Feasibility of measurement-based braiding in the quasi-Majorana regime of semiconductor-superconductor heterostructures
Chuanchang Zeng, Girish Sharma, Tudor D. Stanescu, Sumanta Tewari
FFeasibility of measurement-based braiding in the quasi-Majorana regime ofsemiconductor-superconductor heterostructures
Chuanchang Zeng, Girish Sharma, Tudor D. Stanescu, and Sumanta Tewari Department of Physics and Astronomy, Clemson University, Clemson, SC 29634, USA School of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175005, India Department of Physics and Astronomy, West Virginia University, Morgantown, WV 26506, USA
We discuss the feasibility of measurement-based braiding in semiconductor-superconductor (SM-SC) het-erostructures in the so-called quasi-Majorana regime − the topologically-trivial regime characterized by ro-bust zero-bias conductance peaks (ZBCPs) that are due to partially-separated Andreev bound states (ps-ABSs).These low energy ABSs consist of component Majorana bound states (also called quasi-Majorana modes) thatare spatially separated by a length scale smaller than the length of the system, in contrast with the Majoranazero modes (MZMs) emerging in the topological regime, which are separated by the length of the wire. In thequasi-Majorana regime, the ZBCPs appear to be robust to various perturbations as long as the energy splittingof the ps-ABS is less than the typical width ε w of the low-energy conductance peaks ( ε w ∼ − µeV ).However, the feasibility of measurement-based braiding depends on a different, much smaller, energy scale ε m ∼ . µeV . This energy scale is given by the typical fermion parity-dependent ground state energy shiftdue to virtual electron transfer between the SM-SC system and a quantum dot used for parity measurements. Inthis paper we show that it is possible to prepare the SM-SC system in the quasi-Majorana regime with energysplittings below the ε m threshold, so that measurement-based braiding is possible in principle. However, despitethe apparent robustness of the corresponding ZBCPs, ps-ABSs are in reality topologically unprotected. Startingwith ps-ABSs with energy below ε m , we identify the maximum amplitudes of different types of (local) pertur-bations that are consistent with perturbation-induced energy splittings not exceeding the ε m limit. We argue thatmeasurements generating perturbations larger than the threshold amplitudes appropriate for ε m cannot realizemeasurement-based braiding in SM-SC heterostructures in the quasi-Majorana regime. We find that, if possibleat all, quantum computation using measurement-based braiding in the quasi-Majorana regime would be plaguedwith errors introduced by the measurement processes themselves, while such errors are significantly less likelyin a scheme involving topological MZMs. I. INTRODUCTION
Fault-tolerant quantum computation requires qubits that areprotected against quantum errors. Due to their non-Abeliantopological properties, Majorana zero modes (MZMs) havebeen proposed as an ideal platform for realizing topologicallyprotected qubits . Non-local encoding of quantum informa-tion using spatially separated MZMs makes the storing andprocessing of this information to be immune to local perturba-tions. Spin-orbit coupled semiconductor nanowires with prox-imity induced superconductivity were predicted theoreticallyto support MZMs in the presence of a Zeeman field . Inthis platform, MZMs arise as pairs of zero-energy excitationslocalized at the opposite ends of the nanowire. Braiding theseMZMs, which realizes the Clifford gates in a topologically-protected manner , could be implemented by tuning gatevoltages in a superconducting nanowire network or byperforming parity measurements . Fueled by this sig-nificant potential advantage over “standard” qubits, tremen-dous experimental progress has been made over the past fewyears in realizing topological superconductivity and Majoranamodes in one-dimensional SM-SC heterostructures . Themost recent significant development involves the observationof a quantized ZBCP plateau of height e / (cid:126) in a local chargetunneling measurement of a single topological nanowire .However, in other recent theoretical works it has beenshown that this type of signature, naturally associated withtopological MZMs, is possible even in a topologically triv-ial system due to the presence of so-called partially-separated Andreev bound states (ps-ABSs) or quasi-Majoranas . Ofcourse, gate-controlled braiding cannot be implemented us-ing ps-ABSs, which mimic most of the local phenomenologyof topological MZMs, because they do not obey non-Abelianstatistics.In contrast to gate-controlled braiding, measurement-basedbraiding consists of sequences of projective parity measure-ments of n MZMs ( n = 1 , , . . . ) and has the significantadvantage that it does not involve the actual physical move-ment of the Majorana modes . In the measurement-basedbraiding scheme, quantum information processing could berealized by joint parity measurement of pairs and quartets ofMZMs in the Coulomb-blockade regime. By coupling the(quasi) one-dimensional superconductors (SCs) hosting theMZMs to probing quantum dots, the ground state energy ofthe system is shifted and becomes fermion parity-dependent,which can be read out by suitable energy level spectroscopy.Compared to the braiding schemes based on physically ma-nipulating the Majoranas, the measurement-based braidingavoids serious engineering challenges involving fabricationand implementation and reduces possible thermal errors . InRef. [37] it has been suggested that the measurement-basedbraiding scheme could even be implemented using quasi-Majorana modes (i.e., ps-ABSs), by exploiting the fact thatthe component Majorana bound states (MBSs) of the ps-ABSshave exponentially different couplings to the external quan-tum dot. This is an exciting possibility that would mark a sig-nificant preliminary step toward the realization of a topologi-cal qubit. However, considering the non-topological nature of a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r the quasi-Majoranas, a detailed analysis of how sensitive theyare to local perturbations that may be generated during themeasurement process is indispensable for considering theirusefulness in implementing measurement-based braiding.The ability to perform projective parity measurements restson the controlled realization of non-local couplings to at leasta pair of MZMs . In this scheme, a quantum dot is cou-pled to multiple MZMs hosted by a SC island with nonzerocharging energy, which suppresses the actual transfer of elec-trons between the SC island and the quantum dot (QD). Thevirtual transfer of electrons between the island and the dot in-troduces an energy shift of the island-QD system, which isdependent on the fermion parity of the MZMs. By measuringthis ground state energy shift, e.g., via a frequency shift in atransmon-type measurement, the fermion parity of the MZMsystem can be identified. It can be shown that a sequence ofparity measurements of a group of MZMs is equivalent to aneffective braiding operation . Thus, the measurement se-quence realizes braiding without actually moving the MZMs.It is important to emphasize that the parity-dependent energysplittings of the island-QD ground state due to virtual electrontransfer are required to satisfy the readout condition, i.e, thecorresponding frequency shift should fall within the range ofsensitivity of the transmon-type measurements . There-fore, the feasibility of measurement-based braiding throughprojective parity measurements depends crucially on the ro-bustness of the “intrinsic” energy splittings associated withthe finite MZM overlap, which should not exceed a certainthreshold.In this paper, we explicitly examine the robustness of theenergy splitting of a Majorana wire in the quasi-Majoranaregime, i.e., when the near-zero energy modes are ps-ABSs,rather than topological MZMs. Here, by ps-ABSs we meanlow energy ABSs emerging in the topologically-trivial regimeand being characterized by component MBSs (i.e., quasi-Majorana modes) that are spatially separated by a lengthscale L ∗ (cid:46) ξ , with ξ being the SC coherence length. Bycontrast, topological MZMs are separated by the length ofnanowire L , hence their overlap is exponentially small (i.e.,of order e − L/ξ ). It has been shown recently that ps-ABSsare quite generic in SM-SC heterostructures and can pro-duce ZBCPs in local charge tunneling experiments that arerobust against various local perturbations, giving rise to quan-tized conductance plateaus similar to those generated by topo-logical MZMs . However, despite the apparent robust-ness of the ZBCPs, the ps-ABSs (or quasi-Majoranas) arenot topologically protected. Various local perturbations mayproduce ps-ABS energy splittings that are sufficient to makemeasurement-based braiding unfeasible, in spite of these split-tings not being observable in tunneling conductance experi-ments due to low energy resolution. For instance, any en-ergy splitting less than ε w ∼ − µeV will be consistentwith the observation of robust ZBCPs, but will not necessar-ily be consistent with measurement-based braiding, which re-quires energy splittings less than the typical fermion parity-dependent energy shift due to the coupling of the SC island toan external QD, ε m ∼ . µ eV. Moreover, even when the “un-perturbed” system is characterized by a quasi-Majorana split- ting below the required threshold, δ(cid:15) < ε m , the measurementitself could generate perturbations that enhance the splittingabove the required limit, i.e., the condition δ(cid:15) < ε m couldbreak down during the measurement. In this paper we explic-itly determine the dependence of the quasi-Majorana splittingon different types of local perturbations, including changes inthe spin-orbit coupling, Zeeman field, and effective potential.By comparing these results with the response of topologicalMZMs to similar perturbations, we show that the ps-ABSs areextremely sensitive to local perturbations, particularly thosethat affect the region where the component MBSs overlap. Weidentify the typical amplitudes of the perturbations that drivethe quasi-Majorana splittings above the the readout condition,making measurement-based braiding unfeasible.The remainder of this paper is organized as follows: In Sec.II we briefly describe the basic idea behind measurement-based braiding. In Sec. III we introduce the model Hamil-tonian [Eq. (6)] for the SM-SC heterostructures that can hosttopological MZMs as well as topologically trivial ps-ASBs.We also discuss the role of the confinement potential, effec-tive mass, and spin-orbit coupling strength in the emergenceof ps-ASBs. In Sec. IV, we introduce three different typesof local perturbations corresponding to small variations of thespin-orbit coupling [Eq. (11)], Zeeman field [Eq. (12)], andconfinement potential [Eq. (13)] and discuss the stability ofthe ps-ABSs in the presence of these perturbations. In Sec.V we consider an inhomogeneous system that suports quasi-Majoranas satisfying the measurement-based braiding condi-tion and estimate the maximum amplitudes of local perturba-tions that are consistent with this condition. For comparison,we also calculate the effect of these perturbations on topolog-ical MZMs emerging in the system at a higher value of theZeeman field. We end in Sec. VI with a summary of our find-ings and a discussion of their implications. II. MEASUREMENT-BASED BRAIDING
