# A computational inverse method for constructing spaces of quantum models from wave functions. (arXiv:1802.01590v1 [cond-mat.str-el])

Traditional computational methods for studying quantum many-body systems are

"forward methods," which take quantum models, i.e., Hamiltonians, as input and

produce ground states as output. However, such forward methods often limit

one's perspective to a small fraction of the space of possible Hamiltonians. We

introduce an alternative computational "inverse method," the

Eigenstate-to-Hamiltonian Construction (EHC), that allows us to better

understand the vast space of quantum models describing strongly correlated

systems. EHC takes as input a wave function $|\psi_T\rangle$ and produces as

output Hamiltonians for which $|\psi_T\rangle$ is an eigenstate. This is

accomplished by computing the quantum covariance matrix, a quantum mechanical

generalization of a classical covariance matrix. EHC is widely applicable to a

number of models and in this work we consider seven different examples. Using

the EHC method, we construct a parent Hamiltonian with a new type of triplet

dimer ground state, a parent Hamiltonian with two different targeted degenerate

ground states, and large classes of parent Hamiltonians with the same ground

states as well-known quantum models, such as the Majumdar-Ghosh model, the XX

chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model. EHC gives an

alternative inverse approach for studying quantum many-body phenomena.