发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University; Institute of Natural Sciences, Shanghai Jiao Tong University; MOE-LSC, Shanghai Jiao Tong University; School of Mathematical Sciences, East China Normal University; Key Laboratory of MEA, Ministry of Education, East China Normal University; Shanghai Key Laboratory of PMMP, East China Normal University（上海交通大学数学科学学院; 上海交通大学自然科学研究院; 上海交通大学教育部线性代数与系统控制重点实验室; 华东师范大学数学科学学院; 华东师范大学教育部数学教育重点实验室; 华东师范大学上海偏微分方程与数学物理重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对半经典体系中量子模拟的高昂成本，提出基于Weyl变量、精确Hermite矩和量子奇异值变换的波长均匀量子算法，实现多项式复杂度并突破采样定理限制。
AI 中文摘要
量子模拟的主要挑战之一是在半经典体系中计算其解的代价过高，在该体系中，德布罗意波长与特征长度尺度相比很小，且解具有高度振荡性。通过使用Weyl变量可以克服这一困难，在该变量下解不再振荡。此外，我们利用精确的Hermite矩和量子奇异值变换来处理势能的多项式与傅里叶分量，从而得到一种对所有波长范围都高效的量子算法。具体而言，该算法在空间维度上具有多项式复杂度，其离散化和查询界限不包含可能很小的波长的负幂次，因此即使空间网格无法分辨频率，它也能捕获正确的物理可观测量，从而突破了奈奎斯特-香农采样定理的限制。
英文摘要
One of the main challenges in numerical simulation of quantum dynamics is the prohibitive cost in the semi-classical regime, in which the de Broglie wave length is small compared with the characteristic length scale and the solution is highly oscillatory. For the von-Neumann equation for mixed-state quantum dynamics, this difficulty is overcome by using the Weyl variables, under which the solution is not oscillatory. Furthermore, we use the integral representation of the potential difference, which robustly captures the classical limit as the semi-classical parameter approaches zero. By using exact Hermite moments and quantum singular value transformation to treat the polynomial and sparse coordinate matrices of the dense projected smooth non-polynomial potentials respectively, we obtain a quantum algorithm efficient for {\it all} ranges of wave lengths, with complexity {\it polynomial} in the spatial dimension and discretization and query bounds containing {\it no} negative powers of the small wavelength. Thus it can capture the correct physical observables even if the spatial grid does not resolve the frequency, hence defying the Nyquist-Shannon sampling theorem.
