Analysis of the Error-Correcting Radius of a Renormalisation Decoder for Kitaev's Toric Code
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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 5/6 d^log_2(6/5), where d is the minimum distance of the toric code.
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