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.

翻译：我们研究了在经典和量子设定下，利用NC$^0[\oplus]$电路对受噪声污染的纠错码进行解码的问题。结果表明，当码字通过具有正误码率的噪声信道传输时，任何此类经典电路只能正确恢复极小比例的消息。此前这一结论仅对具有大对偶距离的线性码成立，而我们的结果适用于任意码。相比之下，我们给出了一种简单的量子电路，即使码字中$(1/2 - \varepsilon)$比例的部分受到对抗性破坏，该电路仍能以$\Omega(\varepsilon^2)$的概率正确解码哈达玛码。我们的经典困难性结果基于有限域上多元多项式在偏置输入分布下的等分布现象，这一结论通过结构-随机性策略证明，其中引入了高维多项式映射的新秩概念（该概念本身可能具有独立意义）。我们的量子电路受以下研究启发：Bernstein-Vazirani问题的非局域版本、Watts等人提出的"穷人猫态"生成技术，以及Takahashi与Tani提出的OR函数常量深度量子电路。