二面体陪集算法的傅里叶标签信息损失障碍

中文摘要 AI 辅助

我们针对二面体陪集问题（DCP）的一类广泛量子算法建立了一个不可能性定理。我们考虑了Regev（《SIAM计算杂志》，2004年）提出的傅里叶采样与子集和测量模板，这是求解DCP的主要方法之一。假设在测量子集和的低$n-1$位之后，算法从每个傅里叶标签中丢弃任意$\omega(\log n)$位，那么我们证明该算法无法成功求解DCP。这表明任何遵循此模板的算法都必须充分利用傅里叶标签，从而为开发DCP算法提供了有用的指导。作为主要应用，我们证明了Simon（IACR ePrint:2026/1591，2026年8月11日）的最新算法并不能求解DCP。我们证明，在子集和测量之后，该算法可以仅使用傅里叶标签的最高有效三分之一来实现（误差至多为指数级小），因此受我们的一般不可能性定理约束。为便于验证，我们发布了我们结果的Lean 4代码。

英文摘要

We establish a no-go theorem for a broad class of quantum algorithms for the dihedral coset problem (DCP). We consider the Fourier-sampling and subset-sum-measurement template proposed by Regev (SIAM Journal on Computing, 2004), which is one of the main approaches to solving DCP. Suppose that, after measuring the lower $n-1$ bits of the subset sum, the algorithm discards any $ω(\log n)$ bits from each of the Fourier labels. Then we prove that the algorithm cannot succeed in solving DCP. This shows that any algorithm following this template must make extensive use of the Fourier labels, and thus serves as a useful guide for developing algorithms for DCP. As a main application, we show that the recent algorithm by Simon (IACR ePrint:2026/1591, August 11 2026) does not solve DCP. We show that after the subset-sum measurement, this algorithm can be implemented (up to exponentially-small error) using only the most-significant third of the Fourier labels, and is therefore subject to our general no-go theorem. To help with verifiability, we release Lean 4 code for our results.

发表机构

• MIT（麻省理工学院）
• Google Quantum AI（谷歌量子人工智能）
• Stanford University（斯坦福大学）

机构由 AI 辅助整理，请以论文原文为准。