ReShape: a decoder for hypergraph product codes

TL;DR

This paper introduces ReShape, a decoder for hypergraph product quantum codes that efficiently lifts classical decoders to quantum decoders using algebraic methods, though it is limited to adversarial error models.

Contribution

It presents a novel algebraic approach to construct quantum decoders from classical decoders for hypergraph product codes, with efficiency guarantees.

Findings

Decoder requires only O(k) oracle calls

Works perfectly for adversarial errors

Uses algebraic invariants of hypergraph codes

Abstract

The design of decoding algorithms is a significant technological component in the development of fault-tolerant quantum computers. Often design of quantum decoders is inspired by classical decoding algorithms, but there are no general principles for building quantum decoders from classical decoders. Given any pair of classical codes, we can build a quantum code using the hypergraph product, yielding a hypergraph product code. Here we show we can also lift the decoders for these classical codes. That is, given oracle access to a minimum weight decoder for the relevant classical codes, the corresponding $[[n,k,d]]$ quantum code can be efficiently decoded for any error of weight smaller than $(d-1)/2$. The quantum decoder requires only $O(k)$ oracle calls to the classical decoder and $O(n^2)$ classical resources. The lift and the correctness proof of the decoder have a purely algebraic nature that draws on the discovery of some novel homological invariants of the hypergraph product codespace. While the decoder works perfectly for adversarial errors, it is not suitable for more realistic stochastic noise models and therefore can not be used to establish an error correcting threshold.