Locality is a fundamental feature of many physical time evolutions. Assumptions on locality and related structural properties also underlie recently proposed procedures for learning an unknown Hamiltonian from access to the induced time evolution. However, no protocols to rigorously test whether an unknown Hamiltonian is local were known. We investigate Hamiltonian locality testing as a property testing problem, where the task is to determine whether an unknown $n$-qubit Hamiltonian $H$ is $k$-local or $\varepsilon$-far from all $k$-local Hamiltonians, given access to the time evolution along $H$. First, we emphasize the importance of the chosen distance measure: With respect to the operator norm, a worst-case distance measure, incoherent quantum locality testers require $\tilde{\Omega}(2^n)$ many time evolution queries and an expected total evolution time of $\tilde{\Omega}(2^n / \varepsilon)$, and even coherent testers need $\Omega(2^{n/2})$ many queries and $\Omega(2^{n/2}/\varepsilon)$ total evolution time. In contrast, when distances are measured according to the normalized Frobenius norm, corresponding to an average-case distance, we give a sample-, time-, and computationally efficient incoherent Hamiltonian locality testing algorithm based on randomized measurements. In fact, our procedure can be used to simultaneously test a wide class of Hamiltonian properties beyond locality. Finally, we prove that learning a general Hamiltonian remains exponentially hard with this average-case distance, thereby establishing an exponential separation between Hamiltonian testing and learning. Our work initiates the study of property testing for quantum Hamiltonians, demonstrating that a broad class of Hamiltonian properties is efficiently testable even with limited quantum capabilities, and positioning Hamiltonian testing as an independent area of research alongside Hamiltonian learning.

翻译：局域性是许多物理时间演化的基本特征。局域性假设及相关结构性质也支撑着近期提出的从时间演化中学习未知哈密顿量的算法。然而，目前尚缺乏严格检验未知哈密顿量是否具有局域性的协议。我们将哈密顿量局域性检验作为性质检验问题进行研究，任务是在给定沿$H$的时间演化访问权限时，判断未知$n$量子比特哈密顿量$H$是$k$-局域的还是与所有$k$-局域哈密顿量$\varepsilon$-远离。首先，我们强调所选距离测度的重要性：在算子范数（一种最坏情形距离测度）下，非相干量子局域性检验器需要$\tilde{\Omega}(2^n)$次时间演化查询和$\tilde{\Omega}(2^n / \varepsilon)$的预期总演化时间，即便相干检验器也需要$\Omega(2^{n/2})$次查询和$\Omega(2^{n/2}/\varepsilon)$的总演化时间。相比之下，当采用归一化Frobenius范数（对应平均情形距离）度量距离时，我们给出了一种基于随机测量的高效（样本、时间和计算复杂度）非相干哈密顿量局域性检验算法。事实上，我们的方法可同时检验超出局域性之外的广泛哈密顿量性质。最后，我们证明在此平均情形距离下，学习一般哈密顿量仍然指数困难，从而在哈密顿量检验与学习之间建立了指数级分离。本工作开创了量子哈密顿量性质检验的研究，表明即使只有有限量子能力，一大类哈密顿量性质也可高效检验，并将哈密顿量检验定位为与哈密顿量学习并列的独立研究领域。