babylonpolice.com

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.