Reproducing topological properties with quasi-Majorana states

Andreev bound states in hybrid superconductor-semiconductor devices can have near-zero energy in the topologically trivial regime as long as the confinement potential is sufficiently smooth. These quasi-Majorana states show zero-bias conductance features in a topologically trivial phase, mimicking spatially separated topological Majorana states. We show that in addition to the suppressed coupling between the quasi-Majorana states, also the coupling of these states across a tunnel barrier to the outside is exponentially different for increasing magnetic field. As a consequence, quasi-Majorana states mimic most of the proposed Majorana signatures: quantized zero-bias peaks, the $4\pi$ Josephson effect, and the tunneling spectrum in presence of a normal quantum dot. We identify a quantized conductance dip instead of a peak in the open regime as a distinguishing feature of true Majorana states in addition to having a bulk topological transition. Because braiding schemes rely only on the ability to couple to individual Majorana states, the exponential control over coupling strengths allows to also use quasi-Majorana states for braiding. Therefore, while the appearance of quasi-Majorana states complicates the observation of topological Majorana states, it opens an alternative route towards braiding of non-Abelian anyons and protected quantum computation.


I. INTRODUCTION
One-dimensional topological superconductors support Majorana bound states with zero energy at its endpoints. [1][2][3][4] Because of their non-Abelian exchange statistics and their topological protection to local sources of error, Majorana states are candidates for fault-tolerant qubits in quantum computing. 5,6 In addition to their non-Abelian properties, Majorana states have local signatures, namely 4π-periodicity of the supercurrent in a topological Josephson junction, 7,8 and a quantized zerobias conductance peak in the tunneling spectroscopy of a single topological wire. [9][10][11] Because of the complexity of a braiding experiment demonstrating the non-Abelian statistics, experimental efforts so far focus on observing the local Majorana signatures. [12][13][14] An alternative explanation of the experimental observations is Andreev states with near-zero energy that appear in the topologically trivial phase. [15][16][17] These Andreev states can form at the wire's end, provided the confinement potential is sufficiently smooth. 15 Because smooth confinement potentials are likely to appear due to the separation between metallic gates and nanowires by dielectric layers, these quasi-Majorana states became a focus of recent theoretical research. [18][19][20][21][22][23] In particular, Ref. 18 shows that in case of smooth confinement potentials, trivial zero-bias conductance peaks are commonly appearing in Majorana devices, Ref. 21 demonstrates that nearzero energy Andreev bound states which are partially separated in space can reproduce quantized zero-bias conductance peaks, and Ref. 23 shows that such partially separated states can reproduce the fractional Josephson effect.
We demonstrate that quasi-Majorana states can be either partially separated or spatially fully overlapping, but in both cases these states have an approximately opposite spin. Because quasi-Majorana states are spinpolarised, the couplings across a smooth tunnel barrier within the WKB approximation are equal to: with E the energy, V pot the potential energy, E Z the Zeeman energy, and w 1,2 the spin-dependent width of the tunnel barrier. The ratio of the tunnel probabilities is where γ 1,2 = w1,2 −w1,2 |p 1,2 (x)|dx. Therefore, when the tunnel barrier is smooth and Zeeman splitting is sufficiently large, the quasi-Majorana couplings are exponentially different with the area of the barrier.
Such an exponential difference of the couplings, combined with the exponentially small coupling between the quasi-Majorana states, makes quasi-Majorana states indistinguishable from topological, spatially separated Majorana states, as we illustrate in Fig. 1. Because one of the two quasi-Majorana states is exponentially decoupled from the outside for increasing magnetic field in this regime, any local measurement will give the same result as for a truly topological system. We verify this phenomenon by analyzing tunneling spectroscopy of a quasi-Majorana device, the 4π-periodic Josephson effect, 24,25 and a coupled quantum dot-nanowire system, which has recently been proposed 26,27 and used 28 to measure Majorana nonlocality. Because the exponential suppression of Majorana couplings requires a tunnel barrier, we then analyze the open regime and identify a quantized zero-bias conductance dip instead of a peak as a distinctive feature of topological Majoranas. Couplings in the quasi-Majorana regime with two quasi-Majoranas, located at the tunnel barrier slope and with a suppressed coupling EM, experience a different effective barrier due to their opposite spin and the finite magnetic field. The spin-down quasi-Majorana state (dark yellow) couples strongly with coupling strength Γ1 (thick blue arrow, blue effective barrier), the quasi-Majorana state with spin-up (faint yellow) couples weakly with coupling strength Γ2 (thin orange arrow, orange effective barrier).
