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

Dissipation-enabled bosonic Hamiltonian learning via new information-propagation bounds