Recently proposed Majorana-based qubit architectures con-sist of parallel sets of topological superconducting wires con-nected by a regular superconductor that constitute a Coulombblockaded island hosting four (or six) MZMs . The finitecharging energy exponentially suppresses the quasi-particleexcitations due to electron transfer between the SC islandand the environment (i.e., quasiparticle poisoning processes),rendering the topological superconductor fermion parity-protected. For each pair ( j, k ) of MZMs we can define thefermion annihilation and creation operators c jk = ( γ j − iγ k ) and c † jk = ( γ j + iγ k ) , where γ i are Majorana operators sat-isfying the anticommutation rule { γ j , γ k } = 2 δ jk . The corre-sponding fermion number operator is n jk ≡ c † jk c jk = 12 (1 − iγ j γ k ) , (1)where P jk = iγ j γ k is the fermion parity operator of the MZMpair. Note that the sign of the γ i operators is a matter ofconvention, as it can be changed via a gauge transformation.Here, even fermion parity (i.e., the vacuum) corresponds tothe eigenvalue +1 of P jk , while odd parity corresponds to theeigenvalue − . The total parity of the qubit is fixed becauseof the finite charging energy and, for the simple case of fourMZMs, we assume P P = 1 , i.e., even total parity. Thecorresponding qubit states are || (cid:105)(cid:105) = | P = P = +1 (cid:105) , (2) || (cid:105)(cid:105) = | P = P = − (cid:105) . (3)When one exchanges a pair of MZMs, their associated oper-ators transform into each other, up to a phase. Intuitively, onecan understand this phase as being associated with a MZMcrossing branch cuts emanating from the topological defects(e.g., vortices) hosting the other MZMs, which are associatedwith π changes of the superconducting phase. Each MZMthat crosses such a branch cut will flip sign. Consider, forexample, exchanging the ( j, k ) MZM pair so that γ j crosses(once) the branch cut “carried” by γ k , while γ k does not crossany brach cut (or crosses an even number of times). As a re-sult of this exchange, we have γ j → − γ k and γ k → γ j . Thisbraiding operation can be represented using the unitary oper-ator R jk = (1 + γ j γ k ) / √ , as one can easily verify usingthe Majorana anticommutation relations: R jk γ j R † jk = − γ k , R jk γ k R † jk = γ j . In turn, the braiding operations associatedwith the exchange of MZM pairs can rotate the state of a qubitwithin a fixed total parity subspace. For example, applying R to the state || (cid:105)(cid:105) gives R || (cid:105)(cid:105) = 1 √ || (cid:105)(cid:105) − i || (cid:105)(cid:105) ) . (4)The braiding operations can be realized by physically mov-ing the MZMs, or, alternatively, by performing a sequenceof parity measurements . Consider the operator Π jk =(1 + iγ j γ k ) / − n jk that projects the ( j, k ) MZM pairinto the even fermion parity state (i.e., the vacuum). Usingthe Majorana anticommutation relations one can verify thatthe braiding transformation R corresponding to exchangingthe (1 , pair of MZMs can be implemented by the followingsequence of projections Π Π Π Π = 1 √ R ⊗ Π , (5)where the pair (3 , plays the role of an ancillary pair ofMZMs. For a six MZM qubit (a so-called hexon ), one canshow that the braiding transformations R and R providea sufficient gate set for generating all single-qubit Cliffordgates. Furthermore, a complete set of (multi-qubit) Cliffordgates requires only the additional ability to perform an en-tangling two-qubit Clifford gate between neighboring qubits,which can be implemented by a sequence of projective par-ity measurements on two and four MZMs from two hexons.Note that the projection into the even parity state described bythe operator Π jk corresponds to a parity measurement with anoutcome P jk = +1 . Of course, the outcome of a parity mea-surement is inherently probabilistic. However, one can obtainthe desired outcome using a “forced measurement” protocolinvolving a repeat-until-success strategy . In practice, a parity measurement can be realized by cou-pling the SC island hosting the Majorana modes to a quantumdot, which results in a measurable parity-dependent shift ofthe ground-state energy of the superconductor island-quantumdot system. This parity-dependent energy shift can be deter-mined experimentally using energy level spectroscopy, quan-tum dot charge measurements, or differential capacitancemeasurements. Considering, for example, energy level spec-troscopy, one can couple the MZM island-quantum dot systemto a superconducting transmission line resonator , which willgenerate a parity-dependent resonance frequency shift ∆ ω that can be detected using reflectometry . For realistic pa-rameters, the resonance frequency shift in this transmon-typemeasurement has been estimated as ∆ ω ∼ MHz. Con-sequently, to successfully implement this scheme, the trans-mon sensitivity must exceed
MHz, which limits the MZM(or the quasi-Majorana) energy splitting to δ(cid:15) (cid:46) . µ eV. Asufficiently small MZM energy splitting can be reached in thetopological regime by increasing the length of the nanowiresthat host the Majorana modes. On the other hand, the com-ponent MBSs of a ps-ABSs are separated by a length scale L ∗ (cid:46) ξ that cannot be easily controlled externally. Here,we show that the energy splitting of a ps-ABS can be madesufficiently small (i.e., less than ∼ MHz) by, e.g., en-suring that the confinement potential is sufficiently smooth.However, local perturbations introduced by disorder or by themeasurement process itself will typically increase the energysplitting of the component MBSs, possibly above the trans-mon sensitivity limit. Therefore, it crucial to identify the up-per limits of various types of local perturbations beyond whichmeasurement-based braiding in the quasi-Majorana regimedoes not work. Of course, a related problem concerns themagnitude of these perturbations in the topological regime,which establishes the feasibility of measurement-based braid-ing (and, ultimately, TQC) with MZMs. Note that the keycontrol parameter in the topological regime is the length L ofthe wire, while in the quasi-Majorana regime it is the (aver-age) slope of the confining potential. III. GENERAL PROPERTIES OF QUASI-MAJORANAS
In this section, we consider a perturbation-free quantumdot-semiconductor-superconductor (QD-SM-SC) heterostruc-ture, with the QD representing a short bare segment at the endof the SM wire (i.e., a segment that is not covered by the su-perconductor) where a tunnel gate potential is applied. Notethat an inhomogeneous effective potential is expected to nat-urally arise in the presence of a quantum dot (even withoutan applied gate potential) due to the mismatch of the workfunctions corresponding to the metallic lead/superconductorand the semiconductor wire . For such a system, we showthat ps-ABSs generally arise as the lowest energy states in thetopologically trivial regime and that the characteristic energysplittings of these ps-ABSs can be below the ε m threshold formeasurement-based braiding.A smooth confinement potential is commonly believedto be responsible for the emergence of near-zero-energyABSs in SM-SC heterostructures in the topologically trivialregime . Recently, it was shown that the topologicallytrivial near-zero-energy ABSs can also emerge in a proxim-itized wire coupled to a quantum dot , or in a finite-lengthKitaev chain attached to a QD with a position-dependent step-like potential . A summary of different types of effective po-tential that can induce topologically-trivial low-energy ABSscan be found in Ref. [38]. In this work, we focus on theemergence of the near-zero-energy ps-ABSs in a system char-acterized by a “flat top” Gaussian effective potential in the QDregion, as shown in Fig. 1(b).We start with a model Hamiltonian of the one-dimensionalQD-SM-SC hybrid system given by H = (cid:20) − (cid:126) m ∗ ∂ x − iα ( x ) ∂ x σ y − µ + V ( x ) (cid:21) τ z + Γ( x ) σ x + ∆( x ) τ x , (6)with m ∗ being the effective mass, µ the chemical potential, α ( x ) the spin-orbit coupling (SOC) strength, V ( x ) the con-finement potential, Γ( x ) the externally applied Zeeman field,and ∆( x ) the proximity-induced SC pairing potential. Here, σ i and τ j ( i = x, y, z ) are the Pauli matrices operating inthe spin and particle-hole spaces, respectively. The SOC, theinduced SC pairing, and the Zeeman field are, in general,position-dependent parameters. The consequences of havingposition-dependent effective parameters will be fully investi-gated in Sec. IV, where local perturbations of these parame-ters are considered; in this section we analyze a system withspatially uniform Γ and α . In addition, the inhomogeneousconfinement potential V ( x ) and the position-dependent pair-ing potential ∆( x ) are given by V ( x ) = V max × (cid:40) < x V ,e − ( x − xV )2 δx V if x V < x < L , (7) ∆( x ) = ∆ (cid:32) − e − ( x − x ∆)2 δx (cid:33) . (8)Here, x V defines the width of the “flat-top” region with po-tential V max and δx V describes the smoothness of the de-caying potential barrier. Similarly, x ∆ indicates the length ofthe bare SM region (i.e., the quantum dot) and δx ∆ controlsthe smoothness of ∆( x ) . By discretizing the model given byEq. (6) on a one-dimensional lattice (of lattice constant a ),we obtain the following tight-binding Bogoliubov-de GennesHamiltonian for the the QD-SM-SC structure: H BdG = (cid:88) i (cid:110) Ψ † i [(2 t − µ + V i ) τ z + Γ i σ x + ∆ i τ x ] Ψ i + (cid:104) Ψ † i +1 ( − tτ z + iα i σ y τ x ) Ψ i + h.c. (cid:105)(cid:111) , (9)where Ψ i = (cid:16) c i ↑ , c i ↓ , c † i ↑ , c † i ↓ (cid:17) T are Nambu spinors, with c † iσ ( c iσ ) being the electron creation (annihilation) operator at lat-tice site i . Note that, the position-dependence of the effectiveparameters Γ( x ) , V ( x ) , α ( x ) , and ∆( x ) is now reflected bythe corresponding site-dependence, Γ i , V i , α i , and ∆ i , based FIG. 1. (a) Dependence of the low energy spectrum on the appliedZeeman field for a system described by the Hamiltonian in Eq. (6)with chemical potential µ = 5∆ . The bulk gap has a minimumat Γ c ≈ µ = 5∆ , the critical field associated with the topologicalquantum phase transition. (b) Position-dependent pairing [see Eq.