Because of the exponentially small coupling between quasi-Majoranas and of one quasi-Majorana across a barrier, a smooth tunnel barrier is an alternative approach to addressing individual Majorana states. As a consequence, braiding schemes can also be realized in a topologically trivial phase with quasi-Majoranas, since braiding effectively requires the coupling to a single (quasi-)Majorana state. Therefore, quasi-Majorana states supply an alternative route towards braiding non-Abelian anyons for quantum computing. This paper is organized as follows. In Sec. II, we describe our model and a method to compute coupling strengths. In Sec. III, we discuss quasi-Majorana phase diagrams, wave functions, and couplings across a tunnel barrier. Sec. IV describes quasi-Majorana effects on a coupled quantum dot-nanowire device and on a Josephson junction. We investigate an alternative local measurement in Sec. V and briefly discuss probing a bulk topological phase transition rather than local (quasi-)Majorana modes. To study quasi-Majorana states beyond a simple one-dimensional model, we show in Sec. VI a phase diagram in a 3D nanowire with a smooth potential barrier. In Sec. VII, we discuss braiding with quasi-Majoranas. We give a summary and outlook in Sec. VIII.

A. Hamiltonian
We implement the minimum one-dimensional model, as proposed in Refs. 25 and 29, with a Bogoliubov-De Gennes Hamiltonian given by with m * the effective mass, p x = −i ∂ x the momentum, µ the chemical potential, V pot the potential, α the spin-orbit interaction (SOI) strength, ∆ the superconducting gap and E Z the Zeeman energy due to a parallel magnetic field. The Pauli matrices σ i and τ i (i = x, y, z) act in spin and particle-hole space, respectively. The potential V pot and the position dependence of the superconducting gap ∆(x) vary for different devices, as specified in the following subsection. We choose the following parameter values of the Hamiltonian (3): m * = 0.015m e , corresponding to an InSb nanowire, α = 50 meV nm, and ∆ = 0.5 meV, unless specified otherwise.

B. Devices
We implement the Hamiltonian (3) in three different devices, schematically shown in Fig. 2, that are used to measure local Majorana signatures. The system of Fig. 2(a) is a tunnel spectroscopy setup consisting of a proximitized nanowire of length L SC with a chemical potential µ and constant superconducting gap ∆ connected on the left to a semi-infinite normal lead via a potential barrier V pot (x). The potential in this device is given by a Gaussian-shaped barrier, V pot = V barrier , with with V the height, x 0 = 0 the center and σ the smoothness of the potential barrier. Figure 2(b) shows the second system, a coupled quantum dot-nanowire device, which has been proposed recently as an additional tool for measuring the nonlocality of Majorana states. 26,27 Compared to the setup of Fig. 2(a), we replace the lead by a normal quantum dot (∆ = 0) of length L dot with hard-wall boundary conditions at x = 0. The effective potential is V pot = V barrier + V dot , with V barrier as given in Eq. (4) and V dot describing the chemical potential difference between the dot and the nanowire: with µ dot the chemical potential in the quantum dot, x 0 = L dot the interface between dot and nanowire, and dx the length scale over which the chemical potential varies. This potential allows for different local chemical potentials in the dot and the wire, for example due to local gates in an experiment. Finally, we consider a Josephson junction, consisting of two one-dimensional proximitized nanowires separated by a potential barrier and with a phase difference ϕ, see Fig. 2(c). In this device, V pot = V barrier , with the center of the potential barrier between both superconductors (x 0 = L SC ), and the position-dependent superconducting gap described by with a phase difference ϕ across the junction. In all devices, we fix the nanowire length to L SC = 3 µm. In the coupled quantum dot -nanowire device, we take a quantum dot length of L dot = 250 nm.
In this work, we focus on quasi-Majorana states formed at the monotonously changing slope of a smooth tunnel barrier, specifically as given in Eq. (4). In particular, it should be noted that the quantum dot of the setup in Fig. 2(b) does not play a role in the appearance of quasi-Majorana states -the smooth tunnel barrier slope on the side of the proximitized wire is essential. The tunneling to the dot rather serves as a probe of the quasi-Majorana state.
