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Russian Journal of Physical Chemistry A

Applying the Monte Carlo Method to Problems of Modeling Stationary States of Quantum Systems by Means of Machine Learning

Georgiy K. Ozerov, Aleksandr K. Pupeza, Dmitry S. Bezrukov

Published Oct 28, 2025• 0 citations
Publisher

Abstract

Abstract An overview is presented of recent advances in applying variational, projection, and diffusion Monte Carlo methods (VMC, PMC, and DMC, respectively). The prospects for using the Monte Carlo method to calculate path integrals (PIMC) when modeling molecular systems are also considered. It is emphasized that modern machine learning effectively meets the requirements of parameterizing wave functions of the target space of solutions for a wide range of problems in quantum molecular modeling and functionals of the corresponding observables.

DOI
10.1134/s0036024425702188

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We propose a new method to calculate ground state position dependent observables in quantum many-body systems. The method, which we call the path-integral diffusion Monte Carlo (PI-DMC) method, is essentially a combination of path-integral Monte Carlo (PIMC) and diffusion Monte Carlo (DMC) methods. The distribution resulting from a DMC simulation is further propagated in imaginary time by PIMC sampling. Tests of the new method for simple cases such as the harmonic oscillator, a double well, and a set of ten coupled harmonic oscillators show that the results for several observables converge rapidly to the exact result.
In this article, machine-learning methods are used to solve quantum mechanics problems. The radial basis function network in a discrete basis is used as the variational wave function for the ground state of a quantum system. Variational Monte Carlo (VMC) calculations are carried out for some simple Hamiltonians. The results are in good agreement with theoretical values. The smallest eigenvalue of a Hermitian matrix can also be acquired using VMC calculations. Results are provided to demonstrate that machine-learning techniques are capable of solving quantum mechanical problems.
Computing accurate yet efficient approximations to the solutions of the electronic Schrödinger equation has been a paramount challenge of computational chemistry for decades. Quantum Monte Carlo methods are a promising avenue of development as their core algorithm exhibits a number of favorable properties: it is highly parallel and scales favorably with the considered system size, with an accuracy that is limited only by the choice of the wave function Ansatz. The recently introduced machine-learned parametrizations of quantum Monte Carlo Ansätze rely on the efficiency of neural networks as universal function approximators to achieve state of the art accuracy on a variety of molecular systems. With interest in the field growing rapidly, there is a clear need for easy to use, modular, and extendable software libraries facilitating the development and adoption of this new class of methods. In this contribution, the DeepQMC program package is introduced, in an attempt to provide a common framework for future investigations by unifying many of the currently available deep-learning quantum Monte Carlo architectures. Furthermore, the manuscript provides a brief introduction to the methodology of variational quantum Monte Carlo in real space, highlights some technical challenges of optimizing neural network wave functions, and presents example black-box applications of the program package. We thereby intend to make this novel field accessible to a broader class of practitioners from both the quantum chemistry and the machine learning communities.
Computing accurate yet efficient approximations to the solutions of the electronic Schrödinger equation has been a paramount challenge of computational chemistry for decades. Quantum Monte Carlo methods are a promising avenue of development as their core algorithm exhibits a number of favorable properties: it is highly parallel and scales favorably with the considered system size, with an accuracy that is limited only by the choice of the wave function Ansatz. The recently introduced machine-learned parametrizations of quantum Monte Carlo Ansätze rely on the efficiency of neural networks as universal function approximators to achieve state of the art accuracy on a variety of molecular systems. With interest in the field growing rapidly, there is a clear need for easy to use, modular, and extendable software libraries facilitating the development and adoption of this new class of methods. In this contribution, the DeepQMC program package is introduced, in an attempt to provide a common framework for future investigations by unifying many of the currently available deep-learning quantum Monte Carlo architectures. Furthermore, the manuscript provides a brief introduction to the methodology of variational quantum Monte Carlo in real space, highlights some technical challenges of optimizing neural network wave functions, and presents example black-box applications of the program package. We thereby intend to make this novel field accessible to a broader class of practitioners from both the quantum chemistry and the machine learning communities.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.