Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead

TL;DR

This paper presents a gauge fixing technique to enhance the error correction thresholds of subsystem codes, leading to higher thresholds, lower qubit overhead, and the development of new finite-rate LDPC codes with improved performance under circuit-level noise.

Contribution

It introduces a novel gauge fixing method to improve subsystem code performance and constructs finite-rate LDPC codes with optimal measurement schedules that outperform existing codes.

Findings

Threshold increased from 0.67% to 0.81% under depolarising noise.

Threshold further increased to 2.22% with bias noise.

Finite-rate codes outperform surface and toric codes at 0.2% error rate.

Abstract

We introduce a technique that uses gauge fixing to significantly improve the quantum error correcting performance of subsystem codes. By changing the order in which check operators are measured, valuable additional information can be gained, and we introduce a new method for decoding which uses this information to improve performance. Applied to the subsystem toric code with three-qubit check operators, we increase the threshold under circuit-level depolarising noise from $0.67\%$ to $0.81\%$. The threshold increases further under a circuit-level noise model with small finite bias, up to $2.22\%$ for infinite bias. Furthermore, we construct families of finite-rate subsystem LDPC codes with three-qubit check operators and optimal-depth parity-check measurement schedules. To the best of our knowledge, these finite-rate subsystem codes outperform all known codes at circuit-level depolarising error rates as high as $0.2\%$, where they have a qubit overhead that is $4.3\times$ lower than the most efficient version of the surface code and $5.1\times$ lower than the subsystem toric code. Their threshold and pseudo-threshold exceeds $0.42\%$ for circuit-level depolarising noise, increasing to $2.4\%$ under infinite bias using gauge fixing.