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

TL;DR

This paper introduces the Eigenstate-to-Hamiltonian Construction (EHC), an inverse computational method that derives Hamiltonians from a given wave function, expanding the exploration of quantum model spaces beyond traditional forward methods.

Contribution

The paper presents EHC, a novel inverse approach to identify Hamiltonians from wave functions, applicable to various models and capable of revealing new quantum states and relationships.

Findings

Constructed a parent Hamiltonian with a new antiferromagnetic ground state

Identified Hamiltonians with degenerate ground states for targeted wave functions

Revealed Hamiltonians sharing ground states with well-known quantum models

Abstract

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 antiferromagnetic 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.