Abstract

How well can simple circuits decode corrupted error-correcting codes? We consider this problem for the class of constant-depth circuits with parities (i.e. NC^0[+] circuits) in both 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. By contrast, we give a simple quantum circuit that correctly decodes the Hadamard code with optimal probability 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 under biased input-distributions. This is proved using a structure-versus-randomness strategy based on a new notion of rank for 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.

This talk is based on joint work with Jop Briët, Harry Buhrman and Niels Neumann.