${\it Ab \ Initio}$ Simulation of Non-Abelian Braiding Statistics in Topological Superconductors

TL;DR

This paper numerically investigates the non-Abelian braiding of vortices in topological superconductors, demonstrating geometric phase accumulation and analyzing quantum noise effects on Majorana-based qubits for quantum computation.

Contribution

It provides a numerical simulation of vortex braiding dynamics without assuming the existence of Majorana zero modes, revealing insights into phase accumulation and error sources.

Findings

Vortices acquire a geometric phase of π/2 after braiding.

Dynamical phase errors are negligible regardless of MZM interactions.

Optimal braiding conditions are discussed for high-fidelity quantum gates.

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 quasiparticle interference. Numerically simulating the time-dependent Bogoliubov-de Gennes equation without assuming ${\it a \ priori}$ 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 $\pi/2$, 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 gates operations of Majorana-based qubits.