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arXiv:2008.02997 · Physical review. B./Physical review. B

Ab initio simulation of non-Abelian braiding statistics in topological superconductors

Takumi Sanno, Shunsuke Miyazaki, Takeshi Mizushima, Satoshi Fujimoto

Published Feb 11, 2021• 22 citations
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Abstract

We numerically investigate non-Abelian braiding dynamics of vortices in two-dimensional topological superconductors, such as $s$-wave superconductors with Rashba spin-orbit coupling. Majorana zero modes (MZMs) hosted by the vortices constitute a topological qubit, which offers a fundamental building block of topological quantum computation. As the MZMs are protected by ${\mathbb{Z}}_{2}$ invariant, however, the Majorana qubit and quantum gate operations may be sensitive to intrinsic decoherence caused by MZM hybridization. Numerically simulating the time-dependent Bogoliubov--de Gennes equation without assuming a priori existence of MZMs, we examine quantum noises on the unitary operators of non-Abelian braiding dynamics due to interactions with neighboring MZMs and other quasiparticle states. We demonstrate that after the interchange of two vortices, the lowest vortex-bound states accumulate the geometric phase $\ensuremath{\pi}/2$, and errors stemming from dynamical phases are negligibly small, irrespective of interactions of MZMs. Furthermore, we numerically simulate the braiding dynamics of four vortices in two-dimensional topological superconductors, and discuss an optimal braiding condition for realizing the high performance of non-Abelian statistics and quantum gate operations of Majorana-based qubits.

