Statistical mechanical models for quantum codes with correlated noise

TL;DR

This paper extends a known mapping from quantum error correction to statistical mechanics, enabling threshold estimation and decoding for quantum codes under correlated noise using classical methods like Monte Carlo and tensor networks.

Contribution

It generalizes the mapping to arbitrary stabiliser and subsystem codes with correlated noise, linking error thresholds to phase transitions and enabling new decoding algorithms.

Findings

Threshold for surface code under correlated noise drops to 10.04%.

Mapping allows use of classical statistical methods for quantum decoding.

Provides a linear-time tensor network decoding algorithm for 2D codes.

Abstract

We give a broad generalisation of the mapping, originally due to Dennis, Kitaev, Landahl and Preskill, from quantum error correcting codes to statistical mechanical models. We show how the mapping can be extended to arbitrary stabiliser or subsystem codes subject to correlated Pauli noise models, including models of fault tolerance. This mapping connects the error correction threshold of the quantum code to a phase transition in the statistical mechanical model. Thus, any existing method for finding phase transitions, such as Monte Carlo simulations, can be applied to approximate the threshold of any such code, without having to perform optimal decoding. By way of example, we numerically study the threshold of the surface code under mildly correlated bit-flip noise, showing that noise with bunching correlations causes the threshold to drop to $p_{\textrm{corr}}=10.04(6)\%$, from its known iid value of $p_{\text{iid}}=10.917(3)\%$. Complementing this, we show that the mapping also allows us to utilise any algorithm which can calculate/approximate partition functions of classical statistical mechanical models to perform optimal/approximately optimal decoding. Specifically, for 2D codes subject to locally correlated noise, we give a linear-time tensor network-based algorithm for approximate optimal decoding which extends the MPS decoder of Bravyi, Suchara and Vargo.