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Moscow University Physics Bulletin

Application of Neural Networks for Path Integrals Computation in Relativistic Quantum Mechanics

Dmitry Salnikov, V. V. Chistiakov, A. V. Vasiliev, Aleksey S. Ivanov

Published Dec 1, 2024• 1 citation
Publisher

Abstract

Abstract In quantum theory, the expectation value of an observable can be represented as a path integral. In general, it cannot be computed analytically. There are various approximate methods of lattice calculations, for example, the Monte Carlo method. Currently, an approach to solving this problem using neural networks is being developed. In our research, we calculated path integrals in several models of relativistic quantum mechanics using the normalizing flows algorithm. For fast calculations with high accuracy, this algorithm was used in conjunction with the Markov chain generation method.

DOI
10.3103/s0027134924702096

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Abstract In quantum theory, the expectation value of an observable can be represented as a path integral. In general, it cannot be computed analytically. There are various approximate methods of lattice calculations, for example, the Monte Carlo method. Currently, an approach to solving this problem using neural networks is being developed. In our research, we calculated path integrals in several models of relativistic quantum mechanics using the normalizing flows algorithm. For fast calculations with high accuracy, this algorithm was used in conjunction with the Markov chain generation method.
The Path-Integral Formulation of Quantum Mechanics is introduced along with a detailed mathematical description of how it is used in quantum computations. The important concept of the kernel is explained, along with the free particle and harmonic oscillator as examples. Furthermore, the method for calculating expectation values of quantum operators is explained. The expectation values are naturally calculated by importance sampled Monte Carlo integration and by use of the Metropolis algorithm. This is due to the discretization of the path integral results in an integral with a high number of integration variables. The mathematical concepts of this calculation are explained. Also, a method for obtaining the probability density of the treated system is presented. The calculations are performed by a GPU, due to its high capabilities for numerical operations. This requires the mathematical computations to be parallelized and is done by use of the free software PyOpenCL. A thorough introduction to these concepts are given. The resulting ground state energies and probability densities for many particle systems interacting with harmonic as well as attractive and repulsive gaussian potentials are presented. The calculations worked exceedingly well for many particle systems. Source code is available at https://sourceforge.net/projects/ feynmangpu/files/
These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions, quantization of gauge theories etc.) as well as specialized topics (scattering, dissipative systems, spin \& color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions.
The field of quantum chemistry has long been defined by the central computational challenge of the quantum many-body fermionic system problem, main the electronic Schrödinger equation. This exponential scaling makes it computationally intractable to solve, the exact governing electronic Schrödinger equation. Approximation methods have been developed, ranging from the widely used Density Functional Theory (DFT) to Coupled Cluster (CC) theory. While powerful, these conventional approaches often involve controlled truncations which caused a trade-off between accuracy, leaving many systems—particularly those with strong electron correlation—beyond the reach of ab initio simulation. Furthermore, the famous fermionic sign oscillation problem is particularly severe in stochastic methods like Quantum Monte Carlo (QMC). 1 Traditional Ansätze are also frequently engineered specifically for particular systems, which results in a fragmentation of results and fundamentally limits the transferability. Recent research demonstrates that with carefully designed architectures, ANNs can not only represent ground states of model systems with high precision but also tackle the sign problem, offering a promising path forward for strongly correlated electronic structure calculations.2,3 This review charts the paradigm shift driven by Neural Network Quantum States (NNQS), which leverage the representational power of deep learning to overcome these barriers. We detail the architectural evolution from Restricted Boltzmann Machines to autoregressive models like RNNs and Transformers, which enable exact, uncorrelated sampling and bypass critical bottlenecks.2,4,5Furthermore, we explore the geometric foundation of high-efficiency optimization via the neural networks, which decouples optimization cost from model complexity.6 The frontier of the field is marked by the integration of operator learning and hybrid quantum-classical frameworks, such as Reinforcement Learning for contracted quantum eigen-solvers and Quantum-Enhanced Neural Networks, which leverage quantum processors to enhance expressivity for hardware-efficient Ansätze.7,8 By framing the problem through the lens of computational resource allocation—specifically, trading exponential memory demands for polynomialcomplexity optimization and sampling—this review elucidates how recent advances at the intersection of artificial neural networks and quantum computing have evolved. The emergence of those methods not merely progress into a powerful simulation-tools, but also a hybrid computational interface.9 This convergence offers a versatile and systematically improvable framework for potentially addressing some of the most challenging problems in quantum physics and chemistry.
The field of quantum chemistry has long been defined by the central computational challenge of the quantum many-body fermionic system problem, main the electronic Schrödinger equation. This exponential scaling makes it computationally intractable to solve, the exact governing electronic Schrödinger equation. Approximation methods have been developed, ranging from the widely used Density Functional Theory (DFT) to Coupled Cluster (CC) theory. While powerful, these conventional approaches often involve controlled truncations which caused a trade-off between accuracy, leaving many systems—particularly those with strong electron correlation—beyond the reach of ab initio simulation. Furthermore, the famous fermionic sign oscillation problem is particularly severe in stochastic methods like Quantum Monte Carlo (QMC). 1 Traditional Ansätze are also frequently engineered specifically for particular systems, which results in a fragmentation of results and fundamentally limits the transferability. Recent research demonstrates that with carefully designed architectures, ANNs can not only represent ground states of model systems with high precision but also tackle the sign problem, offering a promising path forward for strongly correlated electronic structure calculations.2,3 This review charts the paradigm shift driven by Neural Network Quantum States (NNQS), which leverage the representational power of deep learning to overcome these barriers. We detail the architectural evolution from Restricted Boltzmann Machines to autoregressive models like RNNs and Transformers, which enable exact, uncorrelated sampling and bypass critical bottlenecks.2,4,5Furthermore, we explore the geometric foundation of high-efficiency optimization via the neural networks, which decouples optimization cost from model complexity.6 The frontier of the field is marked by the integration of operator learning and hybrid quantum-classical frameworks, such as Reinforcement Learning for contracted quantum eigen-solvers and Quantum-Enhanced Neural Networks, which leverage quantum processors to enhance expressivity for hardware-efficient Ansätze.7,8 By framing the problem through the lens of computational resource allocation—specifically, trading exponential memory demands for polynomialcomplexity optimization and sampling—this review elucidates how recent advances at the intersection of artificial neural networks and quantum computing have evolved. The emergence of those methods not merely progress into a powerful simulation-tools, but also a hybrid computational interface.9 This convergence offers a versatile and systematically improvable framework for potentially addressing some of the most challenging problems in quantum physics and chemistry.