Efficient Hamiltonian, structure and trace distance learning of Gaussian states

When learning descriptions of many-body quantum systems, assuming locality of interactions can dramatically reduce the required number of observations and computational time compared to unstructured approaches. In this work, we uncover an infinite-dimensional instance of this phenomenon, initiating the study of Hamiltonian learning for positive-temperature bosonic Gaussian states. We present sample- and computationally efficient protocols for inferring the parameters and graph structure of underlying quadratic Hamiltonians, provided that temperature, squeezing, displacement, and the interaction graph's maximal degree are bounded. Notably, our protocol relies solely on heterodyne measurements, which are often experimentally feasible, and achieves a sample complexity that scales only logarithmically with the number of modes. Furthermore, our techniques allow for the learning of positive-temperature Gaussian states in trace distance with quadratic scaling in precision and polynomial scaling in the number of modes.