Minimising surface-code failures using a color-code decoder

TL;DR

This paper introduces a novel surface-code decoder that leverages color-code decoding algorithms to significantly reduce logical error rates in fault-tolerant quantum computing, especially under depolarising noise.

Contribution

It presents a new decoder for the surface code based on mapping to the color code, achieving exponential improvements in logical error rates over existing methods.

Findings

Decodes all weight d/2 depolarising errors for even code distance d.

Reduces logical error rate exponentially by O(2^{d/2}) compared to traditional decoders.

Demonstrates exponential improvement in correcting independent bit-flip errors in the color code.

Abstract

The development of practical, high-performance decoding algorithms reduces the resource cost of fault-tolerant quantum computing. Here we propose a decoder for the surface code that finds low-weight correction operators for errors produced by the depolarising noise model. The decoder is obtained by mapping the syndrome of the surface code onto that of the color code, thereby allowing us to adopt more sophisticated color-code decoding algorithms. Analytical arguments and exhaustive testing show that the resulting decoder can find a least-weight correction for all weight $d/2$ depolarising errors for even code distance $d$. This improves the logical error rate by an exponential factor $O(2^{d/2})$ compared with decoders that treat bit-flip and dephasing errors separately. We demonstrate this improvement with analytical arguments and supporting numerical simulations at low error rates. Of independent interest, we also demonstrate an exponential improvement in logical error rate for our decoder used to correct independent and identically distributed bit-flip errors affecting the color code compared with more conventional color-code decoding algorithms.