{
  "id": "2309.12165",
  "version": "v1",
  "published": "2023-09-21T15:23:41.000Z",
  "updated": "2023-09-21T15:23:41.000Z",
  "title": "Analysis of the Error-Correcting Radius of a Renormalisation Decoder for Kitaev's Toric Code",
  "authors": [
    "Wouter Rozendaal",
    "Gilles Zémor"
  ],
  "categories": [
    "quant-ph",
    "cs.IT",
    "math.IT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-21T15:23:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kitaevs toric code",
      "renormalisation decoder",
      "error-correcting radius",
      "error pattern",
      "best trade-offs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}