(8)] with ∆ = 0 . meV and effective potential profile [Eq. (7)]with V max = 8∆ . (c) Wave functions of the Majorana componentscorresponding to the ps-ABS marked by the green line in panel (a).The component MBSs are separated by a length scale L ∗ ∼ x ∆ ,the length of the bare SM segment. (d) MZM wave functions corre-sponding to the black line in panel (a). The parameters of the sys-tem are: m = 0 . m e , α ( x ) = α = 0 . eV ˚A, x V = 0 . µ m, δx V = 0 . µ m, δx ∆ = 0 . µ m , and x ∆ = 0 . µ m. on the correspondence x = ia . To obtain the low energy spec-tra and the wave functions of the system, we numerically di-agonalize the Hamiltonian in Eq. (9), i.e., we solve the eigen-value problem H BdG Φ α = E α Φ α .Consider a positive low energy solution E + = ε (cid:28) ∆ withthe eigenfunction Φ + ε ( i ) = ( u i ↑ , u i ↓ , v i ↑ , v i ↓ ) T . A corre-sponding negative energy solution E − = − ε with eigenfunc-tion Φ − ε ( i ) = ( v ∗ i ↑ , v ∗ i ↓ , u ∗ i ↑ , u ∗ i ↓ ) T is guaranteed by particle-hole symmetry. The linear combinations χ A = 1 √ + ε + Φ − ε ) ,χ B = i √ + ε − Φ − ε ) (10)are the corresponding wave functions in the Majorana repre-sentation, i.e., the component MBSs of the BdG states Φ ± ε .Note that the Majorana modes are eigenstates of H BdG onlyfor ε = 0 , while in general they satisfy (cid:104) χ n | H BdG | χ n (cid:105) = 0 and (cid:104) χ A | H BdG | χ B (cid:105) = iε . Throughout this paper we will useEq. (10) to express the near-zero-energy modes as a superpo-sitions of (partially overlapping) MBSs.An example of typical dependence of the low energy spec-trum on the applied Zeeman field for the QD-SM-SC hybridstructure described by the model Hamiltonian (6) is shownin Fig. 1(a). The red lines indicate the (localized) lowestenergy modes, while the blue lines represent bulk states. Atopological quantum phase transition (TQPT) from a topolog-ically trivial to topologically non-trivial phase is indicated bythe bulk gap (nearly) closing at a critical Zeeman field Γ c = (cid:112) ∆ + µ (here, µ = 5∆ and, consequently, Γ c ≈ ).The inhomogeneous effective potential V ( x ) and induced SCpairing potential ∆( x ) are schematically shown in Fig. 1(b).Note that in the presence of the non-uniform potential V ( x ) ,near-zero-energy states emerge within a considerable range ofZeeman field in the topologically trivial regime, Γ < Γ c , asshown in Fig. 1(a). To identify the nature of these low-energystates, we calculate the corresponding wave functions in theMajorana representation. The wave functions χ A and χ B cor-responding to the low-energy mode marked by the green linein Fig. 1(a), i.e. at Γ = 4∆ , are plotted in Fig. 1(c) as thered and yellow lines, respectively. Thus, the low-energy ABSmode can be represented as a pair of (partially) overlappingMBSs located in the quantum dot region – hence, its dubbingas a partially-separated Andreev bound states (ps-ABS). Notethat the two component MBSs are separated by a length scale(given by the distance between the main wave function max-ima) on the order of the QD length. By contrast, the MZMsemerging in the topological regime are separated by the lengthof the nanowire, as shown in Fig. 1(d) for the modes markedby the black line in Fig. 1(a) corresponding to Γ = 6∆ .In the topological regime, the energy splitting induced bythe overlap between the MZMs (which is always nonzero ina finite system) can be exponentially suppressed by increas-ing the length L of the nanowire, (cid:15) ≡ E ∼ e − L/ξ . There-fore, topological MZMs can always satisfy the requirement E (cid:46) ε m for measurement-based braiding in long-enoughwires (as long as the system is free of “catastrophic perturba-tions” that effectively cut the wire in several disjoint pieces).By contrast, the length scale L ∗ of the spatial separation be-tween the component MBSs of a ps-ABS is dictated by thedetails of the effective potential (e.g., by x V , x ∆ , and δx V inour modeling), which cannot be easily controlled. However,as we show explicitly below, one can identify (topologically-trivial) parameter regimes that satisfy the condition E (cid:46) ε m and, considering the rapid developments in the growth andfabrication of SM-SC hybrid devices, it may be possible toproduce topologically trivial ps-ABSs with energy splittingssmall enough to meet the braiding requirement. Note thatthroughout this paper E > will designate the energy of thelowest lying mode in both the trivial and topological regimes.It has been shown that the key parameter that determinesthe energy of the ps-ABS is the “average slope” of the effec-tive potential over a length scale ∼ L ∗ given by the separationof the MBS components. In turn, L ∗ depends not only on the FIG. 2. Dependence of the lowest energy E on the applied Zeemanfield ( Γ ) and the confining potential height ( V max ) for a system withchemical potential (a) µ = 3∆ and (b) µ = 5∆ . The other sys-tem parameters are the same as in Fig. 1. The green dashed lines( Γ / Γ c = 1 ) represent the critical Zeeman field associated with theTQPT. The red regions are consistent with the emergence of ZBCPsand correspond to energy splittings E = 40 −
80 m∆ (light red)and E = 0 . −
40 m∆ (dark red), where m∆ ≡ ∆ / . µ eV, while the light blue areas correspond to energy splittings E < . ∼ ε m , consistent with measurement-based braid-ing. The dark red and light blue regions are associated with robustZBCPs, while the light red areas may be associated with signaturesof zero-bias peak splitting in experiments with high-enough energyresolution. Note that low-energy modes emerge both in the topologi-cal regime ( Γ / Γ c > ), as well as in the topologically-trivial regime( Γ / Γ c > ). The insets show the field dependence of the lowestenergy E for V max = 0 and V max = 1 . µ . details of the effective potential, but also on control parame-ters such as the chemical potential and the Zeeman field. Toillustrate this property, we calculate the lowest energy E ofthe BdG Hamiltonian as a function of the Zeeman field Γ andthe quantum dot potential height V max for a fixed smoothnessparameter, δx V = 0 . µ m. The results are shown in Fig.2. The magnitude of the energy splitting is given by the colorcode and the energy units are m ∆ ≡ ∆ / . µ eV.The (vertical) green dashed lines indicate the critical Zeemanfield associated with the TQPT, i.e., Γ / Γ c = 1 . Note thatthe dark red region characterized by E (cid:46)
40 m∆ , whichsupports robust ZBCPs, extends throughout both the topo-logical ( Γ > Γ c ) and the trivial ( Γ < Γ c ) phases. Thelight red region corresponding to E = 40 −
80 m∆ mayshow signatures of zero-bias peak splitting in experimentswith high-enough energy resolution. The regions character-ized by E (cid:46) ε m ≈ . m ∆ (light blue), which could supportmeasurement-based braiding, are represented by a few narrow FIG. 3. (a) Lowest energy as a function of the effective mass ( m ∗ )and spin-orbit coupling strength ( α ) for a system with Γ = 4∆ < Γ c (i.e. topologically trivial). The color code and the unspecified sys-tem parameters are the same as in Fig. 2. Note that the light blueregion indicates the presence of trivial states with energy splitting E (cid:46) ε m , which, in principle, can be used for measurement-basedbraiding. Line-cuts with fixed values of the spin-orbit coupling( α = 0 . , .
30 eV ˚A) and effective mass ( m ∗ = 0 . , . m e ),are shown in (b) and (c), respectively. parameter windows inside the topological phase in Fig. 2(a)and a finite topologically-trivial area in Fig. 2(b). Of course,these regions can be expanded by appropriately varying thesystem parameters, e.g., increasing the length of the wire orthe smoothness parameter δx V . Regarding the dependenceon the chemical potential, we note that the system with largerchemical potential, µ = 5∆ , [see Fig. 2(b)] is character-ized by a larger region that supports ps-ABSs than the systemwith µ = 3∆ [see Fig. 2(a)]. On the other hand, increasingthe chemical potential increases the MZM energy splitting inthe topological phase, which can be attributed to the largercritical Zeeman fields required for accessing the topologicalregime. Finally, note that, for the confining potential modelconsidered in these calculations, the minimum Zeeman fieldassociated with the emergence of robust ZBCPs generated bytopologically-trivial ps-ABSs occurs at V max ∼ . µ and cor-responds to about . c ≈ . for µ = 5∆ . Again, thisminimum field can be further reduced down to Γ ∗ ∼ ∆ byconsidering, e.g., smoother confining potentials.More realistic modeling of experimentally-available SM-SC hybrid structures has to take into account additional ef-fects such as, for example, the proximity-induced renormal-ization of the effective mass , the gate potential-induced po-sition dependence of the spin-orbit coupling , or the inter-band coupling in systems with multi-band occupancy . We emphasize that many of these effects favor the emergence oftopologically-trivial low-energy states. Consider, for exam-ple, the enhancement of the effective mass due to proximity-coupling to the parent superconductor. In Fig. 3, we showthe dependence of the lowest energy [more specifically, of log( E / ∆) ] on the effective mass ( m ∗ ) and SOC strength ( α )for a system with Γ = 4∆ < Γ c , i.e., in the topologically-trivial phase. The color code and the (unspecified) systemparameters are the same as in Fig. 2. Note that most of theparameter space supports low-energy ps-ABSs with E (cid:46) ε w (red and blue areas), i.e., robust low-field ZBCPs emergingin the topologically trivial regime. Furthermore, there is aconsiderable area characterized by E (cid:46) ε m (light blue),i.e., consistent with measurement-based braiding of quasi-Majoranas. Horizontal and vertical line cuts are shown inpanels (b) and (c). In Fig. 3(b) we show the energy splitting, log( E / ∆) , as a function of the effective mass for two differ-ent SOC values, α = 0 .
15 and 0 .
30 eV ˚A. Also, the depen-dence of the energy splitting on the SOC strength for two val-ues of the effective masse ( m ∗ = 0 . m e and ( m ∗ = 0 . m e )is shown in Fig. 3(c). In general, higher values of the effectivemass allow a larger SOC range consistent with E (cid:46) ε m . Thetypical SOC strength within this range is α ∼ . − . , withsignificantly higher (or lower) strengths being associated withlarger energy splittings (above the measurement threshold).Based on the results discussed above and in agreement withsimilar theoretical results reported in recent years , weconclude that the emergence of low-energy ps-ABSs in thetopologically trivial phase (i.e., at low Zeeman fields) is quitegeneric in SM-SC nanowires coupled to quantum dots similarto the systems investigated experimentally. There is a signifi-cant parameter space region consistent with energy splittings E (cid:46)
40 m∆ , which would result in robust ps-ABS-inducedZBCPs. Furthermore, there are nonzero parameter regionsconsistent with topologically trivial ps-ABSs characterized byan energy scale E (cid:46) ε m ∼ . µ eV, as shown, e.g., inFig. 2(b) and Fig. 3(a). In principle, these low-energy ps-ABSs (or quasi-Majoranas) could enable measurement-basedbraiding . The basic idea is that one of the component MBSsof the ps-ABS [e.g., the “red” Majorana mode in Fig. 1(c)] ischaracterized by an exponentially larger coupling to an end-of-the-wire probe than its partner (e.g., the “yellow” quasi-Majorana). Combined with a sufficiently low energy splitting, E < ε m , this would enable measurement-based braiding .Since the two quasi-Majorana modes have a substantial spatialoverlap, the key questions are: (i) How robust are the quasi-Majoranas against local perturbations (e.g., disorder, differenttypes of inhomogeneity associated with position-dependentpotentials, etc.) inherent in real, less-than-ideal systems orgenerated by the measurement process itself? (ii) What is themaximum amplitude of a given type of perturbation consistentwith the measurement-based braiding condition, E < ε m ?These questions will be examined in the next sections. Wenote that in practice the “robustness” in question (i) could in-volve either the energy scale (cid:15) w (in the context of differen-tial conductance measurements, when it implies robustness ofobserved ZBCPs), or the energy scale ε m (in the context ofmeasurement-based braiding, when it refers to the feasibil- FIG. 4.
