics of teleportation lies be-yond the theory of topological order in the thermody-namic limit. By combining teleportation-based measure-ment and measurement-based braiding, our work unveilsa novel approach to quantum information processing withwell-separated, stationary Majorana zero modes.Our work is especially timely in view of a re-cent groundbreaking experiment on epitaxially grownInAs/Al superconducting nanowires [23], which are theo-retically predicted to host Majorana end modes under anexternal magnetic field [24–26]. Due to charging effectsin the Coulomb blockade regime, transport through thenanowire at zero magnetic field is dominated by Cooperpair tunneling, leading to zero-bias conductance oscilla-tions with the gate voltage that are charge-2 e periodic. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p However, above a critical field and in the presence of a su-perconducting gap, the conductance oscillations becomecharge- e periodic. The observed charge- e transport ina superconducting state supports the theoretically pre-dicted scenario of electron teleportation via Majoranamodes [19, 27, 28]. Another distinctive feature of telepor-tation is that single-electron transport through the super-conducting island is phase coherent [19]. This importantproperty forms the basis for topological qubit readoutin this work. To detect the phase coherence requires anelectron interferometer, which is currently under experi-mental pursuit [29]. Given these exciting developments,we believe teleportation-based braiding without braid-ing is a practical scheme for detecting the non-Abelianstatistics of Majorana zero modes, and offers a promisingprospect for quantum information processing.Our teleportation-based scheme for implementing pro-jective measurements and performing “braiding withoutbraiding” on stationary and spatially-separated Majo-rana zero modes has significant advantages over otherschemes based on physically moving Majoranas to im-plement logical gates or to perform qubit readout. Forbraiding to be feasible, Majoranas must be moved suf-ficiently slowly to obey an adiabaticity condition [13],which is especially stringent in disordered nanowireswithout a hard spectral gap [30, 31]. Qubit readout andgate operation in our proposal are not limited by this con-straint. Moreover, in the process of moving Majoranas,dangerous thermal errors on the topological qubit may beaccumulated, which are extremely difficult (if not impos-sible) to de-code and correct [15]. Finally, teleportation-based measurement of Majorana qubits has advantagesover proposed readout schemes based on charge sensing[13] which can only be performed on pairs of Majoranazero modes that are spatially adjacent.Our paper is organized as follows. We begin by re-viewing the phenomenon of phase-coherent electron tele-portation through Majorana zero modes. We then de-scribe two teleportation-based setups – the Majorana in-terferometer and the Majorana SQUID – for measuring atopological qubit encoded in a pair of well-separated Ma-jorana zero modes, and for detecting their non-Abelianstatistics. Next, we present a general protocol for im-plementing braiding transformations on Majorana zeromodes exclusively through projective measurements. Fi-nally, we provide a concrete experimental realization ofour proposal using proximitized nanowires. Our generalscheme of teleportation-based braiding without braidingis applicable to any Majorana platform, provided that thetopological superconductor hosting the Majoranas has afinite charging energy. I. CONCEPTUAL BASIS
In this section we lay out the theoretical basis ofteleportation-based measurement of a topological qubitencoded in a pair of spatially separated Majorana zero modes. We first elaborate on the transmission phaseshift in electron teleportation via a pair of Majoranasand its dependence on the state of the topological qubit,as pointed out in Ref. [19]. Next, we propose two waysof detecting this phase shift, or equivalently reading outthe topological qubit, by measuring the conductance inan electron interferometer or the persistent current ina closed loop. We then explicitly show the change ofthe teleportation phase shift in the process of physi-cally exchanging two Majorana zero modes. The dif-ference in the phase shift—a physical observable mea-sured by interferometry—before and after the braidingdirectly proves the system has evolved into a new state,thus demonstrating the non-Abelian statistics of Majo-rana zero modes.
