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arXiv:2603.12668

Advancing Machine Learning Applications in Quantum Few-Body Systems

Jin Ziqi, Paolo Recchia, M. Gattobigio

Published Mar 13, 2026• 0 citations
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Abstract

This paper presents a general neural network framework for solving quantum few-body systems, extending prior methods to handle diverse particle masses, interaction types, and system configurations. Our architecture, which combines an adaptive step size with the Metropolis-Adjusted Langevin Algorithm for Monte Carlo sampling, accurately approximates the ground-state wave functions of systems featuring harmonic confinement, Gaussian two-body interactions, and including three-body forces. In ten-particle systems, it achieves lower relative energy errors (with respect to the reference values) than previous machine-learning methods. Leveraging GPU-accelerated computation, the method scales favorably with system size while maintaining robust convergence, reduced hyperparameter sensitivity, and stable training. Beyond accurate energy estimation, the model captures spatial distributions and correlation structures, offering physical insights about inter-particle structure. By unifying applicability across identical and nonidentical particles, the proposed approach establishes a versatile computational tool for exploring complex few-body quantum systems, with significant implications for advancing computational models in few-body quantum systems.

DOI
10.48550/arxiv.2603.12668

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This paper presents a general neural network framework for solving quantum few-body systems, extending prior methods to handle diverse particle masses, interaction types, and system configurations. Our architecture, which combines an adaptive step size with the Metropolis-Adjusted Langevin Algorithm for Monte Carlo sampling, accurately approximates the ground-state wave functions of systems featuring harmonic confinement, Gaussian two-body interactions, and including three-body forces. In ten-particle systems, it achieves lower relative energy errors (with respect to the reference values) than previous machine-learning methods. Leveraging GPU-accelerated computation, the method scales favorably with system size while maintaining robust convergence, reduced hyperparameter sensitivity, and stable training. Beyond accurate energy estimation, the model captures spatial distributions and correlation structures, offering physical insights about inter-particle structure. By unifying applicability across identical and nonidentical particles, the proposed approach establishes a versatile computational tool for exploring complex few-body quantum systems, with significant implications for advancing computational models in few-body quantum systems.
Abstract This paper presents a general neural network framework for solving quantum few-body systems, extending prior methods to handle diverse particle masses, interaction types, and system configurations. Our architecture, which combines an adaptive step size with the Metropolis-Adjusted Langevin Algorithm for Monte Carlo sampling, accurately approximates the ground-state wave functions of systems featuring harmonic confinement, Gaussian two-body interactions, and including three-body forces. In ten-particle systems, it achieves lower relative energy errors (with respect to the reference values) than previous machine-learning methods. Leveraging GPU-accelerated computation, the method scales favorably with system size while maintaining robust convergence, reduced hyperparameter sensitivity, and stable training. Beyond accurate energy estimation, the model captures spatial distributions and correlation structures, offering physical insights about inter-particle structure. By unifying applicability across identical and nonidentical particles, the proposed approach establishes a versatile computational tool for exploring complex few-body quantum systems, with significant implications for advancing computational models in few-body quantum systems.
The simulation of quantum many-body systems poses a significant challenge in physics due to the exponential scaling of Hilbert space with the number of particles. Traditional methods often struggle with large system sizes and frustrated lattices. In this research article, we present a novel algorithm that leverages the power of deep neural networks combined with Markov Chain Monte Carlo simulation to address these limitations. Our method introduces a neural network architecture specifically designed to represent bosonic quantum states on a 1D lattice chain. We successfully achieve the ground state of the Bose-Hubbard model, demonstrating the superiority of the adaptive momentum optimizer for convergence speed and stability. Notably, our approach offers flexibility in simulating various lattice geometries and potentially larger system sizes, making it a valuable tool for exploring complex quantum phenomena. This work represents a substantial advancement in the field of quantum simulation, opening new possibilities for investigating previously challenging systems.
