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Traditional computational methods for studying quantum many-body systems are"forward methods," which take quantum models, i.e., Hamiltonians, as input andproduce ground states as output. However, such forward methods often limitone's perspective to a small fraction of the space of possible Hamiltonians. Weintroduce an alternative computational "inverse method," theEigenstate-to-Hamiltonian Construction (EHC), that allows us to betterunderstand the vast space of quantum models describing strongly correlatedsystems. EHC takes as input a wave function $|\psi_T\rangle$ and produces asoutput Hamiltonians for which $|\psi_T\rangle$ is an eigenstate. This isaccomplished by computing the quantum covariance matrix, a quantum mechanicalgeneralization of a classical covariance matrix. EHC is widely applicable to anumber of models and in this work we consider seven different examples. Usingthe EHC method, we construct a parent Hamiltonian with a new type ofantiferromagnetic ground state, a parent Hamiltonian with two differenttargeted degenerate ground states, and large classes of parent Hamiltonianswith the same ground states as well-known quantum models, such as theMajumdar-Ghosh model, the XX chain, the Heisenberg chain, the Kitaev chain, anda 2D BdG model.

Computational inverse method for constructing spaces of quantum models from wave functions