A. Teleportation-Based Measurement ofTopological Qubit
Let us consider a mesocopic topological superconduc-tor island hosting a number of well-separated Majoranazero modes that have negligible wavefunction hybridiza-tion. Each Majorana zero mode of interest is tunnel cou-pled to a normal metal lead, and the tunnel couplingscan be turned on and off by gates. The superconduct-ing island is capacitively coupled to a nearby gate. Weassume that the charging energy E c is smaller than thesuperconducting gap ∆, but larger than the tunnel cou-pling to the leads, as defined by Eq.(6) below.In the absence of a tunnel coupling to leads, the groundstate energy of the island depends on the total numberof electrons N through the charging energy: E ( N ) = E c ( N − n g ) , (1)where the offset charge n g is continuously tunable bythe gate voltage. Due to the presence of Majorana zeromodes, the superconducting island can accommodate aneven and an odd number of electrons on equal groundwithout paying the energy cost of the superconductinggap. Thus N takes both even and odd integer values.Throughout this work, we assume that the island is inthe Coulomb blockade regime away from the charge de-generacy point, so that the total charge of the island isfixed, denoted by N = N . Under this condition, theisland has 2 M/ − degenerate ground states, where M (an even integer) is the number of Majorana zero modespresent. These degenerate ground states form a topolog-ically protected Hilbert space, which we use to encodequantum information. By detuning the island far awayfrom charge degeneracy, the topological qubits are pro-tected against quasiparticle poisoning from outside theisland at low temperature.A complete basis for this 2 M − -dimensional Hilbertspace is given by the common eigenstates of a setof nonoverlapping Majorana bilinear operators, e.g.,( iγ γ , iγ γ , ..., iγ M − γ M ). A Majorana bilinear oper- G = g + g cos e ( c ) ~ i G = g + g cos e ( c ) ~ (a) (b)FIG. 1. Majorana Interferometer –
Two electron inter-ferometry setups to measure the topological qubit formed byMajorana zero modes γ and γ . In both interferometers, onepath goes through the topological qubit while the other pathgoes through (a) a normal metal with sufficiently long phasecoherence length (blue) and (b) a second Majorana island ini-tialized in a definite parity state iψ ψ = ±
1, which is usedas a reference. ator iγ a γ b has two eigenstates |±(cid:105) ab , defined by iγ a γ b |±(cid:105) ab = ±|±(cid:105) ab . (2)Thus, measuring the topological qubit in this basisamounts to measuring the eigenvalue of iγ a γ b . Note thatany way of partitioning Majoranas into pairs defines acorresponding basis for the topological qubit, and differ-ent bases are related by unitary transformations knownas F -symbols, which are determined by the fusion ruleof Majoranas. It is thus highly desirable to measure theeigenvalue of any Majorana bilinear operator, so that thetopological qubit can be measured equally well in any ba-sis.We now describe a teleportation-based protocol tomeasure the eigenvalue of any Majorana bilinear iγ a γ b by coupling the Majorana island to lead a and to lead b .The bare tunneling Hamiltonian is given by H T = (cid:88) j = a,b t j c † j (0) f ( r j ) + h . c . (3)where c j (0) is the electron operator at the end of lead j ,and f ( r j ) is the electron operator in the island at the tun-neling location r j , where the Majorana zero mode γ j islocated. Next, we expand f ( r j ) in terms of quasiparticle operators in the superconducting island: f † ( r j ) = ξ ∗ j ( r ) e iθ/ γ j + ... (4)Here ξ j ( r ) is the wavefunction associated with the Majo-rana mode operator γ j , defined by γ j = (cid:90) d r (cid:104) ξ j ( r ) e − iθ/ f † ( r ) + ξ ∗ j ( r ) e iθ/ f ( r ) (cid:105) . (5)Here, θ is the phase operator of the superconductor,which is conjugate to the electron number operator N and satisfies the commutation relation [ θ, N ] = 2 i . In theoperator expansion (4) we have neglected all quasiparti-cles above the superconducting gap which are irrelevantto the low energy physics of our interest, as well as Majo-rana zero modes at other locations whose amplitudes at r j are exponentially small. Thus, as shown by (4), in thelow-energy Hilbert space the electron creation operator f † ( r j ) is represented as a product of the Majorana modeoperator γ j and the charge-raising operator e iθ/ whichincreases the charge of the island by 1 e . Physically speak-ing, Eq. (4) describes the charge-statistics separation ofan electron after entering a topological superconductor:the charge of the electron is spread out over the entiresuperconductor, while its Fermi statistics is retained bya localized Majorana fermion that is charge neutral.Substituting (4) into the bare Hamiltonian (3) yieldsan effective tunneling Hamiltonian H T = (cid:88) j = a,b λ j c † j (0) γ j e − iθ/ + h . c ., with λ j = t j ξ j ( r j )We define the tunnel coupling Γ asΓ = (cid:88) j = a,b πρλ j , (6)where ρ is the density of states in the leads. AssumingΓ (cid:28) E ± where E ± ≡ E ( N ± − E ( N ) is the energydifference between the charge states N = N and N = N ±