Top : Spatial profile of the position-dependent spin-orbit cou-pling α ( x ) described by Eq. 11. The blue line represents the bulkvalue α of the SOC strength. Middle : Spatial profile of the position-dependent Zeeman field Γ( x ) . The magenta line represents the uni-form (bulk) Zeeman field Γ , while the gray line represents anotherpossible profile Γ( x ) consistent with Eq. 12. Bottom : Spatial profileof the effective potential perturbation δV ( x ) (orange area) given byEq. 13. The confining potential V ( x ) and the induced pairing ∆( x ) are the same as in Fig. 1(b). ity of this scheme with quasi-Majoranas). We emphasize thatthese energy scales differ by two orders of magnitude. IV. STABILITY OF QUASI-MAJORANAS IN THEPRESENCE OF LOCAL PERTURBATIONS
Partially separated Andreev bound states, or quasi-Majoranas, can mimic the local behavior of topological Majo-rana zero modes, including the generation of robust zero-biasconductance features in a tunneling experiment and the π -Josephson effect . The non-topological quasi-Majoranamodes are even considered suitable for measurement-basedbraiding, as long as the corresponding energy splittings arebelow a certain energy scale ε m . However, it is importantto emphasize that, in contrast to the topological MZMs char-acterized by a spatial separation given by the length of thenanowire, the component Majorana modes of a ps-ABSs aretypically separated by a distance on the order of the lengthscale of the quantum dot, rendering the ps-ABSs topologicallyunprotected against local perturbations. In this section, weconsider three types of local perturbations affecting the quan-tum dot region near the end of the wire, as shown schemat-ically in Fig. 4: local variations of the spin-orbit coupling,local variations of the applied Zeeman field, and local per- turbations of the effective potential. These perturbations canbe viewed as representing either realistic features that haveto be incorporated into the model to accound for character-istics of actual devices, or possible perturbations induced bythe measurement process itself (e.g., in a braiding-type ex-periment). We note that each type of perturbation is char-acterized by a spatial profile (see Fig. 4) and an amplitude(strength). The perturbations that are actually relevant fora given device could be determined by a detailed modelingof the structure; here, we focus on the qualitative aspects ofthe problem, which are expected to be generic. We first in-vestigate the effect of these perturbations on the near-zero-energy states emerging in the topologically trivial phase (sec-tions IV A, IV B, and IV C), then we discuss the limits on theperturbation strength consistent with the ps-ABSs being suit-able for measurement-based braiding (Sec. V). A. Perturbation from step-like spin-orbit coupling
In most of the theoretical calculations, the SOC strength isconsidered to be independent of the position along the wire.However, in the presence of inhomogeneous gate potentialsand position-dependent work function differences (e.g., in aquantum dot region consisting of a wire segment not coveredby the parent superconductor), a non-uniform SOC is possi-ble, even likely. A key question concerns the fate of ps-ABSsin the presence of position-dependent spin-orbit coupling. Re-cently, it has been shown that a step-like SOC (near the endof the wire) can lead to decaying oscillations of the energysplitting as a function of the Zeeman field . To explore theeffect of such inhomogeneity on the ps-ABSs, we consider aposition-dependent SOC of the form α ( x ) = α + α (cid:104) tanh (cid:18) x − x α δx α (cid:19) − (cid:105) , (11)where α is the “bulk” value of the SOC strength and α char-acterizes the suppression near the end of the wire (i.e., in thequantum dot region), with α = α − α representing thestrength of the suppressed SOC. The parameters x α and δx α describe the length scale and the smoothness of the pertur-bation, respectively. A schematic representation of the non-uniform SOC is provided in the top panel of Fig. 4. The effectof the spatially varying SOC defined by Eq. (11) on the ps-ABBs emerging in a QD-SM-SC system is investigated below.First, we consider a system with chemical potential µ =5∆ , quantum dot potential V max = 6∆ , and uniform SOC α = 0 . ˚A ( α = 0 ) and calculate the dependence ofthe low energy spectrum on the Zeeman splitting. The re-sults are shown in Fig. 5(a). Note the robust near-zero en-ergy modes that emerge in the topologically trivial region( Γ < Γ c ∼ ), which are characterized by a typical en-ergy splitting smaller than that of the topological Majoranamodes ( Γ > Γ c ). Next, we switch on the SOC inhomogene-ity described by Eq. (11) and consider a complete suppres-sion of the spin-orbit coupling near the end of the wire, i.e., α = α → α = 0 . As shown in Fig. 5(b), the ps-ABS FIG. 5. Low-energy spectrum as a function of Zeeman field for aQD-SM-SC system with (a) constant spin-orbit coupling α and (b)position-dependent spin-orbit coupling α ( x ) with a profile as shownin Fig. 4(a). Note that the position-dependent SOC generates largeenergy splitting oscillations. The wave function profiles associatedwith the marked lines are given in Fig. 6. The system parametersare µ = 5∆ , m ∗ = 0 . m e , α = 0 . ˚A, ∆ = 0 . meV, x ∆ = 0 . µ m, δx ∆ = 0 . µ m and the confinement potential ischaracterized by x V = 0 . µ m, δx V = 0 . µ m, and V max = 6∆ .The parameters for the position-dependent SOC α ( x ) described byEq. 11 are x α = 0 . µ m, δx α = 0 . µ m, α = 0 . A similarperturbation is applied at the right end of the system. mode is now characterized by large energy splitting oscilla-tions, while the low-energy Majorana mode in the topologi-cal regime ( Γ > ) is only weakly affected. Of course,the effect of the local perturbation on the topological Majo-rana mode could be further reduced by increasing the lengthof the wire. By contrast, the inhomegeneous SOC affects theps-ABS locally, practically independent of the wire length.The Majorana wave function profiles corresponding to thelow-energy states marked by lines in Fig. 5, i.e., the ps-ABSs at Γ = 3 . (green and cyan lines) and the MBSsat Γ = 6 . (black and orange lines), are shown in Fig. 6,along with their corresponding energy splittings. Note thatthe dramatic increase of the ps-ABS energy splitting in thepresence of the SOC inhomogeneity is not accompanied bya major change of the wave function profiles. The relevantchange [see Fig. 6(b)] involves the “yellow” MBS developinga weak oscillatory “tail” within the suppressed SOC region, x (cid:46) x α = 0 . µ m, where the overlap with the “red” MBSwas nearly zero in the uniform SOC case [see Fig. 6(a)]. Bycontrast, the change of the topological MBS wave function FIG. 6. (a) and (b) Majorana wave functions associated with the near-zero energy ps-ABSs marked by the green and cyan lines in Fig. 5(a),and Fig. 5(b), respectively. (c) and (d) wave function profiles for thetopological MBSs marked by the black and orange lines in Fig. 5(a),and Fig. 5(b), respectively. Note that the energy splitting of the ps-ABS is strongly affected by the suppression of the SOC strength atthe end of the wire, from E = 6 . in the system with uniformSOC (a) to E = 48 . in the system with a step-like SOC(b), while the change of the corresponding component MBS wavefunction profiles is rather modest. profiles does not significantly affect the MBS overlap, hencethe energy splitting. Furthermore, this overlap (and the cor-responding energy splitting) can be arbitrarily reduced (e.g.,below the characteristic measurement-based braiding energyscale ε m ) by increasing the length of the wire.An important corollary of our discussion related to Fig. 5is that the stability of ps-ABSs against local perturbations ofthe spin-orbit coupling cannot be directly assessed based onthe dependence of the unperturbed low-energy spectrum onthe Zeeman field. This is in sharp contrast with the behaviorof topological MZMs, when a lower value of the energy split-ting E implies better separated and, implicitly, more robustMZMs. As a consequence, measurement-based braiding us-ing quasi-Majoranas becomes rather problematic, as the per-turbation induced by the measurement itself could result in thetwo component MBSs becoming too strongly coupled. Con-sider, for example, a system similar to that discussed above,but having a larger effective mass, m ∗ = 0 . m e , a smallerSOC strength, α = 0 . ˚A, and a smoother confining poten-tial, δx V = 0 . µ m. The dependence of the correspondinglow-energy spectrum on the Zeeman field is shown in Fig.7(a). Note that the energy splitting associated with the ps- FIG. 7. Low-energy spectrum as a function of Zeeman field for asystem with (a) constant SOC and (b) step-like SOC. The system pa-rameters are m ∗ = 0 . m e , α = 0 . ˚A, ∆ = 0 . meV, µ =6∆ , V max = 8∆ , x V = 0 . µ m, δx V = 0 . µ . The position-dependent SOC is described by Eq. (11) with x α = 0 . µ m, dx α = 0 . µ m, and α /α = 0 . . The Majorana wave functionscorresponding to the ps-ABSs marked by green and cyan lines inpanels (a) and (b) are shown in (c) and (d), respectively. Note thatthe local perturbation generates a huge increase of the ps-ABS en-ergy splitting, as well as a manifest enhancement of the componentMBS overlap. ABSs at a Zeeman field
Γ = 4∆ [green line in Fig. 7(a)] issufficiently small to satisfy the requirement for measurement-based braiding, E = 0 . ≤ ε m ∼ . . Also, com-paring the corresponding Majorana wave functions shown inFig. 7(c) with those in Fig. 6(a) suggests that the lower energysplitting is associated with a larger separation (i.e., lower over-lap) of the MBS components. However, this seemingly “ro-bust” ps-ABS is strongly affected by a a step-like SOC pertur-bation, as revealed by the low-energy spectrum shown in Fig.7(b). Moreover, the “perturbed” wave functions shown in Fig.7(d) confirm our previous observation that the main change in- FIG. 8. Energy E of topologically-trivial ps-ABSs as functionof the SOC strength, α , and the relative amplitude of the step-like SOC perturbation near the end of the wire, α /α , for a sys-tem with Γ = 4∆ < Γ c (i.e. topologically trivial) and dif-ferent values of the effective mass and the smoothness parameter δx α : (a) m ∗ = 0 . m e , δx α = 0 . µ m, (b) m ∗ = 0 . m e , δx α = 0 . µ m, (c) m ∗ = 0 . m e , δx α = 0 . µ m, and (d) m ∗ = 0 . m e , δx α = 0 . µ m. The other system parameters andthe color code are the same as in Fig. 2. Positive (negative) valuesof the parameter α /α correspond to a suppressed (enhanced) SOCwithin the quantum dot region. In general, the SOC inhomogene-ity leads to an increase of the ps-ABS energy splitting. Note that,while the observation of robust (topologically trivial) ZBCPs is pos-sible within a large range of parameters (dark red and cyan areas),the only significant region corresponding to the braiding condition E < ε m (cyan) occurs in (d). duced by the perturbation is the development of an oscillatory“tail” within the suppressed SOC region, x (cid:46) x α = 0 . µ m,where the overlap of the component MBSs was nearly zero inthe uniform SOC case.These example suggest that suppressing the spin-orbit cou-pling in the quantum dot region quickly destabilizes the quasi-Majorana modes, which acquire a finite energy splitting. Tobetter evaluate the effect of the perturbation, we expand theenergy splitting map from Fig. 3(a) along the “direction” α /α corresponding to the strength of the step-like SOC per-turbation [see Eq. 11]. More specifically, we consider cutscorresponding to two different values of the effective mass, m ∗ = 0 . m e and m ∗ = 0 . m e , for a system with position-dependent SOC described