Noisy decoding by shallow circuits with parities: classical and quantum
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We consider the problem of decoding corrupted error correcting codes with NC^0[⊕] 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 non-trivial 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(ε^2) 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.
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