1, transmission through the island is dominated bya second-order process, where a single electron tunnelsinto the island from one lead and a single electron exitsfrom the island to another lead. Therefore, from second-order perturbation in H T , we obtain an effective couplingbetween a Majorana island in the off-resonance Coulombblockade regime and the leads H ab = − λ ∗ a λ b c † b (0) c a (0) (cid:20) (cid:104) N | γ b e − iθ/ | N + 1 (cid:105)(cid:104) N + 1 | γ a e iθ/ | N (cid:105) E ( N + 1) − E ( N ) + (cid:104) N | γ a e iθ/ | N − (cid:105)(cid:104) N − | γ b e − iθ/ | N (cid:105) E ( N − − E ( N ) (cid:21) + h . c . = γ a γ b (cid:104) T ab c † b (0) c a (0) − T ∗ ab c † a (0) c b (0) (cid:105) , (7)where T ab ≡ λ ∗ a λ b (cid:16) E + + E − (cid:17) is the effective single elec-tron tunneling between lead a and b , mediated by a pair of Majorana zero modes γ a , γ b . Due to this entanglementof Majorana degrees of freedom with electron tunneling I ⇠ " ( i ) e ~ sin e ( c ) ~ I ⇠ " ( ) e ~ sin e ( c ) ~ (a) (b)FIG. 2. Majorana SQUID –
When the two Majorana zeromodes γ and γ are connected by a bridge outside the islandto form a closed loop, with the bridge being (a) a normal metalwith sufficiently long phase coherence length or (b) a referenceMajorana island in a definite parity state iψ ψ = ±
1, thetopological qubit defined by iγ γ = ± I in the ground state, whichis a h/e -periodic function of the applied flux Φ. between two leads, H ab enables a direct projective mea-surement of the Majorana bilinear iγ a γ b , even when γ a and γ b are far apart in the superconductor island, as weshow below.Let us first consider the case that the Majorana islandis initialized to be an eigenstate of iγ a γ b , either | + (cid:105) ab or |−(cid:105) ab . It follows from (7) that the single electron tunnel-ing amplitude from lead a to b , which is mediated by γ a and γ b , is equal to − iT ab for the Majorana qubit state | + (cid:105) ab , and + iT ab for the state |−(cid:105) ab . Therefore, the twoMajorana qubit states |±(cid:105) ab are distinguishable by the π difference in the transmission phase shift in electronteleportation via a pair of Majorana zero modes [19].To measure the teleportation phase shift requires quan-tum interference. We now propose two phase measure-ment schemes for Majorana qubit readout. The firstscheme is based on a conductance measurement in atwo-path electron interferometer, with one path goingthrough the Majorana island and the other path servingas a reference. The reference path may be a normal metalwith a sufficiently long phase coherence length [19], or asecond Majorana island in a definite parity state [33, 34],as shown in Fig. 1. The total conductance G then con-tains a term proportional to ( iγ a γ b ) due to the interfer-ence between the two paths, i.e., G (Φ) = g + ig (Φ) γ a γ b ,where g depends periodically on the external magneticflux Φ enclosed by the two interfering paths, with h/e -periodicity. Since the conductance takes different valuesfor the qubit state |±(cid:105) ab , the conductance measurementin such Majorana interferometer provides a projectivemeasurement of the topological qubit in the basis |±(cid:105) ab .The second scheme for qubit readout is based on mea-suring the persistent current in a closed loop. This loopcan be made by connecting Majorana zero modes on theisland to the ends of a normal metal bridge (see Fig. 2a),or to a reference Majorana island in a definite qubit state (see Fig. 2b). Due to the phase coherence of electronmotion around the loop, the energy of the closed sys-tem depends periodically on the external magnetic fluxΦ through the loop with h/e periodicity, E = E + iεγ a γ b cos (cid:20) e (Φ − Φ c ) (cid:126) (cid:21) , (8)where Φ c and ε depend on details of the setup such astunnel couplings between the island and the normal metalbridge. Eq.(8) implies the presence of a persistent circu-lating current in the loop I = ∂E∂ Φ = ( iγ a γ b ) eε (cid:126) sin (cid:20) e (Φ − Φ c ) (cid:126) (cid:21) . (9)This circulating current flows in opposite directions forthe two Majorana qubit states |±(cid:105) ab . Thus the Majoranaqubit is faithfully transferred to the state of the persistentcurrent, which can then be read out by inductive couplingthe system to a SQUID loop.We now estimate the magnitude of the persistent cur-rent in a Majorana SQUID by treating the transmissionthrough a Majorana island as single electron hoppingacross a weak link, as described by the effective Hamilto-nian (7). Details of our calculation are presented in theSupplemental Material [42]. When the Majorana SQUIDis formed by a single island connected to a normal metalbridge, we find that the magnitude of the persistent cur-rent at zero temperature is given by I ∼ e Γ (cid:126) δ (cid:18) E + + 1 E − (cid:19) (10)as explicitly calculated in the Supplemental Material [42].Here, Γ is the tunnel coupling between the island andthe normal metal defined in (6), and δ is the single-particle level spacing in the metal, which is inverselyproportional to the length of the bridge. An order-of-magnitude estimate based on experimental parametersin Ref. [23, 28, 32] yields I ∼