by Eq. 11 with x α = 0 . µ mand two different values of the smoothness parameter, δx α =0 . µ m and δx α = 0 . µ m. The results are shown in Fig.8. Note that the system is characterized by low-energy ps-ABSs consistent with the observation of robust zero-bias con-ductance peaks over a significant range of parameters (dark0red and cyan regions). However, local variations of the thespin-orbit coupling (within the quantum dot region) typicallyenhances the energy splitting E of the ps-ABSs. Conse-quently, the quasi-Majoranas satisfy the braiding condition E < ε m (cyan areas) within a substantial (connected) pa-rameter region only for the conditions corresponding to panel(d), i.e. large effective mass and smooth step-like SOC. Weemphasize that measurement-based braiding using Majoranaor quasi-Majorana modes is feasible only if the system can betuned within a finite (large-enough) domain of the the multi-dimensional space of relevant control and perturbation param-eters characterized by E < ε m . In the case of topologi-cal MZMs, this domain is relatively “isotropic”, in the sensethat enhancing its characteristic “length” scale along one di-rection (e.g., the applied Zeeman field) ensures the expansionof the domain in all directions, including those correspond-ing to local perturbation parameters. This property is a di-rect manifestation of the topological protection enjoyed bythe MZMs. By contrast, the near-zero-energy quasi-Majoranadomain is highly “anisotropic”, the apparent robustness withrespect to some parameters (e.g., Zeeman field and chemi-cal potential) being accompanied by a high susceptibility withrespect to certain local perturbations (e.g., the local suppres-sion/enhancement of SOC). B. Perturbation from a position-dependent Zeeman field
The proximity-coupled semiconductor-superconductor het-erostructure is driven into the topological phase when the ex-ternal Zeeman field parallel to the nanowire (or, more gener-ally, perpendicular to the effective SOC field) exceeds a cer-tain critical value Γ c ( µ ) = (cid:112) ∆ + µ . In this section, we in-vestigate the effect of a local, position-dependent perturbationof the Zeeman field on the energy splitting of topologicallytrivial ps-ABSs. We note that variations of the magnetic fieldnear the end of the wire are expected due to screening by theparent superconductor. Furthermore, the effective g-factor inthe quantum dot region could differ significantly from the g-factor in the segment of the wire covered by the superconduc-tor as a result of the proximity-induced renormalization of thisparameter , which results in a local variation of the Zee-man field. To investigate the effect of a local variation of theZeeman field (within the quantum dot region), we consider thefollowing phenomenological model of a position-dependenteffective Zeeman field Γ( x ) = (cid:2) Γ + Γ sin( ωx ) (cid:3) Θ( x Γ − x ) + ΓΘ( x − x Γ ) , (12)with Γ = Γ − Γ being the value of the Zeeman field at x = 0 and Γ being the field in the absence of the perturba-tion. The parameters /ω and x Γ determine the characteristiclength scales of the perturbation. A schematic representationof the position-dependent Zeeman field described by Eq. (12)is shown in the middle panel of Fig. 4.We start with a QD-SM-SC system in the presence of auniform Zeeman field, Γ( x ) = Γ , hosting robust near-zeroenergy states even in the trivial regime, Γ < Γ c ∼ , asshown in Fig. 9(a). Next, we perturb the Zeeman field in the FIG. 9. Dependence of the low-energy spectrum on the applied Zee-man field for a system with (a) uniform Zeeman field, (b) position-dependent Zeeman field given by Eq.(12) with Γ / Γ = 0 . and ω = − π/ , and (c) position-dependent Zeeman field with Γ / Γ = 0 . and ω = 4 π/ . The position-dependent Zeemanfields are shown in Fig. 4 (middle panel). The Majorana wavefunctions corresponding to the ps-ABSs marked by colored lines areshown in Fig. 10. The chemical potential of the system is µ = 5∆ and the confinement potential is characterised by x V = 0 . µ m, δx V = 0 . µ m and V max = 8∆ . The other parameters are thesame as in Fig. 7. quantum dot region by considering a profile Γ( x ) given byEq.(12) with Γ / Γ = 0 . and ω = − π/ (correspond-ing to the black line in the middle panel of Fig. 4). As aconsequence, the energy splitting associated with the trivialps-ABS increases strongly, while the topological MBS modesare weakly affected, as shown in Fig. 9(b). The same behaviorcharacterizes Fig. 9(c), which represents another example ofperturbed low energy spectrum corresponding to Γ = 0 . and ω = π/ (gray line in the middle panel of Fig. 4).The Majorana wave functions corresponding to the ps-ABSsmarked by lines (at Γ = 4∆ ) in Fig. 9 are shown in Fig. 10.As a result of locally perturbing the Zeeman field, the energyof the ps-ABS increases dramatically from E = 0 .
93 m∆ in1
FIG. 10. Majorana wave functions associated with the near-zero en-ergy ABS modes marked by colored lines in Fig. 9 and the corre-sponding energies. Note that the presence of the local perturbation[panels (b) and (c)] results in a reduced separation (i.e., enhancedoverlap) of MBS components, which generates larger energy split-tings.
Fig. 10(a) to E = 45 m∆ and E = 42 m∆ in panels (b) and(c), respectively. This increase of the energy splitting is dueto an enhancement of the overlap of the corresponding MBScomponents. Note that in the topological regime the perturba-tion affects significantly the wave function of the MBS local-ized near the quantum dot (not shown), but has a weak effecton its overlap with the MBS localized at the opposite end ofthe system. As a result, the energy splitting of the topologicalMajorana modes is weakly affected by the local perturbation,as evident in Fig. 9. Furthermore, this effect can be arbitrarilyminimized by increasing the length of the system, which isnot the case for the ps-ABS.The effect of the perturbation can be understood qualita-tively as a reduction of the wire segment within which the“topological” condition Γ( x ) ≥ Γ c ( x ) is (locally) satisfied,which, in turn, is a result of suppressing the Zeeman field nearthe end of the system. Consequently, the spatial separationof the component MBSs of the emerging ps-ABS decreases,as revealed by the wave functions in Fig. 10, and the higheroverlap results in larger values of the energy splitting. It isnatural to suspect that, perhaps, enhancing the Zeeman fieldnear the end of the system would lower the characteristic en-ergy of the ps-ABS. To test this insight, we calculate the en-ergy splitting E as a function of the applied Zeeman field, Γ / Γ c , and the amplitude of the local perturbation describedby Eq. (12), Γ / Γ . The results, corresponding to two val-ues of the effective mass and two perturbation profiles (seethe middle panel of Fig. 4) are shown in Fig. 11. Remark-ably, a moderate local increase of the Zeeman field can sta-bilize the ps-ABSs, generating (in certain conditions) signif-icant simply connected parameter regions consistent with the FIG. 11. Energy of topologically-trivial ps-ABSs as function of theapplied Zeeman field, Γ / Γ c , and the amplitude of the local perturba-tion described by Eq. (12), Γ / Γ , for a system with different valuesof the effective mass and different perturbation profiles: (a) m ∗ =0 . m e , ω = − π/ , (b) m ∗ = 0 . m e , ω = 4 π/ , (c) m ∗ = 0 . m e , ω = − π/ , (d) m ∗ = 0 . m e , ω = 4 π/ .The other system parameters and the color code are the same as inFig. 2. The yellow dotted line marks the topological phase bound-ary corresponding to Γ = Γ c . Note that, in general, suppressing theZeeman field in the quantum dot region, Γ / Γ > , enhances theenergy splitting of the ps-ABS, while locally increasing the Zeemanfield can stabilize the low-energy (topologically trivial) modes. Formoderate enhancement of the Zeeman field in the quantum dot re-gion, the system with m ∗ = 0 . m e supports a large (connected)region consistent with the braiding condition, E < ε m [cyan areasin (c) and (d)]. braiding condition E < ε m (cyan areas in Fig. 11(c) and(d)]. Also remarkable is the fact that, in the regime charac-terized by Γ / Γ < (i.e., locally-enhanced Zeeman field)and Γ / Γ c < (i.e., topologically-trivial regime), the param-eter regions in Fig. 11(c) and Fig. 11(d) characterized byrobust ps-ABS-induced ZBCPs (dark red and cyan) are com-parable with those defined by the braiding condition (cyan).We note that, in practice, a local increase of the Zeeman fieldin the quantum dot region could be associated with a locally-enhanced value of the effective g-factor. C. Local potential perturbation
As a final example, we consider the effect of perturbationsdue to local potentials on the energy of topologically trivialps-ABSs. For concreteness, we consider a Gaussian-like po-tential perturbation localized near x = x , where x is a point2 FIG. 12. Dependence of the low-energy spectrum on the appliedZeeman field for a QD-SM-SC system with position-dependent ef-fective potential described by Eq. (7) and (a) no local perturba-tion, δV ( x ) = 0 , or (b) local perturbation given by Eq. (13) with δV /V max = 0 . , x = 0 . µ m, and δx = 0 . µ m. Note thatlocal perturbation enhances the characteristic energy of the topolog-ically trivial low-energy mode, but does not affect significantly theenergy splitting of the topological Majorana mode. The Majoranawave functions corresponding to the ps-ABSs marked by the greenand cyan lines in (a) and (b) are shown in panels (c) and (d) respec-tively. Clearly, the local potential perturbation enhances the overlapof the component MBSs. The system parameters are µ = 5 . , m ∗ = 0 . m e , α = 0 .
15 eV ˚A, V max = 8∆ , and x ∆ = 0 . within the quantum dot region. Specifically, we have δV ( x ) = δV exp (cid:20) − ( x − x ) δx (cid:21) , (13)with δV being the amplitude of the potential perturbation and δx representing its characteristic width. A schematic repre-sentation of the local potential is given in the bottom panelof Fig. 4. As in the previous sections, we first consider anunperturbed system that supports low-energy (topologicallytrivial) ps-ABSs, then we apply the local perturbation – heredescribed by δV ( x ) , with δV = 0 . V max and x = 0 . µ m FIG. 13. Energy of topologically-trivial ps-ABSs as function of theapplied Zeeman field, Γ / Γ c , and the amplitude of the local potentialperturbation described by Eq. (13), δV /V max , for a system with dif-ferent values of the effective mass and different characteristic widthsof the perturbing potential: (a) m ∗ = 0 . m e , δx = 0 . µ m, (b) m ∗ = 0 . m e , δx = 0 . µ m, (c) m ∗ = 0 . m e , δx = 0 . µ m,(d) m ∗ = 0 . m e , δx = 0 . µ m. The other system parameters andthe color code are the same as in Fig. 2. The yellow dotted line marksthe topological phase boundary corresponding to Γ = Γ c . Note that,the condition for observing robust ZBCPs in the topologically triv-ial regime is weakly dependent on the perturbing potential (dark redand cyan regions with Γ / Γ c < ). The system with m ∗ = 0 . m e supports a large (connected) region consistent with the braiding con-dition, E < ε m [cyan areas in (c) and (d)]. – and determine its effect on the low-energy modes. The de-pendence of the corresponding low-energy spectra on the ap-plied Zeeman field is shown in Fig. 12. Similar to the pertur-bations studied above, the local variation of the effective po-tential leads to an enhancement of the characteristic ps-ABSenergy [see Fig. 12(b)]. The Majorana wave functions corre-sponding to the unperturbed ps-ABS marked by the green linein Fig. 12(a) are shown in Fig. 12(c). Note that the compo-nent MBSs are fairly well separated, consistent with the lowenergy splitting, E = 5 .
