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arXiv:2505.11442 · Physical Review Letters

Neural Quantum States for Light Nuclei with Chiral Two- and Three-Body Interactions

Pengsheng Wen, Alexandros Gezerlis, Jeremy W. Holt

Published Apr 14, 2026• 2 citations
arXivHTMLPDF

Abstract

Finding high-quality trial wave functions for quantum Monte Carlo calculations of light nuclei requires a strong intuition for modeling the interparticle correlations as well as large computational resources for exploring the space of variational parameters. Moreover, for systems with three-body interactions, the wave function should account for many-body effects beyond simple pairwise correlations. In this Letter, we design neural networks that efficiently incorporate these factors to generate expressive wave function Ansätze for light nuclei using variational Monte Carlo. Our neural-network approach for A=3 nuclei can capture, already at the level of variational Monte Carlo, the overwhelming majority of the ground-state energy estimated by Green's function Monte Carlo (GFMC). It achieves a ground-state energy within 0.45% of the GFMC result for ^{3}H using the softest chiral interaction, representing a substantial improvement over standard variational Monte Carlo, which exhibits a 3.7% deviation. The result indicates the potential of neural networks to construct effective trial wave functions for quantum Monte Carlo calculations.

DOI
10.1103/ygkf-llyp

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Finding high-quality trial wave functions for quantum Monte Carlo calculations of light nuclei requires a strong intuition for modeling the interparticle correlations as well as large computational resources for exploring the space of variational parameters. Moreover, for systems with three-body interactions, the wave function should account for many-body effects beyond simple pairwise correlations. In this Letter, we design neural networks that efficiently incorporate these factors to generate expressive wave function Ansätze for light nuclei using variational Monte Carlo. Our neural-network approach for A=3 nuclei can capture, already at the level of variational Monte Carlo, the overwhelming majority of the ground-state energy estimated by Green's function Monte Carlo (GFMC). It achieves a ground-state energy within 0.45% of the GFMC result for ^{3}H using the softest chiral interaction, representing a substantial improvement over standard variational Monte Carlo, which exhibits a 3.7% deviation. The result indicates the potential of neural networks to construct effective trial wave functions for quantum Monte Carlo calculations.
The complexity of many-body quantum wave functions is a central aspect of several fields of physics and chemistry where nonperturbative interactions are prominent. Artificial neural networks (ANNs) have proven to be a flexible tool to approximate quantum many-body states in condensed matter and chemistry problems. In this work we introduce a neural-network quantum state ansatz to model the ground-state wave function of light nuclei, and approximately solve the nuclear many-body Schrödinger equation. Using efficient stochastic sampling and optimization schemes, our approach extends pioneering applications of ANNs in the field, which present exponentially scaling algorithmic complexity. We compute the binding energies and point-nucleon densities of A≤4 nuclei as emerging from a leading-order pionless effective field theory Hamiltonian. We successfully benchmark the ANN wave function against more conventional parametrizations based on two- and three-body Jastrow functions, and virtually exact Green's function Monte Carlo results.
We introduce an efficient approach to implement neural network quantum states (NNQS) as trial wave functions in auxiliary-field quantum Monte Carlo (AFQMC). NNQS are a recently developed class of variational ansätze capable of flexibly representing many-body wave functions, though they often incur a high computational cost during optimization. AFQMC, on the other hand, is a powerful stochastic projector approach for ground-state calculations, but it normally requires an approximate constraint via a trial wave function or trial density matrix, whose quality affects the accuracy. Recently, it has been shown (Xiao et al., arXiv2505.18519) that a broad class of highly correlated wave functions can be integrated into AFQMC through stochastic sampling techniques. In this work, we apply this approach and present a direct integration of NNQS with AFQMC, allowing NNQS to serve as high-quality trial wave functions for AFQMC with manageable computational cost. We test the NNQS-AFQMC method on the challenging nitrogen molecule (N 2 ) at stretched geometries. Our results demonstrate that AFQMC with an NNQS trial wave function can attain near-exact total energies, highlighting the potential of AFQMC with NNQS to overcome longstanding challenges in strongly correlated electronic structure calculations. We also outline future research directions for improving this promising methodology.
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Deep neural networks can represent very different sorts of functions, including complex quantum many-body states. Tensor networks can also represent these states, have more structure and are easier to optimize. However, they can be prohibitively costly computationally in two or higher dimensions. Here, we propose a generalization of the perceptron -- the perceptrain -- which borrows features from the two different formalisms. We construct variational many-body ansatz from a simple network of perceptrains. The network can be thought of as a neural network with a few distinct features inherited from tensor networks. These include efficient local optimization akin to the density matrix renormalization algorithm, instead of optimizing all the parameters at once; the possibility to dynamically increase the number of parameters during the optimization; the possibility to compress the state; and a structure that remains quantum-inspired. We showcase the ansatz using a combination of variational Monte Carlo (VMC) and Green function Monte Carlo (GFMC) on a 10×1010\times 1010×10 transverse field quantum Ising model with a long-range 1/r61/r^61/r6 antiferromagnetic interaction. The model corresponds to the Rydberg (cold) atoms platform proposed for quantum annealing. We consistently find a very high relative accuracy for the ground state energy, around 10−510^{-5}10−5 for VMC and 10−610^{-6}10−6 for GFMC in all regimes of parameters, including in the vicinity of the quantum phase transition. We use very small ranks (∼2\sim 2∼2-555) of perceptrains, as opposed to multiples of thousand used in matrix product states. The optimization of the energy was very robust. The entire phase diagram was found with a single initial condition and a fixed set of hyperparameters.
We compute ground-state energies and charge radii of light- to medium-mass nuclei with up to A=58A=58A=58 nucleons, leveraging a variational Monte Carlo method based on Pfaffian-Jastrow neural quantum states. To further understand which elements of the nuclear Hamiltonian are "essential" to predict binding energies and charge radii across the nuclear chart with few-percent errors, we consider different interactions inspired by pionless effective field theory. Specifically, in addition to model "o" of [Phys. Rev. C 103, 054003 (2021)], we study the impact of charge-symmetry-breaking and charge-dependent terms in the nucleon-nucleon force, as well as ppp-wave contributions, which have been found to be critical for the stability of ppp-shell nuclei. In addition to its intrinsic interest, our work assesses the performance of neural quantum states in the medium-mass regime and examines the impact of these interaction modifications. Using the resulting ground-state simulations, we analyze the computational scaling of variational Monte Carlo with neural quantum states as a function of system size and computational resources, enabling projections for future large-scale calculations.
Simulating strongly correlated fermionic systems remains a fundamental challenge in quantum physics, largely due to the sign problem in quantum Monte Carlo (QMC) methods. We present a neural network-based variational Monte Carlo (NN-VMC) approach, leveraging a flexible neural network ansatz to represent the many-body wavefunction. Focusing on quantum dots with up to 30 electrons, we demonstrate that NN-VMC significantly reduces variational bias and achieves ground-state energies surpassing those of fixed-node diffusion Monte Carlo (DMC). A key feature is that the neural network adaptively learns and optimizes nodal structures during energy minimization. We provide qualitative insights into the nodal structure of fermionic wavefunctions by comparing the nodal structures generated by NN-VMC with those obtained from traditional trial functions. Additionally, we reveal spin-resolved radial distributions and electron density profiles, highlighting the versatility and accuracy of NN-VMC. This work underscores the potential of machine learning to advance quantum simulations and deepen our understanding of strongly correlated systems.