  We study the inverse problem of constructing an appropriate Hamiltonian from
a physically reasonable set of orthogonal wave functions for a quantum spin
system. Usually, we are given a local Hamiltonian and try to characterize the
relevant wave functions and energies (the spectrum) of the system. Here, we
take the opposite approach; starting from a collection of orthogonal wave
functions, our goal is to characterize the associated parent Hamiltonian, to
see how the wave functions and the energy values determine the structure of the
parent Hamiltonian. Specifically, we obtain (quasi) local Hamiltonians by a
complete set of (multilayer) product states and a local mapping of the energy
values to the wave functions. On the other hand, a complete set of tree wave
functions (having a tree structure) results to nonlocal Hamiltonians and
operators which flip simultaneously all the spins in a single branch of the
tree graph. We observe that even for a given set of wave functions, the energy
spectrum can significantly change the nature of interactions in the
Hamiltonian. These effects can be exploited in a quantum engineering problem
optimizing an objective functional of the Hamiltonian.
