量子伪随机纠错码
Quantum Pseudorandom Error-Correcting Codes
- Hon Hai Research Institute（鸿海研究院）
- Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究提出量子伪随机纠错码（QPRCs），在LPN困难性假设下构造其编码与Haar随机等距或完全退极化信道不可区分，并对局域噪声鲁棒，同时解决CWS框架下非线性码的高效解码开放问题。
AI中文摘要：
伪随机纠错码（PRCs）由Christ和Gunn引入[CRYPTO 2024]，是一类其码字在计算上与均匀随机字符串不可区分的经典纠错码。我们开创了对PRCs量子类似物的研究。我们定义了量子伪随机纠错码（QPRCs）。假设带噪声学习奇偶校验（LPN）问题对$2^{O(\sqrt{n})}$时间的量子算法是困难的，我们构造了其编码与Haar随机等距不可区分的QPRCs。我们称这些为伪随机等距纠错码（PRICs），并且我们的构造对所有$o(n \frac{\log \log n}{\log n})$局域量子噪声具有鲁棒性，其中$n$是物理量子比特的数量。在相同的假设下，我们还构造了其编码与完全退极化信道不可区分且对所有$\alpha n$局域量子噪声（对于某个常数$\alpha>0$）具有鲁棒性的QPRCs。后者QPRCs直接给出了经典PRCs的量子类似物。为了构造PRICs，我们开发了两个独立感兴趣的技术要素。首先，我们引入了一种新的经典密码学原语，称为伪随机函数纠错码（PRFCs），并在相同的LPN假设下构造了它们。其次，我们在[Cross, Smith, Smolin, and Zeng, IEEE ISIT 2008]中引入的码字稳定（CWS）框架内开发了一种新的高效解码过程，该框架是通过组合（可能非线性的）经典纠错码和图来构造量子纠错码的通用框架。这解决了在[Li, Dumer, Grassl, and Pryadko, Physical Review A 2010]中提出的寻找基于非线性经典码的CWS码通用高效解码方法的开放问题。
英文摘要：
Pseudorandom error-correcting codes (PRCs), introduced by Christ and Gunn [CRYPTO 2024], are classical error-correcting codes whose codewords are computationally indistinguishable from uniformly random strings. We initiate the study of quantum analogues of PRCs. We define quantum pseudorandom error-correcting codes (QPRCs). Assuming that Learning Parity with Noise (LPN) is hard for $2^{O(\sqrt{n})}$-time quantum algorithms, we construct QPRCs whose encodings are indistinguishable from Haar-random isometries. We call these pseudorandom isometric error-correcting codes (PRICs), and our construction is robust to all $o(n \frac{\log \log n}{\log n})$-local quantum noise, where $n$ is the number of physical qubits. Under the same assumption, we also construct QPRCs whose encodings are indistinguishable from the completely depolarizing channel and that are robust to all $αn$-local quantum noise for some constant $α>0$. The latter QPRCs give a direct quantum analogue of classical PRCs. To construct PRICs, we develop two technical ingredients of independent interest. First, we introduce a new classical cryptographic primitive which we call pseudorandom functional error-correcting codes (PRFCs) and construct them under the same LPN assumption. Second, we develop a new efficient decoding procedure within the codeword-stabilized (CWS) framework introduced in [Cross, Smith, Smolin, and Zeng, IEEE ISIT 2008], which is a generic framework for constructing quantum error-correcting codes by combining (possibly nonlinear) classical error-correcting codes and graphs. This resolves the open problem of finding a general method for efficiently decoding such CWS codes based on nonlinear classical codes, raised in [Li, Dumer, Grassl, and Pryadko, Physical Review A 2010].