10 nA.When the Majorana SQUID consists of two islandsconnected by two normal metal bridges, we determine thepersistent current by modeling the bridges as mediating adirect electron tunneling between the Majorana islands.In this case, we consider the Hamiltonian H = H T + H c for the full system, where H c = (cid:88) i =1 , E ( i ) c ( N i − n ( i ) g ) (11)describes the charging energy for each of the Majoranaislands ( i = 1, 2). Here, E ( i ) c , N i and n ( i ) g are the charg-ing energies, total charge, and gate charges, respectively,for island i . For simplicity, we let E (1) c = E (2) c = E c forthe remainder of our calculation. Furthermore, as shownin Sec. IA, electron tunneling between the two Majoranaislands is described at low energies by an effective Hamil-tonian in terms of the Majorana operators, as given by H t = it ψ γ e i ( θ − θ ) / + h . c . + it e ie Φ / (cid:126) ψ γ e i ( θ − θ ) / + h . c . (12)with θ , the superconducting phases on each Majoranaisland and Φ, the applied flux through the ring.In the presence of a large charging energy E c (cid:29) t , ,the effective Hamiltonian for the system is, to lowest or-der in perturbation theory, given by H eff = 2 | t t | E + + E − cos (cid:20) e (Φ − Φ c ) (cid:126) (cid:21) ψ ψ γ γ , (13)where the constant Φ c provides an overall shift and ispresent when t , are complex. The magnitude of thepersistent current is then given by I ∼ e (cid:126) (cid:18) | t t | E + + E − (cid:19) (14)In the Supplemental Material [42], we also model the per-sistent current in a Majorana SQUID with two Majoranaislands as single electron hopping in a ring with two weaklinks and determine the magnitude of the persistent cur-rent [42]. B. Detecting non-Abelian Braiding Statistics fromTeleportation Phase Shifts
In this section we explicitly demonstrate the changeof teleportation phase shift due to braiding Majoranazero modes in a two-dimensional topological supercon-ductor. Here, Majorana braiding is realized by adiabati-cally exchanging two identical vortices, which host Majo-rana zero modes in their cores [3]. Since the teleportationphase shift is a physical observable that can be measuredby interferometry, its change before and after braidingimplies a change in the quantum state of the system,thus providing direct proof for non-Abelian statistics.Before proceeding, we first clarify what we mean bynon-Abelian statistics of Majorana-carrying vortices in asuperconductor. We assume that the superconductor iswell described by a Bogoliubov-de Gennes (BdG) Hamil-tonian with a pairing potential ∆( r ) = | ∆( r ) | e iθ ( r ) thatis a complex function of position. We assume that apartfrom the overall phase θ , the pairing potential configura-tion ∆( r ) is non-dynamical and externally set up. On theother hand, we take the overall superconducting phase θ as a quantum mechanical variable, which is conjugate tothe total number of electrons N . Throughout this work,we take N to be fixed due to the large charging energy,so that θ is fluctuating.In this setting, Majorana zero modes are not decon-fined anyons but a type of “twist defect” [39] associatedwith vortices, the point singularities in ∆( r ). A vortexcentered at R corresponds to a ± π winding of the phase θ ( r ) around R . As we adiabatically exchange two vor-tices, ∆( r ) varies slowly. To define non-Abelian statis-tics, it is required that the full function ∆( r ) returns toits original configuration after the vortex exchange. Theevolution of the system into a new quantum state af-ter this process is a defining feature of the non-Abelianstatistics of Majorana zero modes bound to vortex cores. ⇠ ( r ) ! ⇠ ( r ) ⇠ ( r ) ! ⇠ ( r ) ⇠ ( r ) ! ⇠ ( r ) (a) ⇠ ( r ) ! ⇠ ( r ) ⇠ ( r ) ! ⇠ ( r ) ⇠ ( r ) ! ⇠ ( r ) (b)FIG. 3. Teleportation Phase-Shift –
Braiding (a) or ex-changing (b) Majorana zero modes induces a transformationon the wavefunctions as indicated. The overall sign of allof the wavefunctions is ambiguous, and only gauge-invariantquantities, given by products of even numbers of Majoranawavefunctions, appear in physical observables. The shadedlines shown above are physical regions where the supercon-ducting phase rapidly changes by 2 π . The change in signof physical observables, due to the “branch cuts” sweepingthrough the Majorana zero modes [4], provides a signature oftheir non-Abelian statistics that may be detected via telepor-tation. As a warm-up, consider two well-separated vorticescentered at R and R , and denote the correspondingMajorana zero modes localized in the vortex cores by γ and γ . We connect γ and γ by a normal metal bridgeto form an interferometer as discussed in Sec. IA, andconsider how the teleportation phase shift evolves as athird vortex moves around the vortex at R in a full cir-cle (see Fig. 3a).For any given vortex configuration, the wavefunctionassociated with any Majorana zero mode ξ j ( r ), obtainedby solving the BdG Hamiltonian, is defined up to anoverall choice of sign. Since the Majorana zero modeoperator γ j is defined from ξ j ( r ) via Eq. (5), γ j is notgauge-invariant as emphasized in Ref. [40]. Nonetheless,this choice of sign for the wavefunctions ξ , ( r ) does not affect any physical observables, which necessarily corre-spond to gauge-invariant operators such as ξ j ( r ) γ j .For convenience, we now choose the signs of thesewavefunctions such that ξ , ( r ) vary continuously withthe moving position of the third vortex. The eigenvalueof the Majorana bilinear operator iγ γ , taking two pos-sible values ±
1, stays constant during the braiding pro-cess, as the fermion parity of the system is conserved. Asa result, the teleportation amplitude, whose expression(7) contains the product ξ ( R ) ξ ( R ) γ γ , also variescontinuously. As shown by Ivanov [4], after the thirdvortex returns to its original position and the originalvortex configuration is restored, the wavefunction ξ ( r )comes back to itself while ξ ( r ) and ξ ( r ) change sign,as shown in Fig. 3a. Consequently, the phase shift inelectron teleportation via Majorana zero modes γ and γ changes by π before and after the third vortex circlesaround γ . This quantized change of a physical observ-able signals a change in the quantum state of the systeminduced by braiding.The teleportation phase can also detect the change inthe state of the system when two vortices are exchanged ,as shown in Fig. 3b. We assume that the