Though we are focusing on the specific case of a Gaussian potential barrier, previous work has found near-zero energy Andreev bound states also for different types of potentials: a linear potential, 15 a quantum dot in the proximitized wire, 18 and smooth potentials with some sharp features, such as point-like impurities or abrupt changes in the superconducting order parameter. 15 We expect our findings to be similar for such potentials, too.
We discretize the Hamiltonian (3) on a regular onedimensional grid, and diagonalize this Hamiltonian to obtain wave functions and energy spectra. To compute the differential conductance in the tunneling spectroscopy setup of Fig. 2(a) we use the scattering formalism. The scattering matrix, relating incoming and outgoing modes in the normal lead, is where S αβ is the block of the scattering matrix with the scattering amplitudes of incident particles of type β to outgoing particles of type α. The differential conductance is with N e the number of propagating electron modes in the lead and T the transmissions that are related to the scattering matrix by We obtain the discretized Hamiltonian and the scattering matrix (7) numerically using Kwant, 30 see the supplementary material for source code and data. 31 We use adaptive parallel sampling of functions by using the Adaptive package. 32

C. Couplings from Mahaux-Weidenmüller formula
We investigate how the low-energy states in the proximitized nanowire couple to the propagating electron modes in the normal lead in the setup of Fig. 2(a), since this coupling determines the conductance through the leadwire interface. To do so, we write the scattering matrix Eq. (7) in a different form using a generalized form of the Mahaux-Weidenmüller formula derived in Ref. 33: Here, H is the modified Hamiltonian of the scattering region, E is the excitation energy, and W is the matrix containing couplings of the lead modes to the states in the scattering region.
To compute the coupling to the lowest energy eigenstate of the Hamiltonian, ψ + (E), and its particle-hole symmetric partner ψ − (−E) = Pψ + (E), with P the particle-hole operator, we introduce a matrix P = [ψ + , ψ − ]. The product W P contains the coupling of the lead modes to the pair of lowest-energy eigenstates of the Hamiltonian H. To calculate the coupling to the Majorana components ψ 1 , ψ 2 , we write ψ 1 , ψ 2 as linear combinations of ψ + , ψ − , for some arbitrary phase φ. In this Majorana basis, ψ 1 and ψ 2 satisfy ψ 1 = Pψ 1 , ψ 2 = Pψ 2 . The projected coupling matrixŴ in this basis has the form where t γσ is the coupling of Majorana component γ = 1, 2 to a lead electron mode with spin σ =↑, ↓, the complex conjugate t * γσ the coupling to the corresponding lead hole modes, and with t 2 in units of energy. We choose the phase φ such that it minimizes the off-diagonal elements t 1,↓ , t 2,↑ , which results in Majorana components with opposite spin. The computation of the coupling matrix W from the propagating modes in the lead as computed with Kwant 30 is done using the method of Ref. 33.

D. Analytic conductance expressions in different coupling limits
The anti-alignment of the Majorana spins allows for an analytic expression of the conductance Eq. (8). The Hamiltonian in the Majorana basis {ψ 1 , ψ 2 } reads with E M the coupling energy between ψ 1 and ψ 2 . When the spins of the Majorana components are anti-parallel, the projected coupling matrix Eq. (12) simplifies tô where t 1 ≡ t 1,↑ and t 2 ≡ t 2,↓ . For subgap energies, only Andreev reflection processes contribute to conductance, simplifying Eq. (8) to To evaluate this expression, we substitute Eqs. (13) and (14) into Eq. (10) and take out the electron to hole scattering block S eh (see Eq. (7)). To further simplify the resulting expression, we define coupling energies Γ 1 = 2πt 2 1 and Γ 2 = 2πt 2 2 , 34 and study the regime Γ 1 E M , Γ 2 , which describes one strongly coupled and one weakly coupled low-energy state. This approximation yields see App. A for a derivation. So, Eq. (16) gives the subgap conductance through an NS interface expressed in three energy parameters Γ 1 , Γ 2 and E M .
