2026-07-23T13:23:51Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:19549 2024-03-06T11:04:17Z ddc:004 open_access

Noisy Decoding by Shallow Circuits with Parities: Classical and Quantum (Extended Abstract) Briët, Jop Buhrman, Harry Castro-Silva, Davi Neumann, Niels M. P. Coding theory circuit complexity quantum complexity theory higher-order Fourier analysis non-local games 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jop Briët and Harry Buhrman and Davi Castro-Silva and Niels M. P. Neumann 2024 Is Part Of LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2024.21 urn:nbn:de:0030-drops-195490 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.21 eng https://creativecommons.org/licenses/by/4.0/legalcode