local configura-tions of the pairing potential near the vortex centers R and R are identical, so that the wavefunctions of theMajorana zero modes γ and γ are essentially relatedby translation, i.e. ξ ( r − R ) = ξ ( r − R ). After ex-changing vortices 2 and 3 in the manner shown in Fig. 3band restoring the original vortex configuration, the Ma-jorana wavefunctions transform as ξ ( r ) −→ − ξ ( r ), and ξ ( r ) −→ ξ ( r ). To demonstrate the braiding-inducedchange in the quantum state of the system, we connect γ and a reference Majorana zero mode γ by a normalmetal bridge to form an interferometer, and monitor theevolution of the teleportation phase shift in the processof exchanging γ and γ , while keeping one end of thebridge attached to the moving Majorana γ . The initialteleportation amplitude from R to R is given by theproduct ξ ( R ) ξ ( R ) γ γ . After braiding, this interfer-ometer measures the teleportation amplitude from R to R , given by ξ ( R ) ξ ( R ) γ γ . This result should becompared with the teleportation amplitude from R to R before braiding, given by ξ ( R ) ξ ( R ) γ γ and mea-surable by an interferometer containing γ and γ . Thiscomparison shows that braiding γ and γ has the ef-fect of the transformation γ → γ . Repeating the sameanalysis for the teleportation amplitude from R to R shows that same braiding process also has the effect ofthe transformation γ → − γ . Our analysis based onelectron teleportation thus reproduces the “Ivanov rule”for Majorana braiding [4] γ → − γ , γ → γ . (15)It is worth noting that the above braiding transforma-tion (15) per se is non-gauge-invariant, as it is expressedin terms of Majorana operators that suffer from a Z signambiguity. Only after the sign convention for the zeromode wavefunction ξ , is specified, as we did previouslyby choosing ξ ( r − R ) = ξ ( r − R ), do the Majoranaoperators γ , become well-defined, so that the braidingtransformation (15) becomes meaningful.Our analysis, as presented, demonstrates that thebraiding-induced change in the teleportation phase shiftis a physical observable described by a gauge invari-ant operator involving ξ ∗ a ξ b γ a γ b and ξ a ξ ∗ b γ a γ b . Thus thechange is teleportation phase shift is a direct and mea-surable consequence of the non-Abelian statistics of Ma-jorana zero modes. ⌘ ⌘ i⌘ ⌘ = +1 | i i ⇠ | i ⌦ | i⌘ ⌘ = +1 i| f i ⇠ ˆ U | i ⌦ | i⌘ ⌘ = +1 i Measurements I. i ⌘ = +1 II. i⌘ = +1 III. i⌘ ⌘ = +1 FIG. 4.
Measurement-Based Braiding –
Schematic de-piction of the initial state | ψ i (cid:105) , with Majorana zero modes η and η initialized in the state iη η = +1. Performing theindicated sequence of measurements is equivalent to braidingMajorana fermions γ and γ , up to a normalization factor. II. MEASUREMENT-BASED BRAIDING
We now describe the theoretical protocol for perform-ing a braiding transformation on a collection of Majoranazero modes exclusively through a sequence of projectivemeasurements, without needing to move the zero modes.We subsequently describe a teleportation-based measure-ment protocol for realizing our proposal. Consider theschematic setup shown in Fig. 4; Majorana zero modes γ , γ , γ , and γ are used to encode two logical qubits,while η and η will serve, for our purposes, as a singleancilla qubit. We prepare the ancilla qubit in the state iη η = +1 so that the initial state of the system is givenby | ψ i (cid:105) = | φ (cid:105) ⊗ | iη η = +1 (cid:105) (16)with | φ (cid:105) , the logical two-qubit state of the four Majoranas { γ i } that we wish to manipulate.Our measurement-based braiding protocol is based onthe fact that projective measurements of Majorana bilin-ear operators ˆ P ( ± ) γ n η m ≡ ± iγ n η m , (17)may be used to implement a unitary braiding transforma-tion up to an overall normalization factor. Specifically,observe the mathematical identityˆ P (+) η η ˆ P (+) η γ ˆ P (+) γ η | ψ i (cid:105) = 12 / ˆ U | ψ i (cid:105) (18)where the operator ˆ U ≡ γ γ √ Measurement j Measurement j + 1 Measurement j Outcome = ± Outcome = Outcome = +1 (undesirable) (desirable)
Outcome = (undesirable) Outcome = +1 (desirable)
FIG. 5.
Measurement “Decision Tree” –
Summary ofthe measurement protocol. If a measurement yields an un-desirable outcome, the previous measurement step may berepeated (as indicated) to recover the state, before the unde-sirable measurement was performed. braiding operation, starting from the state | ψ i (cid:105) , are sum-marized in Fig. 4.Successfully performing a measurement-based braid-ing transformation crucially relies on the outcomes ofthe measurements that are performed. If a measurementyields an undesirable outcome, however, it is still pos-sible to obtain the desired final state by performing anappropriate sequence of operations. As an example, as-sume that the first measurement yields the undesirableresult that iγ η = − | ϕ (cid:105) ≡ P ( − ) γ η | ψ i (cid:105) . We may recoverthe state of the system before the undesirable measure-ment, | ψ i (cid:105) , by measuring the bilinear iη η . If we findthat iη η = +1, then we recover the initial state P (+) η η | ϕ (cid:105) = 12 | ψ i (cid:105) (20)up to a change in normalization, and we may now re-dothe measurement of the bilinear iγ η . More generally,in order to recover the state | ψ i (cid:105) , we must alternate mea-surements of the bilinears iη η and iγ η until we obtainthe measurement outcome iη η = +1. Observe that P (+) η η P ( s n ) γ η P ( − ) η η · · · P ( s ) γ η P ( − ) η η | ϕ (cid:105) = − s n n | ψ i (cid:105) (21)where s i = ± j is undesirable, we alternate between measurement