Local decoders for the 2D and 4D toric code

Nikolas P. Breuckmann, Kasper Duivenvoorden, Dominik Michels, Barbara M. Terhal

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


We analyze the performance of decoders for the 2D and 4D toric code which are local by construction. The 2D decoder is a cellular automaton decoder formulated by Harrington [1] which explicitly has a finite speed of communication and computation. For a model of independent X and Z errors and faulty syndrome measurements with identical probability, we report a threshold of 0:133% for this Harrington decoder. We implement a decoder for the 4D toric code which is based on a decoder by Hastings [2]. Incorporating a method for handling faulty syndromes we estimate a threshold of 1:59% for the same noise model as in the 2D case. We compare the performance of this decoder with a decoder based on a 4D version of Toom’s cellular automaton rule as well as the decoding method suggested by Dennis et al. [3].

Original languageEnglish (US)
Pages (from-to)181-208
Number of pages28
JournalQuantum Information and Computation
Issue number3-4
StatePublished - Mar 1 2017

Bibliographical note

Funding Information:
We thank Jim Harrington for initially providing us with his software implementation of the decoder. This work is supported by the European Research Council (EQEC, ERC Consol-idator Grant No: 682726). We would like to thank Michael Kastyorano for valuable feedback on the 4D toric code results and Fernando Pastawski for discussing his results on the Toom's rule decoder.

Publisher Copyright:
© Rinton Press.

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Computational Theory and Mathematics


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