Analysis of the Error-Correcting Radius of a Renormalisation Decoder for Kitaev's Toric Code

TL;DR

This paper investigates the worst-case error-correcting capabilities of a renormalisation decoder for Kitaev's toric code, establishing bounds on the smallest uncorrectable error weight and the correctable error threshold.

Contribution

It introduces a study of the decoder's performance against adversarial errors, providing bounds on error weights that are uncorrectable and correctable.

Findings

Uncorrectable error pattern with weight scaling as d^{1/2}

Decoder corrects all errors of weight less than (5/6) * d^{log_{2}(6/5)}

Analysis of the decoder's robustness against worst-case errors

Abstract

Kitaev's toric code is arguably the most studied quantum code and is expected to be implemented in future generations of quantum computers. The renormalisation decoders introduced by Duclos-Cianci and Poulin exhibit one of the best trade-offs between efficiency and speed, but one question that was left open is how they handle worst-case or adversarial errors, i.e. what is the order of magnitude of the smallest weight of an error pattern that will be wrongly decoded. We initiate such a study involving a simple hard-decision and deterministic version of a renormalisation decoder. We exhibit an uncorrectable error pattern whose weight scales like $d^{1/2}$ and prove that the decoder corrects all error patterns of weight less than $\frac{5}{6} d^{\log_{2}(6/5)}$, where $d$ is the minimum distance of the toric code.