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。对局域性及相关结构性质的假设也是近期提出的从诱导时间演化中学习未知哈密顿量方法的基础。然而，此前尚不存在能够严格测试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测试作为一个性质测试问题进行研究，其任务是在给定沿未知$n$-量子比特哈密顿量$H$的时间演化访问权限下，判定该哈密顿量是$k$-局域的，还是与所有$k$-局域哈密顿量$\varepsilon$-远离的。首先，我们强调所选距离度量的重要性：对于算子范数这一最坏情况距离度量，非相干量子局域性测试器需要$\tilde{\Omega}(2^n)$次时间演化查询和$\tilde{\Omega}(2^n / \varepsilon)$的预期总演化时间，即使是相干测试器也需要$\Omega(2^{n/2})$次查询和$\Omega(2^{n/2}/\varepsilon)$的总演化时间。相反，当按照归一化弗罗贝尼乌斯范数（对应于平均情况距离）测量距离时，我们提出了一种基于随机测量的样本、时间和计算高效的非相干哈密顿量局域性测试算法。事实上，我们的方法可用于同时测试超越局域性的广泛哈密顿量性质类别。最后，我们证明了在这种平均情况距离下学习一般哈密顿量仍然具有指数级难度，从而在哈密顿量测试与学习之间建立了指数级分离。我们的工作开创了量子哈密顿量性质测试的研究，证明即使量子能力有限，也能高效测试广泛的哈密顿量性质，并将哈密顿量测试定位为与哈密顿量学习并行的独立研究领域。