发表机构
Inria-Saclay; Harvard University; CNRS; Université Paris Cité; IRIF（法国国家信息与自动化研究所萨克莱分院; 哈佛大学; 法国国家科学研究中心; 巴黎西岱大学; 法国信息学基础研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明在仅访问时间演化算子（无逆算子）的模型中，哈密顿量局域性检验、认证及振幅估计均需Ω(1/ε²)总演化时间，排除了海森堡极限缩放，并通过连续时间对手方法建立下界。
AI 中文摘要
我们为在能够访问时间演化算子但不能访问其逆算子的情况下进行哈密顿量性质检验建立了下界。每次实验可以多次查询时间演化算子，哈密顿量之间的距离以归一化Frobenius范数度量。在该模型中，我们证明检验一个哈密顿量是否为k-局域或与每个k-局域哈密顿量ε-远需要Ω(1/ε²)的总演化时间，与Kallaugher和Liang (TQC'25)的上界匹配。我们还证明检验一个未知哈密顿量是否等于目标哈密顿量或与之ε-远需要Ω(1/ε²)的总演化时间，与Sinha和Tong (2025)的上界匹配。这些是哈密顿量学习和检验中自然问题的首批下界，排除了1/ε的海森堡极限缩放。作为第三个结果，我们证明以精度ε进行振幅估计需要Ω(1/ε²)的总时间演化，在连续时间查询模型中恢复了Tang和Wright (QIP'26)的结果。所有三个结果都源于区分零哈密顿量与适当选择的随机哈密顿量系综的困难性。我们通过将连续时间对手方法适应于前向哈密顿量演化来建立这种困难性。
英文摘要
We establish lower bounds for Hamiltonian property testing with access to the time-evolution operator but not its inverse. Each experiment may query the time-evolution operator multiple times, and distances between Hamiltonians are measured in the normalized Frobenius norm. In this model, we show that testing whether a Hamiltonian is $k$-local or $\varepsilon$-far from every $k$-local Hamiltonian requires $Ω(1/\varepsilon^2)$ total evolution time, matching the upper bound of Kallaugher and Liang (TQC'25). We also prove that testing whether an unknown Hamiltonian equals a target Hamiltonian or is $\varepsilon$-far from it requires $Ω(1/\varepsilon^2)$ total evolution time, matching the upper bound of Sinha and Tong (2025). These are the first lower bounds for natural problems in Hamiltonian learning and testing that rule out Heisenberg-limited scaling of $1/\varepsilon$. As a third result, we show that amplitude estimation to precision $\varepsilon$ requires $Ω(1/\varepsilon^2)$ total time evolution, recovering the result of Tang and Wright (QIP'26) in the continuous-time query model. All three results follow from the hardness of distinguishing the zero Hamiltonian from a suitably chosen ensemble of random Hamiltonians. We establish this hardness by adapting the continuous-time adversary method to forward Hamiltonian evolution.
Comments28 pages
