超越局部性与经典可解码性的OPI变体的量子算法

中文摘要 AI 辅助

Regev的归约是一种量子算法框架，通过解码对偶码来寻找满足非线性约束的码字。迄今为止，尚未被去量子化的应用依赖于高效的经典解码器和逐坐标约束，这些约束为每个坐标指定一组允许的值。我们通过两项独立的贡献分别突破了这两个限制。我们的第一项贡献使用量子解码器来寻找满足$\mathbf{B}\mathbf{y}=0$的解$\mathbf{y}\in(\mathbb{F}_q\setminus\{0\})^m$，其中$\mathbf{B}\in\mathbb{F}_q^{n\times m}$。对于固定的素数$q>2$，Chen、Liu和Zhandry（Eurocrypt 2022）解决了随机矩阵在$m=\Omega(n^2)$情况下的这个问题，而这一范围现在已被经典算法覆盖。我们将他们的模板改编为具有“二重乘法性质”的码（由$\mathbf{B}$的行张成）：码字对的逐坐标乘积张成的空间维度远小于$m$。我们的第二项贡献保留了经典解码，但允许对符号频率施加全局约束。我们研究“直方图局部”约束，这些约束指定每个符号允许出现的次数。对于这些约束的广泛族，我们证明了一个均匀随机的满足向量在重新采样一个均匀随机坐标后，仍以常数概率保持满足性。这种稳定性在可解码的汉明权重处提供了足够的傅里叶质量，从而为最优多项式交集（OPI）的变体产生了高效的量子算法。

英文摘要

Regev's reduction quantumly finds codewords satisfying nonlinear constraints by decoding the dual code. To date, applications that have not been dequantized have relied on efficient classical decoders and coordinate-wise constraints. We overcome these restrictions separately. Our first contribution uses a quantum decoder to find solutions $\mathbf{y}\in(\mathbb{F}_q\setminus\{0\})^m$ to $\mathbf{B}\mathbf{y}=0$, where $\mathbf{B}\in\mathbb{F}_q^{n\times m}$. For fixed prime $q>2$, Chen, Liu, and Zhandry solve this problem for random matrices with $m=Ω(n^2)$, a regime now covered by classical algorithms. We adapt their template to codes (spanned by the rows of $\mathbf{B}$) that satisfy a "two-fold multiplication property": the coordinate-wise products of pairs of codewords span a space of dimension smaller than $m$. Under suitable distance conditions, this allows us to solve instances with $m\leq n^{2-Ω(1)}$, beyond the established guarantees of classical algorithms. We also give an efficient classical algorithm under a stronger three-fold multiplication property, leaving intermediate regimes as candidates for quantum advantage. For Reed-Muller codes punctured at random points, our algorithm finds a word in the unpunctured dual code supported exactly on those points. This works beyond known efficient classical decoding regimes. Our second contribution retains classical decoding but allows global constraints on symbol frequencies. We study "histogram-local" constraints, which specify the allowed numbers of occurrences of each symbol. For broad families, stability under resampling one coordinate yields efficient quantum algorithms for variants of optimal polynomial intersection (OPI) combining coordinate-wise and histogram-local constraints. Adapting Yamakawa-Zhandry, we prove a quantum-classical separation for these problems relative to a classical random oracle.

发表机构

• Google Quantum AI（谷歌量子人工智能）
• Computer Science and Artificial Intelligence Laboratory, MIT（麻省理工学院计算机科学与人工智能实验室）

机构由 AI 辅助整理，请以论文原文为准。