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Towards Local Testability for Quantum Coding

Authors Anthony Leverrier , Vivien Londe, Gilles Zémor

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Author Details

Anthony Leverrier
  • Inria, Paris, France
Vivien Londe
  • Microsoft, Issy-les-moulineaux, France
Gilles Zémor
  • Institut de Mathématiques de Bordeaux, UMR 5251, France

Funding

  • Leverrier, Anthony: Agence Nationale de la Recherche (ANR): QuantERA project QCDA.
  • Zémor, Gilles: Agence Nationale de la Recherche (ANR): ANR-18-CE47-0010 project QUDATA.

Acknowledgements

We would like to thank Benjamin Audoux, Alain Couvreur, Omar Fawzi, Antoine Grospellier and Jean-Pierre Tillich for many fruitful discussions on quantum codes.

Cite As Get BibTex

Anthony Leverrier, Vivien Londe, and Gilles Zémor. Towards Local Testability for Quantum Coding. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 65:1-65:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ITCS.2021.65

Abstract

We introduce the hemicubic codes, a family of quantum codes obtained by associating qubits with the p-faces of the n-cube (for n > p) and stabilizer constraints with faces of dimension (p ± 1). The quantum code obtained by identifying antipodal faces of the resulting complex encodes one logical qubit into N = 2^{n-p-1} binom(n,p) physical qubits and displays local testability with a soundness of Ω(1/log(N)) beating the current state-of-the-art of 1/log²(N) due to Hastings. We exploit this local testability to devise an efficient decoding algorithm that corrects arbitrary errors of size less than the minimum distance, up to polylog factors. 
We then extend this code family by considering the quotient of the n-cube by arbitrary linear classical codes of length n. We establish the parameters of these generalized hemicubic codes. Interestingly, if the soundness of the hemicubic code could be shown to be constant, similarly to the ordinary n-cube, then the generalized hemicubic codes could yield quantum locally testable codes of length not exceeding an exponential or even polynomial function of the code dimension.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Quantum error correcting code

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