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 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 $\Omega(\varepsilon^2)$ 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：我们研究在经典和量子设置下，使用NC$^0[\oplus]$电路对受损纠错码进行解码的问题。我们证明，当码字通过具有正错误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非平凡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，我们给出一个简单的量子电路，即使码字的$(1/2 - \varepsilon)$部分受到敌对性破坏，它仍能以$\Omega(\varepsilon^2)$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变量多项式在偏置输入分布下的等分布现象。该结论通过一种基于高维多项式映射新秩概念的结构-随机性策略证明，这一概念可能具有独立价值。我们的量子电路灵感来源于伯恩斯坦-瓦齐拉尼问题的非局域版本、Watts等人生成"穷人的猫态"的技术以及Takahashi和Tani提出的用于OR函数的常数深度量子电路。