A decoder for the triangular color code by matching on a M\"obius strip

TL;DR

This paper introduces a novel decoder for the triangular color code on a M"obius strip topology, achieving competitive logical failure rates and demonstrating potential for practical fault-tolerant quantum computation.

Contribution

It proposes a new decoder for the planar color code with a M"obius topology, improving error correction performance and generalizing to various noise models and code types.

Findings

Logical failure rate comparable to surface code

Corrects all errors of weight ≤ (d-1)/2 for d ≤ 13

Effective on large error configurations

Abstract

The color code is remarkable for its ability to perform fault-tolerant logic gates. This motivates the design of practical decoders that minimise the resource cost of color-code quantum computation. Here we propose a decoder for the planar color code with a triangular boundary where we match syndrome defects on a nontrivial manifold that has the topology of a M\"{o}bius strip. A basic implementation of our decoder used on the color code with hexagonal lattice geometry demonstrates a logical failure rate that is competitive with the optimal performance of the surface code, $\sim p^{\alpha \sqrt{n}}$, with $\alpha \approx 6 / 7 \sqrt{3} \approx 0.5$, error rate $p$, and $n$ the code length. Furthermore, by exhaustively testing over five billion error configurations, we find that a modification of our decoder that manually compares inequivalent recovery operators can correct all errors of weight $\le (d-1) /2$ for codes with distance $d \le 13$. Our decoder is derived using relations among the stabilizers that preserve global conservation laws at the lattice boundary. We present generalisations of our method to depolarising noise and fault-tolerant error correction, as well as to Majorana surface codes, higher-dimensional color codes and single-shot error correction.