The advantage of quantum control in many-body Hamiltonian learning

We study the problem of learning the Hamiltonian of a many-body quantum system from experimental data. We show that the rate of learning depends on the amount of control available during the experiment. We consider three control models: one where time evolution can be augmented with instantaneous quantum operations, one where the Hamiltonian itself can be augmented by adding constant terms, and one where the experimentalist has no control over the system's time evolution. With continuous quantum control, we provide an adaptive algorithm for learning a many-body Hamiltonian at the Heisenberg limit: $T = \mathcal{O}(\epsilon^{-1})$, where $T$ is the total amount of time evolution across all experiments and $\epsilon$ is the target precision. This requires only preparation of product states, time-evolution, and measurement in a product basis. In the absence of quantum control, we prove that learning is standard quantum limited, $T = \Omega(\epsilon^{-2})$, for large classes of many-body Hamiltonians, including any Hamiltonian that thermalizes via the eigenstate thermalization hypothesis. These results establish a quadratic advantage in experimental runtime for learning with quantum control.