Characterizing the parent Hamiltonians for a complete set of orthogonal wave functions: An inverse quantum problem

Abstract

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 our goal is to characterize the relevant wave functions and energies (the spectrum) of the system. Here, we take the opposite approach; starting from a reasonable collection of orthogonal wave functions, we try to characterize the associated parent Hamiltonians, to see how the wave functions and the energy values affect 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 basis states, 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.

Computational Inverse Method for Constructing Spaces of Quantum Models from Wave Functions

“…To overcome these limitations, we developed a novel inverse method that automates the construction of parent Hamiltonians from wave functions by searching for models in a large space of "physically reasonable" Hamiltonians. More broadly, inverse methods have been successful in applications such as solving machine learning problems [25], targeting many-particle ordering in classical materials [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41], and promoting certain properties in quantum many-body systems [42][43][44][45][46][47].…”