Monte Carlo studies of the properties of the Majorana quantum error correction code: is self-correction possible during braiding?

TL;DR

This study uses Monte Carlo simulations and theoretical analysis to examine the error correction capabilities of Majorana-based quantum codes, revealing limitations in their self-correcting properties during braiding operations.

Contribution

It provides a detailed analysis of thermal error processes in Majorana codes, especially during braiding, and identifies fundamental constraints on their self-correction in 1D structures.

Findings

Lifetime grows logarithmically with wire size at high temperature

Braiding introduces nonlocal errors that limit self-correction

Monte Carlo simulations confirm analytical predictions at low temperature

Abstract

The Majorana code is an example of a stabilizer code where the quantum information is stored in a system supporting well-separated Majorana Bound States (MBSs). We focus on one-dimensional realizations of the Majorana code, as well as networks of such structures, and investigate their lifetime when coupled to a parity-preserving thermal environment. We apply the Davies prescription, a standard method that describes the basic aspects of a thermal environment, and derive a master equation in the Born-Markov limit. We first focus on a single wire with immobile MBSs and perform error correction to annihilate thermal excitations. In the high-temperature limit, we show both analytically and numerically that the lifetime of the Majorana qubit grows logarithmically with the size of the wire. We then study a trijunction with four MBSs when braiding is executed. We study the occurrence of dangerous error processes that prevent the lifetime of the Majorana code from growing with the size of the trijunction. The origin of the dangerous processes is the braiding itself, which separates pairs of excitations and renders the noise nonlocal; these processes arise from the basic constraints of moving MBSs in 1D structures. We confirm our predictions with Monte Carlo simulations in the low-temperature regime, i.e. the regime of practical relevance. Our results put a restriction on the degree of self-correction of this particular 1D topological quantum computing architecture.