02 m∆ . By contrast, the correspond-ing wave functions in the presence of the potential perturba-tion, which are shown in Fig. 12(d), are characterized by alarge overlap, consistent with the increased energy splitting, E = 46 . . Note that the local potential perturbationdoes not visibly affect the energy splitting of the Majoranamodes in the topological regime.To gain a more complete understanding of the effect of thelocal potential perturbation on the trivial low-energy states,we follow the stratedy used in the previous sections and cal-culate the energy E of the lowest energy mode as function3of the applied Zeeman field and the perturbation amplitude, δV /V max . Explicitly, we consider four distinct cases corre-sponding to two values of the effective mass, m ∗ = 0 . m e and m ∗ . m e , and two characteristic widths of the pertur-bation potential, δx = 0 . µ m and δx = 0 . µ m. Theresults are shown in Fig. 13. Our first observation is that thesystem supports low-energy ps-ABSs in the presence of a lo-cal potential perturbation, with a significant parameter rangeconsistent with the observation of (topologically trivial) robustZBCPs (dark red and cyan regions with Γ / Γ c < ). This is anindication that, within and energy window E (cid:46) ε w , the ps-ABSs are relatively insensitive to the details of the effectivepotential in the quantum dot region. Combined with our find-ings from the previous sections, this suggests that the observa-tion of low field, topologically trivial ZBCPs is rather generic.On the other hand, satisfying the condition for measurement-based braiding, E < ε m , depends of the details of the ef-fective potential, e.g., on the sign of the perturbing potential δV . Nonetheless, the system characterized by a large effec-tive mass, which implies a short MBS characteristic lengthscale, has a large (connected) parameter region consistent withps-ABS braiding [cyan areas in Fig. 13(c) and (d)]. Finally,we note that the MBSs emerging in the topological regime( Γ / Γ c > ) typically do not satisfy the braiding condition be-cause of the relatively short length of the wire. Of course, inlonger wire this condition will be satisfied, regardless of thelocal potential in the quantum dot region, provided the sys-tem is uniform enough, e.g., it does not contain “catastrophicperturbations”– bulk perturbations that effectively “cut” thewire into disconnected topological segments. V. AMPLITUDES OF LOCAL PERTURBATIONSCONSISTENT WITH MEASUREMENT-BASED BRAIDING
In the previous sections we have shown that topologicallytrivial ps-ABSs emerging generically in a QD-SM-SC het-erostructure at Zeeman fields below the critical value corre-sponding to the topological quantum phase transition are sen-sitive to local variations of the system parameters, e.g., thelocal effective potential, Zeeman field, and spin-orbit cou-pling strength. Here, we focus on an inhomogeneous sys-tem that supports a ps-ABS satisfying the braiding condition, E < ε m , and evaluate the maximum amplitudes of localperturbations and random disorder potentials that are consis-tent with this condition. This will provide a quantitative es-timate of the susceptibility of topologically trivial ps-ABSsto local perturbations. For comparison, we also calculate thecorresponding variation of the energy splitting associated withtopological MZMs and show that, for a long enough wire, thisvariation does not break the braiding condition.One important feature that characterizes both the ps-ABSsand the topological MZMs is their oscillatory behavior asfunction of the applied Zeeman field. As a consequence, theenergy splitting E (Γ) corresponding to a specific value Γ ofthe Zeeman field provides incomplete information regardingthe robustness of the low-energy mode. In particular, E (Γ) can be made arbitrarily small by moving close to a node, FIG. 14. Position-dependent profiles of the effective potential V ( x ) [see Eq. (7)], pairing potential ∆( x ) [Eq. (8)], Zeeman field Γ( x ) [Eq. (15)], and spin-orbit coupling α ( x ) [Eq. (11)] characteriz-ing an inhomogeneous QD-SM-SC system. The system parame-ters are: L = 2 µ m (wire length), m = 0 . m e (effective mass), ∆ = 0 . meV (induced bulk pairing), µ = 5∆ (chemical po-tential), α = 200 meV ˚A(bulk spin-orbit coupling) and Γ = 4 . (bulk Zeeman field); the position-dependent profiles correspond to V max = 8∆ , x V = 0 . µ m, δx V = 0 . µ m, x ∆ = 0 . µ m, δx ∆ = 0 . µ m, α = 0 . α , x α = 0 . µ m, δx α = 0 . µ m, Γ = − . , x Γ = 0 . µ m, and δx Γ = 0 . µ m. which, of course, does not imply that the corresponding low-energy mode is robust. To better characterize the robustnessof the low-energy mode, we propose the quantity (cid:104) E (cid:105) rep-resenting the average energy splitting over a small range ofZeeman fields, (cid:104) E (cid:105) = 12 δ Γ (cid:90) Γ+ δ ΓΓ − δ Γ d Γ (cid:48) E (Γ (cid:48) ) , (14)where the range δ Γ is determined by the characteristic “wave-length” of the energy splitting oscillations.For concreteness, we consider a QD-SM-SC heterostruc-ture with an inhomogeneous quantum dot region described byan effective potential given by Eq. (7), a pairing potentialprofile described by Eq. (8), a step-like spin-orbit couplingcorresponding to Eq. (11), and a position-dependent Zeemanfield given by Γ( x ) = Γ + Γ (cid:18) tanh x − x Γ δx Γ − (cid:19) , (15)where Γ is the bulk value of the Zeeman field and Γ charac-terizes the suppression (if Γ > ) or enhancement (if Γ < )of the field inside the quantum dot region. The specific valuesof the parameters and the corresponding position-dependentprofiles are given in Fig. 14. Note that the chemical potentialis µ = 5∆ , hence the critical Zeeman field is Γ c ≈ . .Therefore, an applied (bulk) Zeeman field Γ = 4 . , asspecified in the caption of Fig. 14, corresponds to the topo-logically trivial regime. To investigate the properties of thetopological MZMs we will chose Γ = 5 . .4 FIG. 15. Lowest energy mode of the unperturbed QD-SM-SC Majo-rana structure as a function of the (bulk) Zeeman field for two differ-ent wire lengths, L = 2 µ m (top) and L = 3 . µ m (bottom). Theother system parameters are the same as in Fig. 14. The dependence of the lowest energy mode on the appliedZeeman field corresponding to two different wire lengths isshown in Fig. 15. The following remarks are warranted.First, we note that in the topologically-trivial regime ( Γ < Γ c ≈ . ) the low-energy spectrum is practically inde-pendent on the length of the wire. This is a clear indica-tion of the local nature of the ps-ABS responsible for thelow-energy mode. By contrast, the MZM corresponding to Γ > Γ c has a strong (exponential) dependence on the lengthof the wire, the energy splitting oscillations decreasing byabout two orders of magnitude as L increases from µ m to . µ m. Second, the amplitude of the energy splitting oscil-lations associated with the topologically trivial ps-ABS de-creases with the Zeeman field, while the amplitude of thetopological MZM increases with Γ . Third, we notice that the“wavelength” of the MZM energy splitting oscillations cor-responding to L = 3 . µ m (when the system satisfies thebraiding condition in the topological regime) is about . .Consequently, we calculate the characteristic energy splitting (cid:104) E (cid:105) using Eq. (14) with δ Γ = 0 . . The inhomogeneoussystem described by the parameters given in Fig. 14 supportsa trivial ps-ABS with (cid:104) E (cid:105) ≈ . m ∆ , which is below thethreshold ε m ∼ . m ∆ consistent with measurement based braiding. Hence, the (unperturbed) inhomogeneous QD-SM-SC system described above supports topologically-trivial ps-ABSs localized near the quantum dot region that could enablemeasurement-based braiding.Next, we address the critical question regarding the ro-bustness of the low-energy ps-ABS against local perturba-tions. Specifically, we consider the following perturbationsaffecting the QD region. i) Variations of the local effectivepotential corresponding to V max → V max + δV max in Eq.(7). ii) Local variations of the Zeeman field correspondingto Γ → Γ + δ Γ in Eq. (15). iii) Local changes of thespin-orbit coupling corresponding to α → α + δα in Eq.(11). The effects of these local perturbation on the char-acteristic energy splitting (cid:104) E (cid:105) of the quasi-Majorana modeare shown in the top panels of Fig. 16. The topologically-trivial parameter regions consistent with the braiding condi-tion (cid:104) E (cid:105) < ε m ≈ . m ∆ are represented by the black areasin panels (a-c). In general, relatively small variations of thelocal parameters (inside the quantum dot region) away fromthe “unperturbed” values given in Fig. 14 drive the systemoutside the regime consistent with measurement-based braid-ing of quasi-Majoranas. For example, panels (a) and (c) revealthat the system can tolerate variations δV max of the effectivepotential within a typical window δV max ≈ µ eV. Notethat δV max is about of the effective potential V max insidethe quantum dot region. Measurement-based braiding is notpossible in the presence of perturbations (e.g., induced by themeasurement process itself) characterized by δV max outsidethis window, as the corresponding characteristic energy split-ting (cid:104) E (cid:105) becomes larger than ε m . Similarly, panels (a) and(b) show that local perturbations of the spin-orbit couplingstrength δα consistent with the braiding condition have to bewithin a typical window δα ≈ − meV ˚A, which corre-sponds to . − of the bulk spin-orbit coupling α , whilepanels (b) and (c) show that the local variations of the Zeemanfield, δ Γ , should be within a typical window δ Γ ≈ µ eV,corresponding to about of the bulk Zeeman field value.These results suggest that, while measurement-based braid-ing using quasi-Majorana modes is possible in principle, itrequires very precise control of the local parameters, whichhave to be tuned (and maintained) within fairly narrow win-dows. In particular, this imposes strict constraints on themaximum amplitudes of the local perturbations induced bythe measurement process itself. Furthermore, the specific ex-ample discussed in this section assumes a relatively large ef-fective mass, m = 0 . m e . Reducing this value results inthe rapid collapse of the parameter windows consistent withmeasurement-based braiding (see Figs. 8, 11, and 13). Weemphasize that the ps-ABS energy splittings shown in the toppanels of Fig. 16 are about two orders of magnitude smallerthan the characteristic energy (cid:15) w ∼ − µ eV associatedwith the observation of robust zero bias peaks over the entirerange of perturbations explored here. In other words, the sys-tem is characterized by a (topologically trivial) ZBCP that isextremely robust against local perturbations, yet it is not nec-essarily suitable (or, at least, it is not ideal) for measurement-based braiding.For comparison, we have also calculated the characteris-5 FIG. 16. Dependence of the characteristic energy splitting (cid:104) E (cid:105) /m ∆ on local perturbations within the QD region for a (topologically-trivial)quasi-Majorana mode (top panels) and a topological MZM (bottom panels). The parameters of the unperturbed system ( δV max = δ Γ = δα = 0 ) are given in Fig. 14. The quasi-Majorana (ps-ABS) mode is consistent with measurement-based braiding if (cid:104) E (cid:105) < . ,which corresponds to the black regions in the top panels. The typical widths of the black regions represent . − of the bulk values ofthe corresponding parameters (see the main text). By contrast, the characteristic energy splitting of the topological MZM is way below thebraiding threshold ε m over the entire range of local perturbations. tic energy splitting of topological MZMs for a system oflength L = 3 . µ m and a value of the (bulk) Zeeman field Γ = 5 . meV (all other parameters being the same as inFig. 