Equation (16) is a sum of two (semi-)Lorentzian functions, both with a peak height of 2e 2 /h. In the limit Γ 1 Γ 2 , E M , the first Lorentzian, with a peak width of ∼ Γ 1 , is much broader than the second Lorentzian of peak width ∼ Γ 2 . The second, narrower Lorentzian is positive for Γ 2 > 2E 2 M /Γ 1 and negative for Γ 2 < 2E 2 M /Γ 1 , and hence respectively increases the conductance around E = 0 to 4e 2 /h or decreases it to 0, depending on the coupling strength of the second low-energy state. This result explains the numerical findings of Ref. 18. When Γ 2 E 2 M /Γ 1 , the curve shape is similar to the single-mode result of Ref. 35. Temperature broadens the Lorentzian peaks, therefore the second peak is experimentally only observable when k B T Γ 2 . Therefore, in the limit Γ 1 Γ 2 , E M , zero-bias conductance is quantized to 2e 2 /h provided k B T > Γ 2 . Upon increasing Γ 2 , or decreasing temperature, an additional, narrower zero-bias peak is observable, either positive and increasing the overall conductance to 4e 2 /h or negative and decreasing it to zero, depending on the sizes of Γ 1 , Γ 2 and E M . When both Γ 1,2 k B T , both zero-bias conductance peaks are not observable, resulting in a zero subgap conductance.

III. PHASE DIAGRAM, WAVE FUNCTIONS AND COUPLINGS OF QUASI-MAJORANAS
Earlier works have presented Hamiltonian spectra as a function of the magnetic field, where for specific parameter choices quasi-Majorana states occur in the trivial regime. 15,19 To investigate more systematically in which parameter ranges these states occur, we compute a phase diagram as a function of Zeeman energy E Z and chemical potential µ. To do so, we consider the system Fig. 2(a), decoupled from the lead by introducing a hard-wall boundary condition at x = 0 (sufficiently far into the barrier such that quasi-Majoranas can still form at the barrier slope). We compute the energy of the lowest eigenstate of Hamiltonian (3) as a function of E Z and µ, see Fig. 3. In all four panels, inside the topological phase (red line), the lowest energy of the Hamiltonian is exponentially small, indicating the existence of a zero-energy state. This zero-energy state only exists in the topological phase for Fig. 3(a) and (b), when the potential barrier is steep. For a smooth potential, there is a large area of quasi-Majorana states with zero energy outside the topological phase, with a growing area as the SOI weakens, see Fig. 3(c) and (d).
We investigate in Fig. 4 the energy spectra and wave functions corresponding to different regimes of the phase The system goes into a topological phase for a further increase of Zeeman energy, with well-separated Majorana bound states at the system's endpoints ( Fig. 4(g)).
We note that partially separated (i.e. separated on the order of the coherence length ξ or more) have been reported in the literature before.. 19,36 In addition to these works, we find realistic parameter regimes with quasi-Majorana states consisting of spatially almost completely overlapping Majorana components, with a separation much smaller than ξ, that still have an exponentially suppressed near-zero energy. This is shown in the right column of Fig. 4, with Fig 4(b) the corresponding energy spectrum. Figure 4(d) shows that for this choice of parameters, the quasi-Majorana components nearly completely overlap, having a displacement much smaller than ξ. The separation of both quasi-Majorana wave functions is governed largely by the separation of the classical turning points for the spin-split tunnel barrier, controlled by the parameters of the system (such µ or E Z ) and the tunnel barrier. In general, the quasi-Majoranas wave function overlap increases with decreasing SOI strength α and smoothness σ, and increasing barrier height V . As in the partially separated case, also mostly overlapping quasi-Majorana states turn into true topological states on increasing Zeeman splitting (Fig. 4(h)). The origin of the decoupling between two quasi-Majorana states lies in the nearly opposite expectation value of the spin in x-direction, ψ † (x)σ x ψ(x), which we show in Fig. 4(e) and (f). As discussed in Ref. 15, the SOI strength effectively vanishes at the turning point of the smooth potential, leading to two decoupled Majorana states with spins aligned or anti-aligned to the Zeeman field (which is oriented in the x-direction) at the turning point. In this work, we focus on the case of weak spinorbit coupling, as quasi-Majoranas are more prevalent in this limit (see Figs. 3(c) and (d)). In the limit of spinorbit length larger than the wire diameter it is justified to neglect the transverse terms of the spin-orbit coupling, ? as we do in Hamiltonian (3). Hence, we observe that the individual Majorana components have a largely uniform sign of the spin expectation value.
