Myeongjin Shin

We study the problems of certifying and learning local quantum Hamiltonians and their associated Gibbs states. We first address Hamiltonian certification given real-time access to the dynamics of an unknown k-local Hamiltonian. Given oracle access to its time-evolution operator and a fully specified target Hamiltonian, the task is to decide whether the two Hamiltonians are identical or differ by at least a prescribed accuracy in normalized Frobenius norm, while minimizing the total evolution time. We introduce the first certification protocol that achieves optimal performance for all constant-locality Hamiltonians. For general n-qubit, k-local, traceless Hamiltonians, our algorithm succeeds with high probability using total evolution time that scales inversely with the target accuracy, and for constant locality this matches the fundamental lower bound, achieving Heisenberg-limit scaling. In contrast to prior approaches, our method requires neither inverse evolution nor controlled operations, and relies only on forward real-time dynamics. We then turn to thermal states generated by local Hamiltonians. We develop algorithms for both learning and certifying Gibbs states that are fully sample-efficient in all relevant parameters. For polynomially bounded temperature, our methods achieve exponential improvements over general quantum state tomography. While the learning algorithm is inherently time-inefficient due to covering arguments, the certification algorithm is both sample- and time-efficient, resolving a previously open question on efficient Gibbs state testing. Together, these results establish optimal or near-optimal complexity bounds for characterizing local quantum systems in both dynamical and thermal regimes.