  We develop a protocol for learning a class of interacting bosonic
Hamiltonians from dynamics with Heisenberg-limited scaling. For Hamiltonians
with an underlying bounded-degree graph structure, we can learn all parameters
with root mean squared error $\epsilon$ using $\mathcal{O}(1/\epsilon)$ total
evolution time, which is independent of the system size, in a way that is
robust against state-preparation and measurement error. In the protocol, we
only use bosonic coherent states, beam splitters, phase shifters, and homodyne
measurements, which are easy to implement on many experimental platforms. A key
technique we develop is to apply random unitaries to enforce symmetry in the
effective Hamiltonian, which may be of independent interest.