Because quasi-Majorana states are located on the same side of a proximitized nanowire, while topological Majorana states are separated between opposite edges, one might expect local transport measurements to distinguish between both cases. However, this is generally not the case, as shown in Fig. 1: the opposite spin of both quasi-Majorana states result in a different effective barrier, which exponentially suppresses one quasi-Majorana coupling, reproducing the coupling regime of topological Majorana states. Figure 5(a, c) show the coupling parameters for a steep potential barrier (that does not suport the formation of quasi-Majorana states), and Figure 5(b, d) for a smooth barrier with quasi-Majorana states. The energy of the lowest Hamiltonian eigenstate E M is exponentially small for increasing magnetic field only in the topological regime for a steep barrier, see Fig. 5(a), but is suppressed well before the topological phase transition for a smooth barrier with quasi-Majorana states, Fig. 5(b). Likewise, the couplings across the barrier of the Majorana components of the lowest Hamiltonian eigenstate Γ 1 , Γ 2 are exponentially different only in the topological phase for a steep potential barrier, Fig. 5(c). However, for a smooth barrier, the couplings are approximately four orders of magnitude different already in the trivial phase, Fig. 5(d).
Consistently, we find that the exponential suppression of both E M and Γ 2 is stronger in the quasi-Majorana regime than in the topological regime.
The exponential suppression of the coupling between quasi-Majoranas and the coupling of one of the quasi-Majoranas across a tunnel barrier for increasing magnetic field reproduces the topological coupling regime Γ 2 , E M Γ 1 . Hence, the conductance signatures of both the quasi-Majorana regime and the topological Majorana regime are similar. In absence of quasi-Majorana states, a zerobias conductance peak quantized to 2e 2 /h only develops after the topological phase transition, Fig. 5(e), while in presence of quasi-Majorana states, a quantized zerobias peak is also present in the trivial regime, Fig. 5(f). This zero-bias peak quantized to 2e 2 /h coincides with the exponential suppression of the coupling of one of the two zero-bias states, as expected from our analytical formula, Eq. (16). Our calculations also show a narrow conductance dip around E = 0 due to the coupling of the second (quasi-)Majorana Γ 2 as is consistent with Eq. (16), but this is not visible in the color scheme of Coupling energy to the probing lead Γ1, Γ2 of the two lowest states. We do not include the coupling for small Zeeman energies, since the energy of the lowest state is too large compared to the bulk gap in this regime for the approximation of Sec. II C to hold. (e, f): Conductance as a function of bias energy E and Zeeman energy EZ. In all panels, the red vertical line indicates the topological phase transition. The barrier heights and smoothness are V = 18 meV, σ = 10 nm for the left column and V = 11.7 meV, σ = 100 nm for the right column respectively, and the chemical potential is µ = 5 meV for all panels. and quasi-Majorana states.

IV. MAJORANA NON-LOCALITY AND TOPOLOGICAL JOSEPHSON EFFECT
References 26 and 27 express Majorana non-locality as the ratio between the couplings Γ 1 , Γ 2 of the two Majorana states to a probing lead. In Ref. 26, a 'quality factor' q = 1−Γ 2 /Γ 1 is defined, with q = 0 denoting two strongly coupled local Majorana states (Γ 1 = Γ 2 ), and q = 1 denoting complete non-locality (Γ 2 = 0). References 26 and 27 propose a coupled quantum dot-nanowire device, see Fig. 2(b), to determine the quality factor with a local probe, which has been experimentally implemented in Ref. 28. The spectrum of the hybrid quantum dot-nanowire device shows anti-crossing quantum dot states and a flat zero-energy state as a function of the quantum dot chemical potential in case of well-separated Majorana states, with E M , Γ 2 Γ 1 . When the Majorana states are closer together, the increasing coupling of the second Majorana to the quantum dot results in increasingly asymmetric diamond-like shapes in the lowest energy level across the resonance with the quantum dot states. Hence, the measurement of the energy levels in the hybrid device allows to determine the Majorana non-locality with a local probe. Reference 22 pointed out that partially separated Andreev bound states can have different couplings to a quantum dot, mimicking the signatures of spatially separated topological Majorana states. We show that quasi-Majorana states systematically have exponentially different couplings to a quantum dot, hence the quasi-Majorana regime generally exhibits a high degree of non-locality. Figure 6(a, b) show the spectrum in the topological phase as a function of quantum dot chemical potential µ dot and Zeeman energy E Z respectively. The quantum dot and the nanowire are separated by a steep barrier, so no quasi-Majorana states appear in the spectrum of Fig. 6(b). The non-locality of the Majorana states is expressed in Fig. 6(a) by the flat energy level around E = 0 of the nonlocal Majorana state, and spin-dependent anti-crossings of the quantum dot levels coupled to the local Majorana state. A flat energy level around E = 0 and strong anticrossings are absent in Fig. 6(c), where the system is topologically trivial and no single Majorana state couples to the quantum dot. However, in the presence of quasi-Majoranas, Fig. 6(e, f), these characteristics occur in the trivial phase because of the exponentially different coupling of both quasi-Majorana states to the quantum dot. Therefore, since quasi-Majorana states reproduce the topological coupling regime E M , Γ 2 Γ 1 , we observe that quasi-Majorana states can exhibit a high degree of Majorana non-locality, and consequently give rise to high quality factors, while being highly local in space. This makes Majorana non-locality and the Majorana quality factor as proposed in Refs. 26 and 27 unsuitable for distinguishing quasi-Majorana states from topological Majorana states.
