Approximate Hamiltonian Reconstruction from Undercomplete Operator Bases

We examine the effects of incomplete operator bases on the so-called correlation matrix Hamiltonian reconstruction technique [Qi and Ranard; Quantum 3, 159 (2019)]. Our study is motivated by the experimental inaccessibility of many correlators which are needed for an exact reconstruction. We address the fidelity of approximate reconstructions and try to understand when a reconstruction attempt will go from approximately correct to wholly incorrect. We also derive a perturbative expression that relates the degree of failure in the approximate method to the magnitude of the missing terms in the reconstructed Hamiltonian. Our model systems include simple spin chains with non-local but decaying couplings (e.g. the Haldane-Shastry model) and those with higher spin exchanges with diminishing coefficients (like in the Mott limit of the Hubbard model). In the former case, we explore truncations that remove the long-distance or high-momentum physics. In the latter case, we truncate the higher spin exchanges.