  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.