Turning to the 4π-periodic Josephson effect in a device as sketched in Fig. 2(c), we again compare a topological junction to a trivial junction with and without quasi-Majorana states. Figure 7(a) shows a 4π-periodicity of the energy levels corresponding to the Majorana states located at the normal barrier, and a flat zero-energy level corresponding to the Majorana states at the outer edges of the device (with a small splitting due to finite size effects), as is expected theoretically in the topological phase. 24,25 In the trivial phase, as shown in Fig. 7(c), no zero-energy state is present, and energy levels show a 2π-periodicity. When the barrier is smooth, quasi-Majorana states appear in the trivial regime (see Fig. 7(f)), reproducing the flat zero-energy levels and 4π-periodic levels that characterize the topological Josephson junction (Fig. 7(e)). Quasi-Majorana states reproduce the topological phase winding characteristics because two quasi-Majorana states strongly couple across the barrier, resulting in a 4π-periodic level, and two have an exponentially suppressed coupling, resulting in a flat zero-energy level. Therefore, the measurement of a 4π-periodic Josephson current is not a distinctive signature of topological Majorana states, but can be caused by quasi-Majorana states.

V. DISTINCTIVE SIGNATURES OF A TOPOLOGICAL PHASE
Previously discussed measurement setups rely on Majorana modes to determine a topological phase, which makes them inherently sensitive to non-topological local low-energy states. Hence, a better strategy to distinguish a topological from a trivial phase is the measurement of a bulk phase transition rather than the measurement of individual Majorana states, which has been proposed in several earlier works. Reference 37 discusses quantized thermal conductance and electrical shot noise in a proximitized nanowire coupled to two normal leads as signatures of a topological phase transition. Reference 38 proposes the measurement of differences in conductance at one lead connected to a proximitized nanowire when changing the coupling to another lead, while Ref. 39 predicts a sign change of the spin component of bulk bands along the magnetic field as a measure of a topological phase transition. Finally, Ref. 40 proposes the detection of rectifying the behavior of the nonlocal conductance G between two spatially separated leads as a function of the bias E, G(E) ∝ E, as a signature of a bulk phase transition. These proposals all rely on bulk properties and therefore more reliably detect a topological phase than probing a local Majorana state, which might be mimicked by other localized low-energy states. We also suggest an alternative approach relying on local conductance measurements that allows to distinguish topological Majorana states from quasi-Majorana states. According to Eq. (16), when the coupling of the second low-energy subgap state Γ 2 exceeds k B T , an experimentally observable zero-bias conductance peak of 4e 2 /h develops. Hence, our approach does not focus on a quantized conductance peak in the tunneling spectroscopy 41 when Γ 2 is strongly suppressed, but on a conductance measurement in the open regime. We demonstrate the effect of opening the tunnel barrier V → 0 (with V the height of the potential barrier given in Eq. (4)) on the conductance with true Majorana states and with quasi-Majorana states in Fig. 8, using the microscopic model (3). Figure 8(a) shows the conductance as a function of bias energy E and barrier height V in the topological phase with spatially separated Majorana states. In the tunneling regime, the conductance shows a zero-bias peak quantized to 2e 2 /h (see also the light-brown line cut in panel (c)), which broadens to a plateau of 2e 2 /h height upon opening the barrier (purple line cut). When the barrier height is further reduced, the conductance at finite bias increases due to Andreev enhancement, but stays fixed to 2e 2 /h at zero bias due to the presence of a single Majorana state (pink line cut). 11,42 Quasi-Majorana states also exhibit a conductance peak of 2e 2 /h in the tunneling regime and a conductance plateau of 2e 2 /h in the quasi-open regime as shown in Fig. 8(b) and the line cuts in Fig. 8(d). However, upon further opening the barrier, both quasi-Majorana states couple to the lead, resulting in a conductance peak of 4e 2 /h which broadens to a plateau when further reducing V .
