Union-Find Decoders For Homological Product Codes

TL;DR

This paper introduces union-find decoders for homological product codes, specifically for augmented surface codes, improving low-noise performance despite a lower threshold.

Contribution

It develops a novel decoding method combining union-find and brute force decoders for homological product codes, enhancing low-noise error correction.

Findings

Decoders successfully decode errors up to half the code distance.

Low noise performance is improved despite a lower threshold.

Numerical simulations validate the decoding approach.

Abstract

Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a $[[4,2,2]]$ code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.