Manipulation of Majorana Kramers Qubit and its Tolerance in Time-Reversal-Invariant Topological Superconductor

TL;DR

This paper studies the robustness of Majorana Kramers pairs' non-Abelian braiding in topological superconductors, showing it withstands certain symmetry-breaking perturbations but can fail due to inhomogeneous potentials.

Contribution

It demonstrates the conditions under which Majorana Kramers pairs maintain their non-Abelian braiding properties despite symmetry-breaking effects.

Findings

Braiding is robust against magnetic fields if combined symmetry is preserved.

Non-Majorana states can interfere with MKPs, causing braiding failure.

Stability depends on the length scale of inhomogeneous potentials.

Abstract

We investigate non-Abelian statistics of Majorana Kramers pairs (MKPs) in a network system of one-dimensional time-reversal invariant topological superconductors by using numerical simulations of braiding dynamics, and examine the tolerance against various perturbations which may cause decoherence of MKPs. We, first, consider effects of a magnetic field which breaks time-reversal symmetry. In contrast to a naive expectation, the non-Abelian braiding of MKPs is robust against applied magnetic fields provided that the initial and final states of a braiding process are invariant under the combination of a time reversal and a mirror reflection, even when intermediate states break the combined symmetry. Secondly, we investigate the stability of non-Abelian braidings in the case with gate-induced inhomogeneous potentials at junctions between superconducting nanowires, which generally generate a non-Majorana nearly zero-energy Andreeev bound state at the junctions. It is found that the non-Majorana nearly zero-energy states interfere with MKPs, resulting in the failure of non-Abelian braidings, when the length scale of the inhomogeneous potentials is comparable to the coherence length of the superconductors.