Note that in this argument we rely on a strong coupling of both quasi-Majorana states to the lead, tunable by e.g. an external tunnel gate. Should the coupling be limited by intrinsic effects such as local disorder, the conductance may stay below 4e 2 /h. In the topological case, the conductance exceeds 2e 2 /h for voltages away from zero. This finite bias conductance value is not universal, and depend on details of the system, such as potential shapes. In contrast, the zero-bias conductance must always stay quantized at 2e 2 /h as long as not more than two conductance channels are opened in the tunnel barrier, due to particle-hole symmetry. 11 Therefore, while a zero-bias conductance peak or conductance plateau quantized to 2e 2 /h does not distinguish quasi-Majorana states from topological Majorana states, a quantized zero-bias dip in the conductance in the open regime does.

VI. QUASI-MAJORANA STATES IN A 3D NANOWIRE
Because quasi-Majorana states so far have been studied in one-dimensional systems, 15,16,18,19,21,22 it is uncertain how likely quasi-Majoranas are to appear in realistic situations. While currently doing a fully realistic simulation of a three-dimensional device is beyond state of the art, we do a 3D simulation that includes the orbital effect of magnetic field, 43,44 transverse spin-orbit coupling, multiple modes mixed by an inhomogeneous potential in the direction perpendicular to the wire axis, and an external superconducting shell proximitizing the nanowire (see App. B for a detailed description of the model).
We show the phase diagram of this 3D device as a function of µ and E Z in Fig. 9. The upper panels, with a steep potential barrier (σ → 0), show that the emergence of a zero-energy state coincides with the topological phase, which has a more complicated shape compared to Fig. 3 due to multiple modes and the orbital effect of the magnetic field. In Fig. 9(b), the gap outside the topological phase is weaker due to a weaker spin-orbit coupling, but no robust trivial zero-energy state emerges. However, Fig. 9(c, d) show that for a smooth potential barrier (σ = 200 nm) a region of zero-energy quasi-Majorana states emerges, especially prominent for weak spin-orbit strength α = 20 meV nm, Fig. 9(d). Figure 9 is qualitatively similar to Fig. 3: for a smooth potential, regions of zero-energy quasi-Majoranas emerge, increasing in size for decreasing spin-orbit strength. Thus, we find that quasi-Majorana states are also present in realistic 3D systems with smooth potentials that are close to currently available experimental devices.

VII. BRAIDING OPERATIONS WITH QUASI-MAJORANA STATES
Braiding schemes that demonstrate and utilize the non-Abelian statistics of Majorana states are subdivided into gate-controlled braiding in T-junctions, [45][46][47] Coulomb-assisted braiding in Josephson junctions, 48,49 or measurement-based braiding in topological nanowires coupled to quantum dots. [50][51][52] Having quasi-Majorana states in the topologically trivial phase in these devices still admits braiding. Gate-controlled braiding requires microscopically precise manipulation of electrostatic potentials, and therefore we leave gate-controlled braiding with quasi-Majorana states as a topic for future research. On the other hand, the other two schemes only rely on the coupling to individual Majorana states, which is possible in the quasi-Majorana regime, since quasi-Majorana states couple exponentially different across a tunnel barrier. The presence of the second, uncoupled quasi-Majorana state still allows these braiding schemes to work. 53 We show a possible setup for a measurement-based braiding scheme with quasi-Majorana states in Fig. 10(a), where only one quasi-Majorana state of each pair couples to the adjacent quantum dot, and for a Coulomb-assisted braiding scheme in Fig. 10(b), where only one quasi-Majorana state of each pair couples to one other quasi-Majorana state of the two other nanowires.
