Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract)

Authors Jop Briët , Harry Buhrman, Davi Castro-Silva , Niels M. P. Neumann

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Author Details

Jop Briët
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Harry Buhrman
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands
Davi Castro-Silva
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Niels M. P. Neumann
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
  • The Netherlands Organisation for Applied Scientific Research (TNO), Den Haag, The Netherlands


We thank Richard Cleve for helpful discussions on the proof of Theorem 2, Emanuele Viola for helpful pointers to the literature and comments on an earlier version of this manuscript, Tamar Ziegler for encouragement to explore the high-characteristic setting, as well as anonymous referees for helpful comments.

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Jop Briët, Harry Buhrman, Davi Castro-Silva, and Niels M. P. Neumann. Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract). In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 21:1-21:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We consider the problem of decoding corrupted error correcting codes with NC⁰[⊕] circuits in the classical and quantum settings. We show that any such classical circuit can correctly recover only a vanishingly small fraction of messages, if the codewords are sent over a noisy channel with positive error rate. Previously this was known only for linear codes with large dual distance, whereas our result applies to any code. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with probability Ω(ε²) even if a (1/2 - ε)-fraction of a codeword is adversarially corrupted. Our classical hardness result is based on an equidistribution phenomenon for multivariate polynomials over a finite field under biased input-distributions. This is proved using a structure-versus-randomness strategy based on a new notion of rank for high-dimensional polynomial maps that may be of independent interest. Our quantum circuit is inspired by a non-local version of the Bernstein-Vazirani problem, a technique to generate "poor man’s cat states" by Watts et al., and a constant-depth quantum circuit for the OR function by Takahashi and Tani.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Coding theory
  • Theory of computation → Circuit complexity
  • Coding theory
  • circuit complexity
  • quantum complexity theory
  • higher-order Fourier analysis
  • non-local games


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