Numerical and analytical bounds on threshold error rates for hypergraph-product codes
Alexey A. Kovalev, Sanjay Prabhakar, Ilya I. Dumer, Leonid P. Pryadko · Physical Review A · 2018
We study analytically and numerically decoding properties of finite-rate hypergraph-product quantum low density parity-check codes obtained from random (3,4)-regular Gallager codes, with a simple model of independent $X$ and $Z$ errors. Several nontrivial 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 $7%$.