Disorder-assisted error correction in Majorana chains

TL;DR

This paper demonstrates that disorder in Majorana chains can exponentially enhance quantum memory storage times at zero temperature by inducing localization, supported by theoretical analysis and numerical simulations.

Contribution

It provides a detailed analysis of how disorder-induced localization improves quantum memory stability in Majorana chains, including theoretical estimates and numerical evidence.

Findings

Disorder causes exponential growth of storage time with system size.

Without disorder, storage time grows only logarithmically.

Pseudorandom potentials can outperform purely random disorder.

Abstract

It was recently realized that quenched disorder may enhance the reliability of topological qubits by reducing the mobility of anyons at zero temperature. Here we compute storage times with and without disorder for quantum chains with unpaired Majorana fermions - the simplest toy model of a quantum memory. Disorder takes the form of a random site-dependent chemical potential. The corresponding one-particle problem is a one-dimensional Anderson model with disorder in the hopping amplitudes. We focus on the zero-temperature storage of a qubit encoded in the ground state of the Majorana chain. Storage and retrieval are modeled by a unitary evolution under the memory Hamiltonian with an unknown weak perturbation followed by an error-correction step. Assuming dynamical localization of the one-particle problem, we show that the storage time grows exponentially with the system size. We give supporting evidence for the required localization property by estimating Lyapunov exponents of the one-particle eigenfunctions. We also simulate the storage process for chains with a few hundred sites. Our numerical results indicate that in the absence of disorder, the storage time grows only as a logarithm of the system size. We provide numerical evidence for the beneficial effect of disorder on storage times and show that suitably chosen pseudorandom potentials can outperform random ones.