To estimate whether quasi-Majorana states are realistic candidates for braiding, we compare quasi-Majorana en- ergy and length scales to braiding requirements. Coulombassisted and measurement-based braiding involves a fermion parity measurement in a transmon, 54 where the parity shift is expressed in a resonance frequency shift ∆ω, which has been estimated in Ref. 52 for realistic parameters as ∆ω ∼ 100 MHz. Hence, the transmon sensitivity must exceed 100 MHz, which limits the quasi-Majorana energy splitting to ∆ω ∼ 0.1 µeV. The energy splitting E M for the parameters of Fig. 5 does not meet this requirement (see Fig. 5(b)), but we find that for increasing barrier smoothness σ and SOI strength α (while keeping the wire length fixed to L SC = 3 µm), the splitting is reduced to a value below the braiding requirement. As an example, for experimentally realistic values of σ = 150 nm and α = 100 meV nm, we find a quasi-Majorana splitting of 0.1 µeV. Additionally, we consistently observe that the coupling energy E M in the quasi-Majorana regime is an order of magnitude smaller than in the topological regime. The smaller quasi-Majorana coupling compared to the topological Majorana coupling is due to the lower magnetic fields in the quasi-Majorana regime, which results in a smaller coupling of quasi-Majorana states to the other end of the wire. The suppression of the coupling of the second quasi-Majorana state to the outside is orders of magnitude smaller, Γ 2 ∼ 10 −3 µeV, and again we find this suppression stronger in the quasi-Majorana regime than in the topological regime, see Fig. 5(d). Consequently, using quasi-Majorana states may be an attractive approach to demonstrate braiding properties.

VIII. SUMMARY AND OUTLOOK
Experimental setups to measure Majorana states in hybrid semiconductor-superconductor nanowire devices contain electrostatic gates that can generate smooth potential profiles, which give rise to non-topological quasi-Majorana states, that have an exponentially suppressed energy as a function of the magnetic field. Additionally, one of the quasi-Majorana states has an exponentially suppressed coupling across the tunnel barrier. This makes quasi-Majoranas mimic all local Majorana signatures, specifically a quantized zero-bias peak conductance in the tunneling spectroscopy, the resonance spectrum in a coupled nanowire -quantum dot device, and 4π-periodicity of the energy levels as a function of phase in a Josephson junction. Therefore, it is impossible to categorise signatures in current Majorana experiments into topological Majorana states or trivial quasi-Majorana states.
A measurement of a bulk phase transition, rather than a measurement of the presence of local (quasi-)Majorana states, can experimentally distinguish non-topological quasi-Majorana states from topological Majorana states. Additionally, we propose to measure conductance in the open regime, which results in a plateau at G = 4e 2 /h around zero bias in the conductance in presence of quasi-Majoranas, and in a conductance dip to G = 2e 2 /h at zero bias in presence of topological, spatially separated Majoranas.
While quasi-Majorana states make it harder to unambiguously demonstrate topological Majorana states, they reproduce topological properties such as braiding. Quasi-Majorana states lack true topological protection and are hence sensitive to magnetic impurities or other short-range disorder mechanisms that break the smoothness of the potential barrier. However, due to the progress in device design, the current experimental devices are likely to be in the ballistic regime required to support robust quasi-Majorana states. [55][56][57][58] Also, for a given chemical potential, quasi-Majorana states emerge for smaller magnetic fields, which reduces the coupling to the opposite end of the wire compared to topological Majorana states, resulting in smaller energy splittings. Furthermore, combined with topological Majorana states, quasi-Majorana states increase the overall phase space in which protected quantum computing can be performed. Therefore, it may be an interesting direction of further research to engineering quasi-Majorana states to study topological properties. Here p = −i ∇+eAτ z is the canonical momentum, where e is the electron charge, and A = [B y z − B z y, 0, B x y] T is the vector potential chosen such that it does not depend on x, which we include in the tight-binding system using the Peierls substitution. 59 Further, m * is the effective mass, µ is the chemical potential controlling the number of occupied subbands in the wire, α is the strength of the SOI, g is the Landé g-factor, µ B is the Bohr magneton, and ∆ is the superconducting pairing potential. The Pauli matrices σ and τ act in spin space and electron-hole space respectively. We assume a Gaussian potential V (x, z) inside the wire centered around x = 0, with different peak heights at the top (z = R) and bottom (z = −R) of the wire, and linearly interpolated for −R < z < R: V (x, z) = V bottom exp 1 2 where R is the wire radius, V bottom and V top are the heights of the Gaussian peaks at the bottom and top respectively, V offset is the difference in potential between the top and bottom, and σ the width of the peaks. We perform numerical simulations of the Hamiltonian (B1) on a 3D lattice using Kwant. 30 The source code and the specific parameter values are available in the Supplemental Material . 31 The full set of materials, including the computed raw data and experimental data, is available in Ref. 31.