  Reliable quantum technology requires knowledge of the dynamics governing the
underlying system. This problem of characterizing and benchmarking quantum
devices or experiments in continuous time is referred to as the Hamiltonian
learning problem. In contrast to multi-qubit systems, learning guarantees for
the dynamics of bosonic systems have hitherto remained mostly unexplored. For
$m$-mode Hamiltonians given as polynomials in annihilation and creation
operators with modes arranged on a lattice, we establish a simple moment
criterion in terms of the particle number operator which ensures that learning
strategies from the finite-dimensional setting extend to the bosonic setting,
requiring only coherent states and heterodyne detection on the experimental
side. We then propose an enhanced procedure based on added dissipation that
even works if the Hamiltonian time evolution violates this moment criterion:
With high success probability it learns all coefficients of the Hamiltonian to
accuracy $\varepsilon$ using a total evolution time of
$\mathcal{O}(\varepsilon^{-2}\log(m))$. Our protocol involves the
experimentally reachable resources of projected coherent state preparation,
dissipative regularization akin to recent quantum error correction schemes
involving cat qubits stabilized by a nonlinear multi-photon driven dissipation
process, and heterodyne measurements. As a crucial step in our analysis, we
establish our moment criterion and a new Lieb-Robinson type bound for the
evolution generated by an arbitrary bosonic Hamiltonian of bounded degree in
the annihilation and creation operators combined with photon-driven
dissipation. Our work demonstrates that a broad class of bosonic Hamiltonians
can be efficiently learned from simple quantum experiments, and our bosonic
Lieb-Robinson bound may independently serve as a versatile tool for studying
evolutions on continuous variable systems.