steps j − j ; this cycle is repeated until measurement step j yields the desired outcome. The number of steps requiredto recover from an undesirable measurement only changesthe normalization of the final state, as can be seen fromEq. (21).We have assumed in our analysis that undesirable mea-surements only arise due to the inherently probabilis-tic nature of the measurement-based braiding protocol. However, an undetected error event (e.g. quasiparticlepoisoning) that occurs during the measurement proce-dure can also lead to an undesirable final state. If themeasurements are performed sufficiently rapidly relativeto the poisoning time, these errors may be significantlysuppressed, due to the quantum Zeno effect [38]. Aconcrete scheme for incorporating error correction intomeasurement-based braiding will the subject of a forth-coming work [41]. III. EXPERIMENTAL REALIZATIONA. Experimental Setup
We now propose a teleportation-based scheme for real-izing measurement-based braiding; our proposal is sum-marized in Fig. 6. Consider a superconducting nanowire;gate voltages may be applied along the length of thenanowire to introduce an interface between the topo-logical and trivial superconducting regions, which local-izes a Majorana zero mode. In our setup, we applygate voltages so that six Majorana zero modes appear( γ , γ , γ , γ , η and η ) at points along the wire. Thedistance between the Majoranas is assumed to be suffi-ciently large, so that hybridization between adjacent Ma-joranas may be neglected. We initialize η and η in thestate iη η = +1 by nucleating the two Majorana zeromodes in a topologically trivial region of the nanowirewhere the total fermion parity is fixed.Parallel to the existing nanowire in our setup, we nowplace either (i) a normal metal strip or (ii) a single, prox-imitized nanowire with gate voltages applied appropri-ately so that the gated region is topological and hoststwo Majorana zero modes ( ψ and ψ ) at its two ends.In both setups, four metal bridges are used to connectthe existing nanowire to the normal metal or the secondnanowire. In the following section, we will describe theimplementation of setup (ii). A similar protocol may beused to implement setup (i), involving the same sequenceof interferometric or flux-based measurements, as long asthe metal strip has a sufficiently long phase coherencelength.To implement our “braiding without braiding” pro-tocol using setup (ii), we will tune gate voltages tore-position ψ and ψ along the length of the secondnanowire; hence we will refer to this nanowire as the“Majorana bus”. We initialize the Majorana zero modesin the bus in the state iψ ψ = +1. The Majorana busand the remaining Majoranas in our setup are coupledtogether by four metallic bridges, and each coupling canbe tuned on or off, as indicated schematically by the“switches” in Fig. 6. Each lead is chosen to be shorterthan the phase coherence length of the metal, to allow forMajorana qubit readout based on electron teleportationand intereference.We may perform projective measurements of Majo-rana bilinears in the top nanowire using the interfer- ⌘ ⌘ ( i⌘ ⌘ = +1) Normal metal or Majorana island (a) Initialization ⌘ ⌘ (b) Measurement I ( iγ η = +1) ⌘ ⌘ (c) Measurement II ( iη γ = +1) ⌘ ⌘ (d) Measurement III ( iη η = +1)FIG. 6. Experimental Realization—
Protocol forteleportation-based braiding without braiding is illustratedin a nanowire-based Majorana platform. A nanowire hostssix Majorana zero modes at the interface between topologi-cal and trivial superconducting regions. γ , ..., γ are used astopological qubits and η , η as an ancilla qubit. The greenstrip can be either a normal metal with a long phase coherencelength or a Majorana island in a definite parity state (a Majo-rana bus). We begin by initializing the the ancilla qubit in (a),before performing measurements of the appropriate Majoranabilinears in the top nanowire. The coupling between the topo-logical superconductor wire to the normal metal or Majoranabus through the metallic strips may be turned on and off, asindicated schematically by the “switches”. Fluxes may be ap-plied through appropriate loops for topological qubit readoutvia conductance or persistent current measurement. ence of electron trajectories through our setup. Thiscan be achieved by measuring the persistent current in aclosed loop, or by measuring the two-terminal conduc-tance across the Majorana bus. To implement eithermeasurement procedure, we first align the Majorana busso that ψ and ψ are across from the pair Majorana zeromodes that we wish to measure, respectively. By turningon the switches on the metallic strips, we introduce elec-tron tunneling between aligned Majorana zero modes onthe bus and on the top wire. In the presence of a largecharging energy on both the bus and the wire, the effec-tive Hamiltonian describing the full system is given byEq.(13); it depends periodically on the flux through theloop Φ and on the eigenvalue of the Majorana bilinearoperator in the top wire. A flux- or conductance-basedreadout of the Majorana bilinear may then be performed,as detailed in the following section. B. Measurement Procedures
