Efficient decoding of random errors for quantum expander codes

TL;DR

This paper demonstrates that quantum expander codes can efficiently correct a linear number of random errors with high probability, advancing quantum error correction with constant-rate, low-overhead codes.

Contribution

It introduces a decoding method for quantum expander codes that corrects many random errors efficiently, a first for constant-rate quantum LDPC codes.

Findings

Corrects a constant fraction of random errors with high probability

First to achieve linear error correction in constant-rate quantum LDPC codes

Uses a novel notion of $oldsymbol{ extalpha}$-percolation for analysis

Abstract

We show that quantum expander codes, a constant-rate family of quantum LDPC codes, with the quasi-linear time decoding algorithm of Leverrier, Tillich and Z\'emor can correct a constant fraction of random errors with very high probability. This is the first construction of a constant-rate quantum LDPC code with an efficient decoding algorithm that can correct a linear number of random errors with a negligible failure probability. Finding codes with these properties is also motivated by Gottesman's construction of fault tolerant schemes with constant space overhead. In order to obtain this result, we study a notion of $\alpha$-percolation: for a random subset $W$ of vertices of a given graph, we consider the size of the largest connected $\alpha$-subset of $W$, where $X$ is an $\alpha$-subset of $W$ if $|X \cap W| \geq \alpha |X|$.