DOI
10.1103/physrevb.103.054504

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We numerically investigate non-Abelian braiding dynamics of vortices in two-dimensional topological superconductors, such as sss-wave superconductors with Rashba spin-orbit coupling. Majorana zero modes (MZMs) hosted by the vortices constitute a topological qubit, which offers a fundamental building block of topological quantum computation. As the MZMs are protected by Z2{\mathbb{Z}}_{2}Z2​ invariant, however, the Majorana qubit and quantum gate operations may be sensitive to intrinsic decoherence caused by MZM hybridization. Numerically simulating the time-dependent Bogoliubov--de Gennes equation without assuming a priori existence of MZMs, we examine quantum noises on the unitary operators of non-Abelian braiding dynamics due to interactions with neighboring MZMs and other quasiparticle states. We demonstrate that after the interchange of two vortices, the lowest vortex-bound states accumulate the geometric phase \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2, and errors stemming from dynamical phases are negligibly small, irrespective of interactions of MZMs. Furthermore, we numerically simulate the braiding dynamics of four vortices in two-dimensional topological superconductors, and discuss an optimal braiding condition for realizing the high performance of non-Abelian statistics and quantum gate operations of Majorana-based qubits.
Majorana zero modes (MZMs) localized in superconducting vortices are predicted to realize Ising anyon non-Abelian statistics. In finite-size vortex lattices, overlapping MZM wavefunctions generate exponentially decaying energy splittings. We investigate whether these finite-size effects merely perturb the ideal Ising braid group representation or fundamentally alter it into a distinct, dynamically dependent representation. By modeling a two-dimensional topological superconductor with an effective low-energy Hamiltonian incorporating exponentially decaying overlap terms, we numerically evaluate the time-evolution operator during adiabatic braiding. Our analysis shows that finite-size effects introduce a non-commuting dynamical phase that modifies the ideal Ising braid matrix. This results in a time-dependent braid matrix that alters expected interferometric conductance oscillations. Specifically, the conductance oscillations exhibit a phase shift proportional to the braiding time and inter-vortex distance, scaling as δϕ∝τΔe−R/ξ\delta \phi \propto \tau \Delta e^{-R/\xi}δϕ∝τΔe−R/ξ [to verify]. The non-Abelian nature appears degraded, with the braid matrix depending on the adiabatic trajectory and duration rather than solely on its topological class. This suggests interferometry can dynamically detect these finite-size modifications. ## 1. Introduction Majorana zero modes (MZMs) in topological superconductors are predicted to exhibit non-Abelian braiding statistics, forming the basis for topological quantum computation. In a p+ipp+ipp+ip superconductor or similar platforms, vortices trap MZMs whose braiding is described by the Ising anyon model. However, realistic systems are finite, and MZMs localized at neighboring vortices have overlapping wavefunctions, leading to exponentially small energy splittings. While often treated as a negligible perturbation, the impact of these splittings on the braid group representation during finite-time braiding remains an open question. Here, we analyze the
We analyze the topological protection of a Clifford gate set generated by braiding four Majorana zero modes (MZMs) in a two-dimensional class-D topological superconductor. Encoding a single logical qubit in four MZMs bound to π\piπ-flux vortices, we show that elementary exchanges generate the single-qubit Clifford group up to a global phase. The residual gate error arises from finite-size Majorana hybridization and quasiparticle poisoning, scaling as ϵ∼exp⁡(−L/ξ)\epsilon \sim \exp(-L/\xi)ϵ∼exp(−L/ξ), where LLL is the inter-vortex separation and ξ\xiξ is the superconducting coherence length. We propose an interferometric measurement scheme to falsify this topological protection. By threading an Aharonov-Bohm loop around two MZMs, the conductance exhibits 4π4\pi4π-periodic oscillations with phase jumps of π/2\pi/2π/2 per braid. Any deviation from this quantized phase beyond the exp⁡(−L/ξ)\exp(-L/\xi)exp(−L/ξ) envelope at large L/ξL/\xiL/ξ falsifies the topological protection mechanism, indicating gap closing or quasiparticle poisoning. ## 1. Introduction Topological quantum computation seeks to process information non-locally to achieve intrinsic fault tolerance. In two-dimensional topological superconductors belonging to symmetry class D (such as a px+ipyp_x + i p_ypx​+ipy​ paired state or a proximitized semiconductor heterostructure), Majorana zero modes (MZMs) bind to π\piπ-flux vortices. Exchanging these vortices implements non-Abelian unitaries on the degenerate ground-state manifold. The central question addressed here is whether braiding four MZMs generates a topologically protected Clifford gate set with a residual error rate scaling as exp⁡(−L/ξ)\exp(-L/\xi)exp(−L/ξ), and whether this claim is falsifiable via interferometric measurement of the braiding phase. ## 2. Background The non-Abelian statistics of MZMs in class-D superconductors were established by Ivanov (2001), demonstrating that exchanging vortices realizes a braid group representation with π/2\pi/2π/2 rotations in Majorana Hilbert space [arXiv:cond-mat/0105040]. The topologi
Qubits in topological quantum computation are built from non-Abelian anyons. Adiabatic braiding of anyons is exploited as topologically protected logical gate operations. Thus, the adiabaticity upon which the notion of quantum statistics is defined plays a fundamental role in defining the non-Abelian anyons. We study the nonadiabatic effects in braidings of Ising-type anyons, namely, Majorana fermions in topological superconductors, using the formalism of time-dependent Bogoliubov-de Gennes equations. Using this formalism, we consider nonadiabatic corrections to non-Abelian statistics from (1) tunneling splitting of anyons imposing an additional dynamical phase to the transformation of ground states and (2) transitions to excited states that are potentially destructive to non-Abelian statistics since the nonlocal fermion occupation can be spoiled by such processes. However, if the bound states are localized and being braided together with the anyons, non-Abelian statistics can be recovered once the definition of Majorana operators is appropriately generalized taking into account the fermion parity in these states. On the other hand, if the excited states are extended over the whole system and form a continuum, the notion of local fermion parity no longer holds. We then quantitatively characterize the errors introduced in this situation.
Braiding Majorana zero modes (MZMs) is the key procedure toward topological quantum computation. We show such braiding can be well performed in a parallel semiconductor-superconductor nanowire structure. Considering the fact that the low-energy Andreev bound states (ABSs) usually mix with the MZMs in the present setup, we further investigate the braiding properties of MZMs when an ABS is presented. Our numerical simulation suggests that ABS can be regarded as a pair of weakly coupled MZMs. The dynamical hybridization of MZMs plus the non-Abelian braiding of MZMs would induce an arbitrary rotation on the Bloch sphere of a single qubit. Remarkably, such rotation is manipulable since the rotation parameters could be individually modulated. Thus, the dynamic evolution can be eliminated and the non-Abelian braiding statistics, independent of the braiding time, retrieves.
FINDING: Majorana zero modes (MZMs) in topological superconductors obey non-Abelian braid statistics, with exponential suppression of coupling between spatially separated modes enabling topologically protected quantum gates. | MATH: The Kitaev chain Hamiltonian H=−t∑ci†ci+1+Δ∑cici+1+h.c.−μ∑ci†ci H = -t\sum c^\dagger_i c_{i+1} + \Delta\sum c_i c_{i+1} + \text{h.c.} - \mu\sum c^\dagger_i c_i H=−t∑ci†​ci+1​+Δ∑ci​ci+1​+h.c.−μ∑ci†​ci​ yields unpaired Majorana operators γ1,γ2 \gamma_1, \gamma_2 γ1​,γ2​ with γi†=γi \gamma_i^\dagger = \gamma_i γi†​=γi​, {γi,γj}=2δij \{\gamma_i,\gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​. Braiding exchanges γi→γj \gamma_i \to \gamma_j γi​→γj​ via unitary Uij=exp⁡(π4γiγj) U_{ij} = \exp(\frac{\pi}{4}\gamma_i\gamma_j) Uij​=exp(4π​γi​γj​), giving non-Abelian representation of the braid group Bn B_n Bn​ (e.g., U12U23≠U23U12 U_{12}U_{23} \neq U_{23}U_{12} U12​U23​=U23​U12​). Splitting energy ΔE∝e−L/ξ \Delta E \propto e^{-L/\xi} ΔE∝e−L/ξ where ξ=ℏvF/Δ \xi = \hbar v_F/\Delta ξ=ℏvF​/Δ is the coherence length — exponential suppression with wire length L L L. | CONNECTION: The braid group Bn B_n Bn​ is generated by transpositions σi \sigma_i σi​ satisfying \( \sigma_i\s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We simulate the non-Abelian exchange of Majorana zero modes (MZMs) on a quantum computer. Rather than utilizing MZMs at the boundaries of quantum Ising chains, which are typically represented as nonlocal operators on a quantum computer, using a Kitaev lattice allows us to exploit a local representation of MZMs. We detail the protocol for braiding two and four MZMs in terms of a spin Hamiltonian, i.e. physical qubit Hamiltonian. Projecting this onto a subspace of states, we extract an effective Hamiltonian which drives a non-Abelian Berry phase. Using several approximations, we construct a set of gates which mimics this accumulation of non-Abelian phase and process this construction on a quantum computer. For two and four MZMs, we realize braiding fidelities of approximately 85 and 47%, respectively.