发表机构
University of Copenhagen; Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP（哥本哈根大学; 里昂大学、里昂高等师范学院、里昂第一大学、法国国家科学研究中心、法国国家信息与自动化研究所、里昂信息实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出量子算法，以海森堡标度估计量子哈密顿量的对数配分函数，通过自适应冷却调度和递归倍增恒等式，将查询复杂度改进至Õ(n^{5/4}/ε)，并证明在受控制备下可达到最优Õ(n/ε)。
AI 中文摘要
我们给出了量子算法，用于估计n量子比特量子哈密顿量的对数配分函数，达到加性误差ε，并具有海森堡标度，同时最小化对热态纯化制备的调用次数。输入是H的块编码，在选定的逆温度下制备热场双态的酉算子，及其逆算子和受控版本，以及一个自适应缓慢变化的冷却调度。以下系统大小界限假设0⪯H⪯hI，其中h=O(n)，块编码归一化α=Θ(h)。对于恒定逆温度，我们证明了长度为O(√n)的这样的调度总是存在，并且可以从重叠估计中生成，额外需要Õ(√n)次制备。给定这样的调度和经典选择的温度，我们估计对数配分函数，期望调用状态制备和块编码的次数为Õ(n^{5/4}/ε)，相比非自适应策略改进了n^{1/4}倍。为实现我们的结果，我们使用一个递归倍增恒等式，将每个调度增量表示为热场双体重叠的对数和由量子奇异值变换实现的短虚时间步长的对数的加权和，并结合先前开发的方差减少技术。如果热场双态可以在温度寄存器上受控制备，则结合量子均值估计的量子逆二项对数估计器将两种查询次数都减少到Õ(n/ε)。我们证明了后者在块编码查询中达到最优，直到多对数因子，并讨论了一维配分函数估计的端到端复杂度。
英文摘要
We give quantum algorithms that estimate the log partition function of an $n$-qubit quantum Hamiltonian to additive error $ε$ with Heisenberg scaling, while minimizing calls to thermal-state purification preparation. The input is a block encoding of $H$, unitaries preparing thermofield-double states at chosen inverse temperatures, with their inverses and controlled versions, and an adaptive slowly-varying cooling schedule. The system-size bounds below assume $0\preceq H\preceq hI$ with $h=O(n)$ and block-encoding normalization $α=Θ(h)$. For constant inverse temperature, we show that such a schedule of length $O(\sqrt n)$ always exists and can be generated from overlap estimates with $\widetilde O(\sqrt n)$ further preparations. Given such a schedule and classically chosen temperatures, we estimate the log partition function with $\widetilde O(n^{5/4}/ε)$ expected calls to state preparation and to the block encoding, improving by $n^{1/4}$ on non-adaptive strategies. To achieve our result, we use a recursive-doubling identity that expresses each schedule increment as a weighted sum of logarithms of thermofield-double overlaps and of short imaginary-time steps implemented by quantum singular value transformation combined with previously developed variance reduction techniques. If thermofield doubles can be prepared controlled on a temperature register, a quantum inverse-binomial log estimator combined with quantum mean estimation reduces both query counts to $\widetilde O(n/ε)$. We show that the latter scaling is optimal in block-encoding queries up to polylogarithmic factors and discuss end-to-end complexities for one-dimensional partition-function estimation.
Comments65 pages, 1 figure
