Hamiltonian Reconstruction: the Correlation Matrix and Incomplete Operator Bases

We explore the robustness of the correlation matrix Hamiltonian reconstruction technique with respect to the choice of operator basis, studying the effects of bases that are undercomplete and over-complete -- too few or too many operators respectively. An approximation scheme for reconstructing from an undercomplete basis is proposed and performed numerically on select models. We discuss the confounding effects of conserved quantities and symmetries on reconstruction attempts. We apply these considerations to a variety of one-dimensional systems in zero- and finite-temperature regimes.