Variational wave functions, ground state and their overlap

TL;DR

This paper introduces a formula to compute the overlap between a variational wave function and the ground state using imaginary time quantum dynamics, simplifying assessment of wave function quality in quantum Monte Carlo methods.

Contribution

It derives a general formula linking the wave function overlap to the energy curve in imaginary time, facilitating easier evaluation in simulations.

Findings

Overlap relates to the area under the energy vs. imaginary time curve.

Good variational wave functions have high overlap with the ground state.

Overlap calculation can be integrated into routine quantum Monte Carlo simulations.

Abstract

An intrinsic measure of the quality of a variational wave function is given by its overlap with the ground state of the system. We derive a general formula to compute this overlap when quantum dynamics in imaginary time is accessible. The overlap is simply related to the area under the $E(\tau)$ curve, i.e. the energy as a function of imaginary time. This has important applications to, for example, quantum Monte-Carlo algorithms where the overlap becomes as a simple byproduct of routine simulations. As a result, we find that the practical definition of a good variational wave function for quantum Monte-Carlo simulations, {\it i.e.} fast convergence to the ground state, is equivalent to a good overlap with the actual ground state of the system.