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arXiv:2603.15460

Introduction to the artificial neural network-based variational Monte Carlo method

William Freitas

Published Mar 16, 2026• 0 citations
arXivHTMLPDF

Abstract

The construction of trial wave functions based on neural networks combined with the variational Monte Carlo method is discussed. The mathematical formulation for representing quantum states as artificial neural networks is introduced. The advantages of employing such trial states and how machine learning works are discussed. It is shown that the variational method is a kind of unsupervised learning algorithm, where the multiple minima landscape is used as an asset that leads to a stable optimization procedure. The feature representation plays an important role on interpretability and on extracting physical insights from nontrivial trial wave functions. The algorithm is illustrated for the Yukawa potential and the hydrogen molecule.

DOI
10.48550/arxiv.2603.15460

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The construction of trial wave functions based on neural networks combined with the variational Monte Carlo method is discussed. The mathematical formulation for representing quantum states as artificial neural networks is introduced. The advantages of employing such trial states and how machine learning works are discussed. It is shown that the variational method is a kind of unsupervised learning algorithm, where the multiple minima landscape is used as an asset that leads to a stable optimization procedure. The feature representation plays an important role on interpretability and on extracting physical insights from nontrivial trial wave functions. The algorithm is illustrated for the Yukawa potential and the hydrogen molecule.
The construction of trial wave functions based on neural networks combined with the variational Monte Carlo method is discussed. The mathematical formulation for representing quantum states as neural networks is introduced. The advantages of employing such trial states and how machine learning works are considered. It is shown that the variational method is a kind of unsupervised learning algorithm, where the multiple minima landscape is used as an asset that leads to a stable optimization procedure. The feature representation plays an important role on interpretability and on extracting physical insights from nontrivial trial wave functions. The algorithm is illustrated for the Yukawa potential and the hydrogen molecule.
Abstract Inspired by the universal approximation theorem and widespread adoption of artificial neural network techniques in a diversity of fields, feed‐forward neural networks are proposed as a general purpose trial wave function for quantum Monte Carlo simulations of continuous many‐body systems. Whereas for simple model systems the whole many‐body wave function can be represented by a neural network, the antisymmetry condition of non‐trivial fermionic systems is incorporated by means of a Slater determinant. To demonstrate the accuracy of the trial wave functions, an exactly solvable model system of two trapped interacting particles, as well as the hydrogen dimer, is studied.
In this note, variational Monte Carlo method based on neural quantum states for spin systems is reviewed. Using a neural network as the wave function allows for a more generalized expression of various types of interactions, including highly non-local interactions, which are closely related to its non-linear activation functions. Additionally, neural networks can represent relatively complex wave functions with relatively small computational resources when dealing with higher-dimensional systems, which is undoubtedly a "flattening" advantage. In quantum-state tomography, the representation method of neural quantum states has already achieved significant results, hinting at its potential in handling larger-sized systems.
Carleo and Troyer [3] have recently pointed out the possibility of solving quantum many-body problems by using Artificial Neural Networks (ANN). Their work is based on minimizing a variational wave function to obtain the ground states for various spin-dependent systems. This work is primarily focused on developing efficient method using ANN to solve the ground state wave function for atomic systems. We have developed a theoretical groundwork to represent the wave function of a many-electron atom by using artificial neural network while still preserving its antisymmetric property. By using the Metropolis algorithm, Variational Monte Carlo (VMC), and Stochastic Reconfiguration (SR) methods for minimization, we were able to obtain a highly accurate ground state wave function for the He atom. To verify our optimization algorithm, we reproduced the results for the ground state of a three dimensional Simple Harmonic Oscillator (SHO) given by Teng [18].
In this study, we explore the similarities and differences between variational Monte Carlo techniques that employ conventional and artificial neural network representations of the ground-state wave function for fermionic systems. Our primary focus is on shallow neural network architectures, specifically the restricted Boltzmann machine, and we examine unsupervised learning algorithms that are appropriate for modeling complex many-body correlations. We assess the advantages and drawbacks of conventional and neural network wave functions by applying them to a range of circular quantum dot systems. Our findings, which include results for systems containing up to 90 electrons, emphasize the efficient implementation of these methods on both homogeneous and heterogeneous high-performance computing facilities.