To perform a flux-based measurement one of the Ma-jorana bilinears, we tune gate voltages in the Majoranabus so that ψ and ψ are across from the Majorana zeromodes in the top row that we wish to measure. For con-creteness, consider a measurement of iγ η , as shown inFig. 6a. After aligning the Majorana bus, we turn offthe couplings in the last two metallic strips; this is indi-cated schematically by the closed and open switches inFig. 6a. Furthermore, we insert a flux Φ through theloop formed by ψ , ψ , η and γ , as shown.In the presence of a charging energy that removes thedegeneracy between even and odd charge-states on boththe Majorana bus and on the top nanowires, the Hamil-tonian for the system is H = H t + H c – with H t = it ψ γ e i ( θ − θ ) / + it e ie Φ / (cid:126) ψ η e i ( θ − θ ) / + h . c . describing the coupling between the Majorana bus to γ and η through the metallic strips, while H c is the charg-ing energy on the nanowire and Majorana bus, as givenpreviously in Eq. (11). A measurement of the persistentcurrent may be used to determine the Majorana bilinear iγ η as detailed in Sec. IIA.To perform a conductance measurement, we may intro-duce a weak tunnel coupling between the Majorana busand two external leads. A similar protocol for measur-ing stabilizer operators for the Majorana fermion surfacecode [34–37] has also been proposed [34, 37]. The tun-neling Hamiltonian takes the form H T = t L c † L ψ e − iθ / + t R c † R ψ e − iθ / + h . c . (22)where t L,R are the tunnel couplings to the left and rightleads. Here, e ± iθ / is the charge- e raising (lowering)operator on the bus, while c † L and c † R are the electroncreation operators in the left and right lead, respectively.When the charging energy is large, we may derive aneffective Hamiltonian that takes the form H eff = H + H to lowest order, where H = 2 | t t | E + + E − cos (cid:20) e (Φ − Φ c ) (cid:126) (cid:21) ψ ψ γ η (23) and H = t ∗ L t R (cid:18) E + + 1 E − (cid:19) ψ ψ c † R c L + 2 | t t | t ∗ L t R (cid:20) E + ) + 1( E − ) (cid:21) cos (cid:20) e (Φ − Φ c ) (cid:126) (cid:21) ( iγ η ) c † R c L + h . c . The tunneling conductance depends sensitively on themeasured value of the bilinear iγ η and is determinedto be G = g + 2 πe (cid:126) g cos (cid:20) e (Φ − Φ c ) (cid:126) (cid:21) ψ ψ γ η . (24)Here g is a constant contribution to the conductancethat is independent of the measurement outcome, while g = 4 | t L | | t R | | t t | (cid:20) E + ) + 1( E − ) (cid:21) (cid:20) E + + 1 E − (cid:21) ρ L ρ R with ρ L,R , the density of states in the left and rightleads, respectively.
ACKNOWLEGEMENTS
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To estimate the magnitude of the persistent current ina Majorana interferometer with a metallic arm or anotherMajorana island, we compute the persistent current in afree electron ring with one and two weak links.
1. Single Weak Link
Consider a free electron ring of length L with a singleweak link. The bosonized form of the action S = S + S weak where S is [1, 2] S = 12 v F (cid:90) L dx (cid:90) β dτ (cid:2) ( ∂ τ φ ) + v F ( ∂ x φ ) (cid:3) (A1)with v F is the Fermi velocity of the metal. The “weaklink” is modeled by a weak hopping between the ends ofthe ring [3, 4] S weak = − (cid:82) dτ (cid:2)(cid:101) t ψ † ( L, τ ) ψ (0 , τ ) + h . c . (cid:3) ,with ψ and ψ † the electron creation/annihilation oper-ators respectively. After bosonizing, the most relevantterm in the action for the weak hopping given by S weak = − (cid:101) t (cid:90) β dτ cos (cid:2) √ π ( φ ( L, τ ) − φ (0 , τ )) + Θ (cid:3) with Θ ≡ π ΦΦ (A2)the flux through the ring, in units of the flux quantumΦ = h/e .We observe that the fields φ ( x, τ ) for x (cid:54) = 0, L maybe integrated out to obtain an effective action that onlyinvolves the phase difference ϑ ( τ ) ≡ [ φ ( L, τ ) − φ (0 , τ )][3, 4]. Expanding the phase difference as ϑ ( τ ) = 1 β (cid:88) iω n e iω n τ ϑ ( ω n ) (A3) with Matsubara frequencies ω n ≡ πn/β , and integratingout the bulk fields results in the effective action for afinite-size system S eff = 1 β (cid:88) iω n | ω n | coth (cid:20) | ω n | L v F (cid:21) | ϑ ( ω n ) | − (cid:101) t (cid:90) β dτ cos (cid:2) √ π ϑ ( τ ) + Θ (cid:3) (A4)We may determine the persistent current I = β − ( ∂ ln Z/∂
Φ) perturbatively in the coupling (cid:101) t , where Z = (cid:82) Dφ e − S eff is the path integral. The leading contri-bution to the persistent current comes at O ( (cid:101) t ); observethat I = 1 β ∂ ln Z∂ Φ = e (cid:101) tβ (cid:90) β dτ (cid:10) sin (cid:2) √ π ϑ ( τ ) + Θ (cid:3)(cid:11) + · · · = e (cid:101) t sin (cid:18) π ΦΦ (cid:19) e − π (cid:104) ϑ ( τ ) (cid:105) + · · · (A5)where (cid:104)· · · (cid:105) denotes the expectation value with respectto the Gaussian part of the action. From the two-pointcorrelation function (cid:104) ϑ ( τ ) ϑ (0) (cid:105) = 1 β (cid:88) iω n e iω n τ | ω n | coth (cid:104) | ω n | L v F (cid:105) (A6)we determine that the persistent current at zero temper-ature is given, to leading order in the weak hopping, andafter restoring factors of (cid:126) , by I = − e (cid:101) t (cid:18) π (cid:126) v F (cid:15) F L (cid:19) e − γ sin (cid:18) π ΦΦ (cid:19) + O ( t ) (A7)where γ is the Euler-Mascheroni constant. Here, (cid:126) (cid:101) t – thehopping strength across the Majorana island – is givenby (cid:126) (cid:101) t = t (1 /E + + 1 /E − ) with E ± the energy differ-ence between adjacent charge states on the Majoranaisland as defined in the main text, and t , the tunnel-ing amplitude into a single Majorana zero mode. Wemay re-express the persistent current in terms of E ± aswell as the tunnel-coupling Γ and the level-spacing in themetallic wire δ . The persistent current is then given by I = I sin(2 π Φ / Φ ) with I = 2 e Γ δ (cid:126) (cid:18) E + + 1 E − (cid:19) e − γ (A8)Taking the tunnel-coupling Γ to be approximately one-tenth of the superconducting gap ∆, which determinesthe level-broadening of the Majorana edge-states we findthat I ≈
10 nA.
