Quantum computing for classical problems: Variational Quantum Eigensolver for activated processes

TL;DR

This paper demonstrates how a Variational Quantum Eigensolver can be applied to classical stochastic problems, specifically the Fokker-Planck-Smoluchowski equation, enabling quantum computers to address classical dynamics.

Contribution

It introduces a novel application of VQE to classical stochastic eigenvalue problems, expanding quantum computing's scope beyond quantum chemistry.

Findings

Successfully computed conformational transition rates for a rotor chain.

Achieved results consistent with classical benchmarks on noisy quantum hardware.

Provided a scalable encoding method for probability distributions on quantum computers.

Abstract

The theory of stochastic processes impacts both physical and social sciences. At the molecular scale, stochastic dynamics is ubiquitous because of thermal fluctuations. The Fokker-Plank-Smoluchowski equation models the time evolution of the probability density of selected degrees of freedom in the diffusive regime and it is therefore a workhorse of physical chemistry. In this paper we report the development and implementation of a Variational Quantum Eigensolver procedure to solve the Fokker-Planck-Smoluchowski eigenvalue problem. We show that such an algorithm, typically adopted to address quantum chemistry problems, can be applied effectively to classical systems paving the way to new applications of quantum computers. We compute the conformational transition rate in a linear chain of rotors experiencing nearest-neighbour interaction. We provide a method to encode on the quantum computer the probability distribution for a given conformation of the chain and assess its scalability in terms of operations. Performance analysis on noisy quantum emulators and quantum devices (IBMQ Santiago) is provided for a small chain showing results in good agreement with the classical benchmark without further addition of any error mitigation technique.