Steady-State Statistics of Classical Nonlinear Dynamical Systems from Noisy Intermediate-Scale Quantum Devices

TL;DR

This paper explores using NISQ quantum computers with QPE and VQE algorithms to directly compute steady-state probability distributions of classical nonlinear systems, offering a promising alternative to traditional simulation methods.

Contribution

It demonstrates the feasibility of using quantum algorithms on NISQ devices to solve the Fokker-Planck Equation for steady-state distributions, a novel approach in this context.

Findings

Quantum algorithms produce PDFs in reasonable agreement with classical solutions.

NISQ devices can be employed to solve steady-state problems in classical dynamical systems.

Potential for extending quantum methods to higher-dimensional systems.

Abstract

Classical nonlinear dynamical systems are often characterized by their steady-state probability distribution functions (PDFs). Typically, PDFs are accumulated from numerical simulations that involve solving the underlying dynamical equations of motion using integration techniques. An alternative procedure, direct statistical simulation (DSS), solves for the statistics directly. One approach to DSS is the Fokker-Planck Equation (FPE), which can be used to find the PDF of classical dynamical systems. Here, we investigate the utility of Noisy Intermediate-Scale Quantum (NISQ) computers to find steady-state solutions to the FPE. We employ the Quantum Phase Estimation (QPE) and the Variational Quantum Eigensolver (VQE) algorithms to find the zero-mode of the FPE for one-dimensional Ornstein-Uhlenbeck problems enabling comparison with exact solutions. The quantum computed steady-state probability distribution functions (PDFs) are demonstrated to be in reasonable agreement with the classically computed PDFs. We conclude with a discussion of potential extensions to higher-dimensional dynamical systems.