Heisenberg-Limited Quantum Hamiltonian Learning via Randomly Spread Product-States

We show how one can asymptotically reach the Heisenberg limit in quantum Hamiltonian learning without entanglement, globally coherent measurements, or dynamical control, using only local quantum operations. Our protocol uses ensemble-averaging over the outcomes of experiments initialized in Haar-random local product states, accompanied by random Pauli measurements, leading to the effective cancellation of interference terms so that a Heisenberg-limited regime emerges for short-time experiments. Furthermore, we show that the act of ensemble averaging makes unbiased estimation data, meaning all Hamiltonian parameters can be simultaneously estimated from the same data-set, removing the need for parameter isolation. We supplement the theoretical results by showing empirically that, even away from the asymptotic limit, one can surpass the SQL using randomly spread product-state ensembles. We do so numerically by learning a selection of different disordered multi-qubit Hamiltonians in a black-box learning scenario.