Numerical and analytical bounds on threshold error rates for hypergraph-product codes

TL;DR

This paper analytically and numerically investigates the decoding limits of hypergraph-product quantum LDPC codes derived from Gallager codes, providing bounds on their error correction capabilities.

Contribution

It introduces new analytical bounds and numerical thresholds for the decodable region of hypergraph-product quantum LDPC codes.

Findings

Decodable region bounds derived analytically from homological difference analysis

Numerical minimum weight decoding threshold of approximately 7%

Upper bounds from specific heat calculations in associated Ising models

Abstract

We study analytically and numerically decoding properties of finite rate hypergraph-product quantum LDPC codes obtained from random (3,4)-regular Gallager codes, with a simple model of independent X and Z errors. Several non-trival lower and upper bounds for the decodable region are constructed analytically by analyzing the properties of the homological difference, equal minus the logarithm of the maximum-likelihood decoding probability for a given syndrome. Numerical results include an upper bound for the decodable region from specific heat calculations in associated Ising models, and a minimum weight decoding threshold of approximately 7%.