Noisy decoding by shallow circuits with parities: classical and quantum

TL;DR

This paper demonstrates classical intractability of decoding error-correcting codes with shallow circuits and introduces a quantum decoding circuit that outperforms classical limits for the Hadamard code, highlighting fundamental differences between classical and quantum decoding.

Contribution

It proves classical hardness for decoding with shallow circuits for any code and presents a quantum circuit that successfully decodes the Hadamard code under adversarial noise.

Findings

Classical shallow circuits can decode only a negligible fraction of messages over noisy channels.

A quantum circuit can decode the Hadamard code with probability proportional to the square of the error rate.

The classical proof uses a new polynomial rank concept and an equidistribution phenomenon.

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.