使用奇偶校验的浅电路进行噪声解码：经典与量子
Noisy decoding by shallow circuits with parities: classical and quantum
AI总结：
研究证明正错误率下任意码都无法由 NC$^0[\oplus]$ 经典浅电路有效解码，并提出可在近半码字遭对抗性损坏时解码 Hadamard 码的量子电路。
摘要：
我们考虑在经典和量子环境中用 NC$^0[\oplus]$ 电路解码受损纠错码的问题。我们证明，如果码字通过具有正错误率的噪声信道发送，那么任何此类经典电路都只能正确恢复极小一部分消息，且该比例趋于零。此前这一点仅针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，我们给出了一个简单的量子电路，即使码字中有 $(1/2 - \varepsilon)$ 比例遭到对抗性损坏，它仍能以 $\Omega(\varepsilon^2)$ 的概率正确解码 Hadamard 码。我们的经典困难性结果基于有限域上多元多项式在有偏输入分布下的等分布现象。这是通过一种结构与随机性对立的策略证明的，该策略基于为高维多项式映射提出的新秩概念，可能具有独立意义。我们的量子电路灵感来自 Bernstein-Vazirani 问题的非局部版本、Watts 等人用于生成“穷人的猫态”的技术，以及 Takahashi 和 Tani 提出的用于 OR 函数的恒定深度量子电路。
英文摘要：
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 $Ω(\varepsilon^2)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