14) in the presence of the same type of local perturba-tions. The results are shown in the lower panels of Fig. 16.Note that (cid:104) E (cid:105) is way below the measurement-based braidinglimit ε m over the entire perturbation range explored here. Thisis a direct consequence of the topological protection that theMZMs enjoy, unlike their quasi-Majorana counterparts. Wenote that the characteristic energy (cid:104) E (cid:105) of the MZM dependsstrongly (exponentially) on the length of the system, as clearlyillustrated in Fig. 15. If the system is long-enough, the MZMis practically immune against local perturbations that do noteffectively break the system into disjoint topological regions.To further emphasize the difference between thetopologically-induced robustness of the MZMs againstlocal disorder and the relative fragility of the quasi-Majoranas(ps-ABSs), we consider the inhomogeneous QD-SM-SCsystem described by the parameters given in Fig. 14 inthe presence of a random onsite potential V ( i ) = V dis ζ i ,where i labels the lattice sites, V dis is the amplitude of therandom potential, and ζ i is a site-dependent random numberbetween − and +1 . Note that, unlike the local perturbationsconsidered above, which were localized within the QDregion, the random potential V ( i ) is defined throughoutthe entire system, including the region near the moddle ofthe wire where the MZMs have an exponentially-small,but finite overlap. To evaluate the effect of disorder on theenergy splitting, we average the characteristic splitting (cid:104) E (cid:105) FIG. 17. Disordered averaged splitting energy (cid:104)(cid:104) E (cid:105)(cid:105) as a functionof disorder strength. The length of the wire is L = 3 . µ m, while therest of the parameters are the same as in Fig. 14. We note that the ps-ABS states are sensitive to even a small amount of disorder, in sharpcontrast with the Majorana modes. The red shaded area indicates theregime apt for measurement based braiding. defined by Eq. (14) over 100 disorder realizations. Thedependence of the disorder-averaged characteristic energy (cid:104)(cid:104) E (cid:105)(cid:105) on the amplitude V dis of the random potential isshown in Fig. 17. While the MZM is practically unaf-fected by weak disorder, the characteristic energy of the6quasi-Majorana (ps-ABS) exceeds the braiding threshold ε m even in the presence of a random potential with anamplitude V dis representing only of the induced gap. Ofcourse, this is a direct consequence of the ps-ABSs not beingtopologically protected, but the rather small values of V dis consistent with measurement-based braiding re-emphasizethe difficulty of practically realizing conditions consistentwith braiding of quasi-Majoranas. Finally, we note that, forthe disorder strengths considered in Fig. 17, the characteristicenergy (cid:104)(cid:104) E (cid:105)(cid:105) of the ps-ABS is still well below the limit (cid:15) w associated with the observation of robust ZBCPs. Again,the robustness of the observed ZBCP provides no relevantinformation regarding the feasibility of measurement-basedbraiding.As a final comment, we point out that the local perturba-tions considered in this study do not include “catastrophicperturbations” that effectively cut the wire into disjoint (pos-sibly topological) segments. In the presence of such perturba-tions, the “topological” regime will be characterized by thepresence of multiple pairs of MBSs distributed throughoutthe system and characterized by separation lengths that arecontrolled by the concentration of catastrophic perturbations,rather than the size of the system. Since the characteristicMZM energy depends critically on the separation length (see,e.g., Fig. 15), a concentration of catastrophic local perturba-tions in excess of one per several microns may completelyprohibit the realization of (topological) measurement-basedBraiding with MZMs. By contrast, if the concentration ofthese perturbations is not too high, ps-ABSs emerging near theend of the wire (e.g., inside a QD region) are weakly affectedby their presence in the bulk of the system, as demonstratedby the weak size-dependence of the ps-ABS mode in Fig.15. Nonetheless, while these quasi-Majoranas can produceextremely robust zero-bias conductance peaks, they are nottopologically protected; using them for measurement-basedbraiding is possible, in principle, but requires fine tuning andexquisite control of the local parameters. VI. DISCUSSION AND CONCLUSIONS
In this paper, we have investigated the feasibility ofmeasurement-based braiding using quasi-Majorana modesemerging in the quantum dot region of a quantum dot-semiconductor-superconductor (QD-SM-SC) structure. Wehave shown that such modes, which represent the Majoranacomponents of a partially-separated Andreev bond state (ps-ABS), emerge rather generically in this type of system atZeeman fields below the critical value associated with thetopological quantum phase transition (TQPT), i.e. in thetopologically-trivial phase, and we have investigated in detailtheir behavior in the presence of local perturbations, such aslocal variations of the effective potential, spin-orbit coupling,and Zeeman field in the quantum dot region and random dis-order potentials.The robustness of the quasi-Majorana (ps-ABS) modes canbe evaluated based on two different experimentally-relevantcriteria: (i) the ability to generate robust zero-bias conduc- tance peaks (ZBCPs) in a charge tunneling experiment and(ii) the ability to generate energy splittings that do not exceeda certain threshold that enables measurement-based braiding.According to criterion (i), the quasi-Majorana mode is robustif its characteristic energy splitting is less that the character-istic width of a ZBCP, (cid:15) w ∼ µ eV, while criterion (ii) in-volves an energy scale determined by the parity-dependent en-ergy shift due to the coupling of (quasi-) Majorana modes toexternal quantum dots, ε m ∼ . µ eV. The key observationis that the two energy scales differ by about two orders ofmagnitude. Consequently, robustness with respect to criterion(i) – the ability to generate robust ZBCPs – does not implyrobustness with respect to criterion (ii), hence suitability formeasurement-based braiding.Considering these observations and based on the resultsof our detailed numerical analysis, we can formulate thefollowing conclusions. (1) In a QD-SM-SC system theemergence of near-zero-energy ps-ABSs (quasi-Majoranas) israther generic, with these modes satisfying criterion (ii), i.e.having characteristic energies E < ε w , over large rangesof system parameters (see Figs. 2, 11, and 13). Practically,the low-field region of the topological phase diagram is domi-nated by topologically trivial ps-ABSs that are virtually indis-tinguishable from topological Majorana zero-energy modes(MZMs) under local probes. The only systematic qualita-tive difference between the trivial and the nontrivial modesis that the energy oscillations of the ps-ABSs typically de-cay with the Zeeman field, while the amplitude of the MZMsincreases with Γ (see Figs. 5, 9, 12, and 15, as well asRefs. ). (2) The quasi-Majoranas (ps-ABSs) are not topo-logically protected and, consequently, they are susceptible tolocal perturbations. This susceptibility to local perturbationshas to be judged differently with respect to criteria (i) and (ii).While within an energy resolution ε w the quasi-Majoranasare as robust to local perturbations as the genuine topologi-cal MZMs, with respect to the measurement-based braidingcriterion they are rather fragile, unlike the MZMs (see Figs.16 and 17). Furthermore, while the robustness of MZMs is“isotropic” – robustness with respect to one type of perturba-tion guaranties robustness with respect to other types of lo-cal perturbations, the stability of quasi-Majoranas is highlyanisotropic. For example, the quasi-Majorana mode analyzedin Fig. 16 can tolerate, according to criterion (ii), varia-tions up to of the local potential, but only up to ofthe (bulk) SOC strength. (3) From the perspective of cri-terion (ii) – i.e., the feasibility of measurement-based braid-ing – the quasi-Majoranas are highly susceptible to relativelyweak local perturbations, while the topological MZMs aresusceptible to rare “catastrophic perturbations”, i.e. pertur-bations that effectively cut the wire into disjointed topolog-ical regions. This suggests two possible near-term paths to-ward the demonstration of measurement-based braiding withMajorana modes. The topological route , based on MZMs,can lead to a genuine topological qubit, but has to overcomethe requirement of no-catastrophic-perturbation over possiblymulti-micron length scales. The poor man’s route , based onquasi-Majoranas, can significantly relax the no-catastrophic-perturbation requirement, but involves exquisite control of the7local properties of the system. Furthermore, it imposes dras-tic limits on the local perturbations induced by the measure-ment process itself. Realistically, this route cannot be success-ful based on spontaneously-produced quasi-Majoranas, whichare ubiquitous within an energy window ∼ ε w , but are uselessfor measurement-based braiding; if successful, this route hasto involve a systematic effort to design and control the localproperties of the system near the end of the wire. VII. ACKNOWLEDGMENTS
C. Z. and S. T. acknowledge support from ARO Grant No.W911NF-16-1-0182. G. S. acknowledges IIT Mandi startupfunds. T.D.S. acknowledges NSF DMR-1414683. A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires,”
Physics Uspekhi , vol. 44, p. 131, Oct 2001. S. B. Bravyi and A. Y. Kitaev, “Fermionic quantum computation,”