2. Double Weak Link
A similar answer is obtained in the case of a metallicwire with two weak links, which we model as two inde-pendent wires, each of length L/
2, whose ends are weakly1coupled. In this case, the bosonized action is given by S = S + S weak with S = 12 v F (cid:90) L/ dx (cid:90) β dτ (cid:88) (cid:96) =1 , (cid:2) ( ∂ τ φ (cid:96) ) + v F ( ∂ x φ (cid:96) ) (cid:3) (A9)and S weak = − (cid:101) t (cid:90) β dτ cos (cid:2) √ π ( φ (0 , τ ) − φ ( L/ , τ )) + Θ (cid:3) − (cid:101) t (cid:90) β dτ cos (cid:2) √ π ( φ (0 , τ ) − φ ( L/ , τ )) (cid:3) (A10) with Θ again given by the flux through the ring Θ =2 π Φ / Φ .We now integrate out the bulk fields after expandingthe phase differences ϑ ( τ ) ≡ [ φ (0 , τ ) − φ ( L/ , τ )] / ϑ ( τ ) ≡ [ φ (0 , τ ) − φ ( L/ , τ )] / ϑ (cid:96) ( τ ) = β − (cid:80) iω n e iω n τ ϑ (cid:96) ( ω n ) to obtainthe effective action S = S (eff)0 + S (eff)weak where S (eff)0 = 1 β (cid:88) iω n | ω n | coth (cid:18) | ω n | L v F (cid:19) (cid:26) | ϑ ( ω n ) | + | ϑ ( ω n ) | + sech (cid:18) | ω n | L v F (cid:19) [ ϑ ∗ ( ω n ) ϑ ( ω n ) + c . c . ] (cid:27) (A11) S (eff)weak = − (cid:101) t (cid:90) β dτ cos (cid:2) √ π ϑ ( τ ) + Θ (cid:3) − (cid:101) t (cid:90) β dτ cos (cid:2) √ π ϑ ( τ ) (cid:3) (A12)We again compute the persistent current perturbativelyin the hoppings (cid:101) t , . We begin by observing that in termsof the variable ϕ ( ω n ) = ϑ ( ω n ) + sech (cid:18) | ω n | L v F (cid:19) ϑ ( ω n ) (A13)the Gaussian part of the action is now diagonal S (eff)0 = 1 β (cid:88) ω n | ω n | (cid:104) coth (cid:18) | ω n | L v F (cid:19) | ϕ ( ω n ) | + tanh (cid:18) | ω n | L v F (cid:19) | ϑ ( ω n ) | (cid:105) . (A14)The two-point correlation function (cid:10) ϑ ( τ ) (cid:11) = 1 β (cid:88) ω n | ω n | − coth (cid:18) | ω n | L v F (cid:19) (A15)diverges at low frequencies, so that (cid:10) cos (cid:2) √ π ϑ ( τ ) (cid:3)(cid:11) = 0 . (A16)As a result, the leading contribution to the persistentcurrent comes at O ( (cid:101) t (cid:101) t ), as expected. To leading order in the weak hopping, the persistentcurrent now takes the form I = e (cid:101) t (cid:101) t β (cid:90) β dτ dτ (cid:48) (cid:10) sin (cid:2) √ π ϑ ( τ ) + Θ (cid:3) cos (cid:2) √ π ϑ ( τ (cid:48) ) (cid:3)(cid:11) We may explicitly evaluate the above the expression, af-ter using the correlation function (A15) and the fact that (cid:104) ϑ ( τ ) ϑ (0) (cid:105) = − β (cid:88) ω n e iω n τ | ω n | sinh (cid:16) | ω n | L v F (cid:17) . (A17)At zero temperature, however, the perturbative expres-sion for the persistent current does not converge, as thesmall parameter in the perturbative expansion turns outto be (cid:101) t (cid:101) t β . Instead, we find that at low temperatures T ,the persistent current takes the form I = I sin(2 π Φ / Φ )where I , after re-writing in terms of the level-spacing δ , the charging-energy cost for the islands E ± , and thetunnel-coupling for each Majorana island Γ i is given by I = 8 πe (cid:126) Γ Γ k B T δ (cid:18) E + + 1 E − (cid:19) e − γ . (A18) [1] F. D. M. Haldane, J. Phys. C , 2585 (1981).[2] F. D. M. Haldane, Phys. Rev. Lett. , 1840 (1981).[3] C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. , 1220(1992) [4] C. L. Kane and M. P. A. Fisher, Phys. Rev. B46