发表机构
Center for Theoretical Physics — a Leinweber Institute, MIT; Department of Computer Science, University of Warwick; QuICS, University of Maryland; Department of Computer Science and Technology, University of Cambridge（麻省理工学院莱因韦伯理论物理中心; 华威大学计算机科学系; 马里兰大学量子信息与计算科学中心; 剑桥大学计算机科学与技术系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明时间反转是缩小哈密顿量性质测试中演化时间差距的必要条件，通过傅里叶分析方法建立下界和匹配上界，确立其为影响量子测量精度与算法鲁棒性的关键资源。
AI 中文摘要
量子算法在传感和学习中的许多任务中实现了海森堡极限缩放，使用 $O(1/\varepsilon)$ 的总演化时间即可达到精度 $\varepsilon$。然而，在哈密顿量性质测试中，已知的最佳算法需要 $O(1/\varepsilon^2)$ 的总演化时间，除非能够在哈密顿量下反转时间的流动。在这项工作中，我们证明了时间反转是弥合这一差距所必需的。我们针对哈密顿量认证和局域性测试，在从平均情况到最坏情况的一系列连续误差度量下，建立了演化时间下界。我们还证明了在所有情况下，无论有无时间反转，哈密顿量认证的匹配上界，表明时间反转带来的优势持续存在。我们的结果源于一种新的傅里叶分析方法，用于连续时间中的量子下界，该方法对时间反转敏感，并且适用于哈密顿量性质测试之外。这为振幅放大和估计需要逆运算提供了另一种证明，改进了Tang和Wright的下界，并扩展到分数查询。我们还表明，对于无结构搜索，即使预言机是固定且相干的，没有逆查询时，预言机相位的不确定性也会破坏量子加速。我们的工作严格确立了时间反转作为控制量子测量精度和量子算法鲁棒性的一种资源。
英文摘要
Quantum algorithms enable Heisenberg-limited scaling across many tasks in sensing and learning, achieving precision $\varepsilon$ using $O(1/\varepsilon)$ total evolution time. However, in Hamiltonian property testing, the best known algorithms have $O(1/\varepsilon^2)$ total evolution time, unless it is possible to $\textit{reverse}$ the flow of time under the Hamiltonian. In this work, we prove that time reversal is necessary to close this gap. We establish evolution time lower bounds for Hamiltonian certification and locality testing, for a continuous family of error metrics ranging from average- to worst-case. We also prove matching upper bounds for Hamiltonian certification in all cases, both with and without time reversal, showing that the advantage from time reversal persists. Our results follow from a new Fourier-analytic approach to quantum lower bounds in continuous time, which is sensitive to time reversal and applicable beyond Hamiltonian property testing. This gives an alternative proof that inverses are required for amplitude amplification and estimation, improving the lower bounds of Tang and Wright and extending to fractional queries. We also show that, for unstructured search, uncertainty in the phase of the oracle destroys the quantum speedup without inverse queries, even when the oracle is fixed and coherent. Our work rigorously establishes time reversal as a resource governing both the precision of quantum measurement and the robustness of quantum algorithms.
Comments36 pages, 2 figures
