Predicting quantum ground-state energy by data-driven Koopman analysis of variational parameter nonlinear dynamics

TL;DR

This paper introduces a data-driven Koopman analysis approach to estimate quantum ground-state energies by analyzing nonlinear variational parameter dynamics, extending traditional variational methods with eigenvalue-based predictions.

Contribution

It develops a novel framework combining Koopman theory with variational quantum dynamics, enabling energy estimation even outside the variational manifold.

Findings

Successfully applied to four-site transverse-field Ising model

Estimated ground-state energy using extended dynamic mode decomposition

Framework applicable to infinite chain systems with matrix product states

Abstract

In recent years, the application of machine learning to physics has been actively explored. In this paper, we study a method for estimating the ground-state energy of quantum Hamiltonians by applying data-driven Koopman analysis within the framework of variational wave functions. Koopman theory is a framework for analyzing the nonlinear dynamics of vectors, in which the dynamics are linearized by lifting the vectors to functions defined over the original vector space. We focus on the fact that the imaginary-time Schr\"{o}dinger equation, when restricted to a variational wave function, is described by a nonlinear time evolution of the variational parameter vector. We collect sample points of this nonlinear dynamics at parameter configurations where the discrepancy between the true imaginary-time dynamics and the dynamics on the variational manifold is small, and perform data-driven continuous Koopman analysis. Within our formulation, the ground-state energy is reduced to the leading eigenvalue of a differential operator known as the Koopman generator. As a concrete example, we generate samples for the four-site transverse-field Ising model and estimate the ground-state energy using extended dynamic mode decomposition (EDMD). Furthermore, as an extension of this framework, we formulate the method for the case where the variational wave function is given by a uniform matrix product state on an infinite chain. By employing computational techniques developed within the framework of the time-dependent variational principle, all the quantities required for our analysis, including error estimation, can be computed efficiently in such systems. Since our approach provides predictions for the ground-state energy even when the true ground state lies outside the variational manifold, it is expected to complement conventional variational methods.