Annals of Physics , vol. 298, no. 1, pp. 210 – 226, 2002. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma,“Non-abelian anyons and topological quantum computation,”
Rev.Mod. Phys. , vol. 80, pp. 1083–1159, Sep 2008. J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, “Genericnew platform for topological quantum computation using semi-conductor heterostructures,”
Phys. Rev. Lett. , vol. 104, p. 040502,Jan 2010. J. D. Sau, S. Tewari, R. M. Lutchyn, T. D. Stanescu, andS. Das Sarma, “Non-abelian quantum order in spin-orbit-coupledsemiconductors: Search for topological majorana particles insolid-state systems,”
Phys. Rev. B , vol. 82, p. 214509, Dec 2010. S. Tewari, J. D. Sau, and S. D. Sarma, “A theorem for the exis-tence of majorana fermion modes in spinorbit-coupled semicon-ductors,”
Annals of Physics , vol. 325, no. 1, pp. 219 – 231, 2010.January 2010 Special Issue. R. M. Lutchyn, J. D. Sau, and S. Das Sarma, “Majoranafermions and a topological phase transition in semiconductor-superconductor heterostructures,”
Phys. Rev. Lett. , vol. 105,p. 077001, Aug 2010. J. Alicea, “Majorana fermions in a tunable semiconductor device,”
Phys. Rev. B , vol. 81, p. 125318, Mar 2010. Y. Oreg, G. Refael, and F. von Oppen, “Helical liquids and ma-jorana bound states in quantum wires,”
Phys. Rev. Lett. , vol. 105,p. 177002, Oct 2010. T. D. Stanescu, R. M. Lutchyn, and S. Das Sarma, “Majoranafermions in semiconductor nanowires,”
Phys. Rev. B , vol. 84,p. 144522, Oct 2011. J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher,“Non-Abelian statistics and topological quantum information pro-cessing in 1D wire networks,”
Nature Physics , vol. 7, pp. 412–417, May 2011. S. Das Sarma, M. Freedman, and C. Nayak, “Majorana ZeroModes and Topological Quantum Computation,” arXiv e-prints ,p. arXiv:1501.02813, Jan. 2015. D. Litinski and F. von Oppen, “Quantum computing with majo-rana fermion codes,”
Phys. Rev. B , vol. 97, p. 205404, May 2018. J. D. Sau, S. Tewari, and S. Das Sarma, “Universal quantum com-putation in a semiconductor quantum wire network,”
Phys. Rev.A , vol. 82, p. 052322, Nov 2010. J. D. Sau, D. J. Clarke, and S. Tewari, “Controlling non-abelianstatistics of majorana fermions in semiconductor nanowires,”
Phys. Rev. B , vol. 84, p. 094505, Sep 2011. D. J. Clarke, J. D. Sau, and S. Tewari, “Majorana fermion ex-change in quasi-one-dimensional networks,”
Phys. Rev. B , vol. 84,p. 035120, Jul 2011. B. I. Halperin, Y. Oreg, A. Stern, G. Refael, J. Alicea, and F. vonOppen, “Adiabatic manipulations of majorana fermions in a three- dimensional network of quantum wires,”
Phys. Rev. B , vol. 85,p. 144501, Apr 2012. D. Aasen, M. Hell, R. V. Mishmash, A. Higginbotham, J. Danon,M. Leijnse, T. S. Jespersen, J. A. Folk, C. M. Marcus, K. Flens-berg, and J. Alicea, “Milestones toward majorana-based quantumcomputing,”
Phys. Rev. X , vol. 6, p. 031016, Aug 2016. M. A. Nielsen and I. L. Chuang, “Programmable quantum gatearrays,”
Phys. Rev. Lett. , vol. 79, pp. 321–324, Jul 1997. M. A. Nielsen, “Quantum computation by measurement andquantum memory,”
Physics Letters A , vol. 308, no. 2, pp. 96 –100, 2003. L. Fu, “Electron teleportation via majorana bound states in amesoscopic superconductor,”
Phys. Rev. Lett. , vol. 104, p. 056402,Feb 2010. P. Bonderson, “Measurement-only topological quantum computa-tion via tunable interactions,”
Phys. Rev. B , vol. 87, p. 035113, Jan2013. S. Vijay and L. Fu, “Teleportation-based quantum informationprocessing with majorana zero modes,”
Phys. Rev. B , vol. 94,p. 235446, Dec 2016. L. P. Rokhinson, X. Liu, and J. K. Furdyna, “The fractional a.c.Josephson effect in a semiconductor-superconductor nanowireas a signature of Majorana particles,”
Nature Physics , vol. 8,pp. 795–799, Nov 2012. V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M.Bakkers, and L. P. Kouwenhoven, “Signatures of majoranafermions in hybrid superconductor-semiconductor nanowire de-vices,”
Science , vol. 336, no. 6084, pp. 1003–1007, 2012. A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrik-man, “Zero-bias peaks and splitting in an Al-InAs nanowire topo-logical superconductor as a signature of Majorana fermions,”
Na-ture Physics , vol. 8, pp. 887–895, Dec 2012. M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, andH. Q. Xu, “Anomalous Zero-Bias Conductance Peak in a Nb-InSbNanowire-Nb Hybrid Device,”
Nano Letters , vol. 12, pp. 6414–6419, Dec 2012. M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon, M. Leijnse,K. Flensberg, J. Nyg˚ard, P. Krogstrup, and C. M. Marcus, “Majo-rana bound state in a coupled quantum-dot hybrid-nanowire sys-tem,”
Science , vol. 354, no. 6319, pp. 1557–1562, 2016. S. M. Albrecht, A. P. Higginbotham, M. Madsen, F. Kuemmeth,T. S. Jespersen, J. Nyg˚ard, P. Krogstrup, and C. M. Marcus, “Ex-ponential protection of zero modes in Majorana islands,”
Nature(London) , vol. 531, pp. 206–209, Mar 2016. J. Chen, P. Yu, J. Stenger, M. Hocevar, D. Car, S. R. Plissard, E. P.A. M. Bakkers, T. D. Stanescu, and S. M. Frolov, “Experimentalphase diagram of zero-bias conductance peaks in superconduc-tor/semiconductor nanowire devices,”
Science Advances , vol. 3,no. 9, 2017. H. Zhang, ¨O. G¨ul, S. Conesa-Boj, M. P. Nowak, M. Wimmer,K. Zuo, V. Mourik, F. K. de Vries, J. van Veen, M. W. A. de Moor, J. D. S. Bommer, D. J. van Woerkom, D. Car, S. R.Plissard, E. P. A. M. Bakkers, M. Quintero-P´erez, M. C. Cas-sidy, S. Koelling, S. Goswami, K. Watanabe, T. Taniguchi, andL. P. Kouwenhoven, “Ballistic superconductivity in semiconduc-tor nanowires,”
Nature Communications , vol. 8, p. 16025, Jul2017. H. Zhang, C.-X. Liu, S. Gazibegovic, D. Xu, J. A. Logan,G. Wang, N. van Loo, J. D. S. Bommer, M. W. A. de Moor,D. Car, R. L. M. Op Het Veld, P. J. van Veldhoven, S. Koelling,M. A. Verheijen, M. Pendharkar, D. J. Pennachio, B. Shojaei, J. S.Lee, C. J. Palmstrøm, E. P. A. M. Bakkers, S. D. Sarma, and L. P.Kouwenhoven, “Quantized Majorana conductance,”
Nature (Lon-don) , vol. 556, pp. 74–79, Apr 2018. C. Moore, C. Zeng, T. D. Stanescu, and S. Tewari, “Quantizedzero-bias conductance plateau in semiconductor-superconductorheterostructures without topological majorana zero modes,”
Phys.Rev. B , vol. 98, p. 155314, Oct 2018. C.-X. Liu, J. D. Sau, T. D. Stanescu, and S. Das Sarma, “An-dreev bound states versus majorana bound states in quantumdot-nanowire-superconductor hybrid structures: Trivial versustopological zero-bias conductance peaks,”
Phys. Rev. B , vol. 96,p. 075161, Aug 2017. F. Setiawan, C.-X. Liu, J. D. Sau, and S. Das Sarma, “Elec-tron temperature and tunnel coupling dependence of zero-biasand almost-zero-bias conductance peaks in majorana nanowires,”
Phys. Rev. B , vol. 96, p. 184520, Nov 2017. C. Moore, T. D. Stanescu, and S. Tewari, “Two-terminal chargetunneling: Disentangling majorana zero modes from partially sep-arated andreev bound states in semiconductor-superconductor het-erostructures,”
Phys. Rev. B , vol. 97, p. 165302, Apr 2018. A. Vuik, B. Nijholt, A. R. Akhmerov, and M. Wimmer, “Repro-ducing topological properties with quasi-Majorana states,”
Sci-Post Phys. , vol. 7, p. 61, 2019. T. D. Stanescu and S. Tewari, “Robust low-energy andreev boundstates in semiconductor-superconductor structures: Importance ofpartial separation of component majorana bound states,”
Phys.Rev. B , vol. 100, p. 155429, Oct 2019. F. L. Pedrocchi and D. P. DiVincenzo, “Majorana braiding withthermal noise,”
Phys. Rev. Lett. , vol. 115, p. 120402, Sep 2015. T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hast-ings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M.Marcus, and M. H. Freedman, “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with majo-rana zero modes,”
Phys. Rev. B , vol. 95, p. 235305, Jun 2017. P. Bonderson, M. Freedman, and C. Nayak, “Measurement-onlytopological quantum computation,”
Phys. Rev. Lett. , vol. 101,p. 010501, Jun 2008. T. Hyart, B. van Heck, I. C. Fulga, M. Burrello, A. R. Akhmerov,and C. W. J. Beenakker, “Flux-controlled quantum computationwith majorana fermions,”
Phys. Rev. B , vol. 88, p. 035121, Jul 2013. C. Knapp, M. Zaletel, D. E. Liu, M. Cheng, P. Bonderson, andC. Nayak, “The nature and correction of diabatic errors in anyonbraiding,”
Phys. Rev. X , vol. 6, p. 041003, Oct 2016. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schus-ter, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J.Schoelkopf, “Charge-insensitive qubit design derived from thecooper pair box,”
Phys. Rev. A , vol. 76, p. 042319, Oct 2007. J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson,J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wall-raff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf,“Coupling superconducting qubits via a cavity bus,”
Nature (Lon-don) , vol. 449, pp. 443–447, Sept. 2007. G. Kells, D. Meidan, and P. W. Brouwer, “Near-zero-energy endstates in topologically trivial spin-orbit coupled superconductingnanowires with a smooth confinement,”
Phys. Rev. B , vol. 86,p. 100503, Sep 2012. E. Prada, P. San-Jose, and R. Aguado, “Transport spectroscopyof ns nanowire junctions with majorana fermions,” Phys. Rev. B ,vol. 86, p. 180503, Nov 2012. T. D. Stanescu and S. Tewari, “Nonlocality of zero-bias anomaliesin the topologically trivial phase of majorana wires,”
Phys. Rev. B ,vol. 89, p. 220507, Jun 2014. C. Zeng, C. Moore, A. M. Rao, T. D. Stanescu, and S. Tewari,“Analytical solution of the finite-length kitaev chain coupled to aquantum dot,”
Phys. Rev. B , vol. 99, p. 094523, Mar 2019. T. D. Stanescu and S. Das Sarma, “Proximity-induced low-energyrenormalization in hybrid semiconductor-superconductor majo-rana structures,”
Phys. Rev. B , vol. 96, p. 014510, Jul 2017. Z. Cao, H. Zhang, H.-F. L¨u, W.-X. He, H.-Z. Lu, and X. C. Xie,“Decays of majorana or andreev oscillations induced by steplikespin-orbit coupling,”
Phys. Rev. Lett. , vol. 122, p. 147701, Apr2019. J. Chen, B. D. Woods, P. Yu, M. Hocevar, D. Car, S. R. Plissard,E. P. A. M. Bakkers, T. D. Stanescu, and S. M. Frolov, “Ubiqui-tous non-majorana zero-bias conductance peaks in nanowire de-vices,”
Phys. Rev. Lett. , vol. 123, p. 107703, Sep 2019. B. D. Woods, J. Chen, S. M. Frolov, and T. D. Stanescu, “Zero-energy pinning of topologically trivial bound states in multi-band semiconductor-superconductor nanowires,”
Phys. Rev. B ,vol. 100, p. 125407, Sep 2019. C.-K. Chiu and S. Das Sarma, “Fractional josephson effectwith and without majorana zero modes,”
Phys. Rev. B , vol. 99,p. 035312, Jan 2019. W. S. Cole, S. Das Sarma, and T. D. Stanescu, “Effects oflarge induced superconducting gap on semiconductor majoranananowires,”
Phys. Rev. B , vol. 92, p. 174511, Nov 2015. G. Sharma, C. Zeng, T. Stanescu, and S. Tewari, “Majo-rana versus Andreev bound state energy oscillations in a 1Dsemiconductor-superconductor heterostructure,” arXiv e-printsarXiv e-prints