Neural networks are emerging as a powerful tool for determining the quantum states of interacting many-body fermionic systems. The standard approach trains a neural-network ansatz by minimizing the mean local energy estimated from Monte Carlo samples. However, this typically results in large sample-to-sample fluctuations in the estimated mean energy and thus slow convergence of the energy minimization. We propose that minimizing a logarithmically compressed variance of the local energies can dramatically improve convergence. Moreover, this loss function can be adapted to systematically obtain the energy spectrum across multiple runs. We demonstrate these ideas for spin-1/2 particles in a 2D harmonic trap with attractive Poschl-Teller interactions between opposite spins.
Quantum many-body problems are some of the most challenging problems in science and are central to demystifying some exotic quantum phenomena, e.g., high-temperature superconductors. The combination of neural networks (NN) for representing quantum states, coupled with the Variational Monte Carlo (VMC) algorithm, has been shown to be a promising method for solving such problems. However, the run-time of this approach scales quadratically with the number of simulated particles, constraining the practically usable NN to - in machine learning terms - minuscule sizes (<10M parameters). Considering the many breakthroughs brought by extreme NN in the +1B parameters scale to other domains, lifting this constraint could significantly expand the set of quantum systems we can accurately simulate on classical computers, both in size and complexity. We propose a NN architecture called Vector-Quantized Neural Quantum States (VQ-NQS) that utilizes vector-quantization techniques to leverage redundancies in the local-energy calculations of the VMC algorithm - the source of the quadratic scaling. In our preliminary experiments, we demonstrate VQ-NQS ability to reproduce the ground state of the 2D Heisenberg model across various system sizes, while reporting a significant reduction of about ×10{\times}10×10 in the number of FLOPs in the local-energy calculation.
Quantum Monte Carlo coupled with neural network wavefunctions has shown success in computing ground states of quantum many-body systems. Existing optimization approaches compute the energy by sampling local energy from an explicit probability distribution given by the wavefunction. In this work, we provide a new optimization framework for obtaining properties of quantum many-body ground states using score-based neural networks. Our new framework does not require explicit probability distribution and performs the sampling via Langevin dynamics. Our method is based on the key observation that the local energy is directly related to scores, defined as the gradient of the logarithmic wavefunction. Inspired by the score matching and diffusion Monte Carlo methods, we derive a weighted score matching objective to guide our score-based models to converge correctly to ground states. We first evaluate our approach with experiments on quantum harmonic traps, and results show that it can accurately learn ground states of atomic systems. By implicitly modeling high-dimensional data distributions, our work paves the way toward a more efficient representation of quantum systems.
The study of interacting quantum many-body systems is constrained by the exponential growth of the Hilbert space with system size, which limits both numerical simulations and theoretical understanding. This thesis explores how the interplay between disorder and interactions shapes localization and ergodicity in quantum many-body systems, using the exact diagonalization method, and how machine learning can serve as an alternative approach for studying these systems. We first investigate the many-body localization problem in a one-dimensional SU(2)-invariant random antiferromagnetic Heisenberg chain using exact and sparse matrix diagonalization methods. By analyzing energy level gap-ratio statistics, participation ratio, entanglement entropy and the entanglement spectral parameter as a function of disorder strength under di!erent random coupling distributions, we identify clear qualitative di!erences from the widely studied random field Heisenberg chain. For system sizes up to L = 24, signatures of localized behaviour are much less pronounced than in the case of the random field model. Motivated by the limitations of conventional numerical methods, we examine machine learning as a scalable paradigm for many-body physics. Using the Bose- Hubbard model in one and two dimensions as a benchmark, we test HUBBARDNET, a deep neural network architecture, by replacing standard stochastic gradient descent with adaptive moment estimation (ADAM), adjusting the learning rate for improved optimization, and introducing physics-informed output activations that can resolve extremely small wave-function amplitudes. This approach yields ground-state energy errors reduced by orders of magnitude and wave function fidelities exceeding 99%, remaining robust across four decades of interaction strength and over superfluid, Mott-insulating, and quantum chaotic regimes. Beyond energy prediction, the trained models accurately reproduce generalized inverse participation ratios and multifractal dimensions for ground and excited states, capturing localization, delocalization, and multifractality across the spectrum, with excited-state training explicitly guided by fractal dimensions that encode the multifractal properties of the wave functions.