Noisy decoding by shallow circuits with parities: classical and quantum

• URL: http://arxiv.org/abs/2302.02870v2
• Date: Tue, 19 Dec 2023 10:35:13 GMT
• Title: Noisy decoding by shallow circuits with parities: classical and quantum
• Authors: Jop Bri\"et, Harry Buhrman, Davi Castro-Silva and Niels M. P. Neumann
• Abstract summary: We show that any 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. We give a simple quantum circuit that correctly decodes the Hadamard code with probability $Omega(varepsilon2)$ even if a $(1/2 - varepsilon)$-fraction of a codeword is adversarially corrupted.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of decoding corrupted error correcting codes with NC$^0[\oplus]$ 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 $\Omega(\varepsilon^2)$ even if a $(1/2 - \varepsilon)$-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.

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