Indirect Quantum Tomography of Quadratic Hamiltonians

A number of many-body problems can be formulated using Hamiltonians that are quadratic in the creation and annihilation operators. Here, we show how such quadratic Hamiltonians can be efficiently estimated indirectly, employing very few resources. We find that almost all properties of the Hamiltonian are determined by its surface, and that these properties can be measured even if the system can only be initialized to a mixed state. Therefore our method can be applied to various physical models, with important examples including coupled nano-mechanical oscillators, hopping fermions in optical lattices, and transverse Ising chains.