Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.25872

Scaling universal Fermi network toward ground states: A diffusion-Monte-Carlo assessment

Yu-Sheng Li, Saskia Põldmaa, Tze Tzen Ong, Ahmed Abouelkomsan, Tay-Rong Chang, Hsin Lin, Liang Fu

Published Jul 28, 2026• 0 citations
arXivHTMLPDF

Abstract

In this work, we show that Fermi Sets---a provably universal neural network architecture for fermionic wavefunctions---can be systematically scaled up to find interacting ground states through energy minimization in a variational Monte Carlo framework. By further performing fixed-phase diffusion Monte Carlo (DMC) on the optimized neural network wavefunction, we demonstrate that as the network size increases, the variational energy systematically decreases while the energy improvement from DMC collapses monotonically to zero, indicating convergence to the ground state. We illustrate the scaling of Fermi Sets accompanied by the DMC assessment for interacting electrons in jellium and in a quantum dot under high magnetic fields.

DOI
10.48550/arxiv.2607.25872

Related papers

Towards the ground state of molecules via diffusion Monte Carlo on neural networks

Weiluo Ren, Weizhong Fu, Xiaojie Wu +1 more

Diffusion Monte Carlo (DMC) based on fixed-node approximation has enjoyed significant developments in the past decades and become one of the go-to methods when accurate ground state energy of molecules and materials is needed. However, the inaccurate nodal structure hinders the application of DMC for more challenging electronic correlation problems. In this work, we apply the neural-network based trial wavefunction in fixed-node DMC, which allows accurate calculations of a broad range of atomic and molecular systems of different electronic characteristics. Our method is superior in both accuracy and efficiency compared to state-of-the-art neural network methods using variational Monte Carlo (VMC). We also introduce an extrapolation scheme based on the empirical linearity between VMC and DMC energies, and significantly improve our binding energy calculation. Overall, this computational framework provides a benchmark for accurate solutions of correlated electronic wavefunction and also sheds light on the chemical understanding of molecules.

arXiv:2204.13903•Nature Communications•Apr 3, 2023•60 citations

Neural network-based nodal structures optimization for interacting fermionic systems

Freitas, William, B De Abreu, S. A. Vitiello

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.

arXiv:2411.02257•Nov 4, 2024•0 citations

Neural network based nodal structure optimization for interacting fermionic systems

William Freitas, Bruno Ricardi de Abreu, S. A. Vitiello

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.

Physical review. B./Physical review. B•Sep 9, 2025•1 citation

Discovering Quantum Phase Transitions with Fermionic Neural Networks

Gino Cassella, Halvard Sutterud, Sam Azadi +4 more

Deep neural networks have been very successful as highly accurate wave function Ansätze for variational Monte Carlo calculations of molecular ground states. We present an extension of one such Ansatz, FermiNet, to calculations of the ground states of periodic Hamiltonians, and study the homogeneous electron gas. FermiNet calculations of the ground-state energies of small electron gas systems are in excellent agreement with previous initiator full configuration interaction quantum Monte Carlo and diffusion Monte Carlo calculations. We investigate the spin-polarized homogeneous electron gas and demonstrate that the same neural network architecture is capable of accurately representing both the delocalized Fermi liquid state and the localized Wigner crystal state. The network converges on the translationally invariant ground state at high density and spontaneously breaks the symmetry to produce the crystalline ground state at low density, despite being given no a priori knowledge that a phase transition exists.

arXiv:2202.05183•Physical Review Letters•Jan 20, 2023•79 citations

Enhancing Neural-Network Variational Monte Carlo through Basis Transformation

Zhixuan Liu, Dongheng Qian, Jing Wang

Neural-network variational Monte Carlo (NNVMC) has emerged as a powerful tool for solving quantum many-body problems, yet systematic pathways for improving its accuracy remain largely heuristic. Here, we introduce a physically motivated basis transformation for NNVMC that enhances variational expressivity without increasing the complexity of the neural-network ansatz itself. By formulating the many-body wave function in a Gaussian basis, we introduce a single learnable locality parameter, $α$, that reshapes the target ground state into a more learnable representation. This approach introduces minimal computational overhead and can be readily combined with existing neural-network architectures. Using the three-dimensional homogeneous electron gas as a benchmark, we show that the optimized basis transformation consistently lowers the variational energy for both FermiNet and message-passing neural-network architectures. Notably, for the latter, it enables a more precise determination of the Fermi liquid to Wigner crystal phase transition. More broadly, our results highlight basis transformation as a new route to improving NNVMC in continuous space, showing that accuracy can be enhanced not only by refining the ansatz but also by making the target ground state easier to represent.

cond-mat.str-elCross-listedarXiv:2604.15888•Apr 20, 2026•0 citations

Fermi Sets: Universal and interpretable neural architectures for fermions

Liang Fu

We introduce Fermi Sets, a universal and physically interpretable neural architecture for fermionic many-body wavefunctions. Building on a ``parity-graded'' representation [1], we prove that any continuous fermionic wavefunction on a compact domain can be approximated to arbitrary accuracy by a linear combination of K antisymmetric basis functions--such as pairwise products or Slater determinants--multiplied by symmetric functions. A key result is that the number of required bases is provably small: K=1 suffices in one-dimensional continua (and on lattices in any dimension), K=2 suffices in two dimensions, and in higher dimensions K grows at most linearly with particle number. The antisymmetric bases can be learned by small neural networks, while the symmetric factors are implemented by permutation-invariant networks whose width scales only linearly with particle number. Thus, Fermi Sets achieve universal approximation of fermionic wavefunctions with minimal overhead while retaining clear physical interpretability. As a numerical illustration, a single Fermi Sets model applied to metallic solid hydrogen in three dimensions, trained simultaneously across multiple nuclear geometries, surpasses all diffusion Monte Carlo benchmarks.

arXiv:2601.02508•Jan 5, 2026•1 citation
In this work, we show that Fermi Sets---a provably universal neural network architecture for fermionic wavefunctions---can be systematically scaled up to find interacting ground states through energy minimization in a variational Monte Carlo framework. By further performing fixed-phase diffusion Monte Carlo (DMC) on the optimized neural network wavefunction, we demonstrate that as the network size increases, the variational energy systematically decreases while the energy improvement from DMC collapses monotonically to zero, indicating convergence to the ground state. We illustrate the scaling of Fermi Sets accompanied by the DMC assessment for interacting electrons in jellium and in a quantum dot under high magnetic fields.
Diffusion Monte Carlo (DMC) based on fixed-node approximation has enjoyed significant developments in the past decades and become one of the go-to methods when accurate ground state energy of molecules and materials is needed. However, the inaccurate nodal structure hinders the application of DMC for more challenging electronic correlation problems. In this work, we apply the neural-network based trial wavefunction in fixed-node DMC, which allows accurate calculations of a broad range of atomic and molecular systems of different electronic characteristics. Our method is superior in both accuracy and efficiency compared to state-of-the-art neural network methods using variational Monte Carlo (VMC). We also introduce an extrapolation scheme based on the empirical linearity between VMC and DMC energies, and significantly improve our binding energy calculation. Overall, this computational framework provides a benchmark for accurate solutions of correlated electronic wavefunction and also sheds light on the chemical understanding of molecules.
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.
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.
Deep neural networks have been very successful as highly accurate wave function Ansätze for variational Monte Carlo calculations of molecular ground states. We present an extension of one such Ansatz, FermiNet, to calculations of the ground states of periodic Hamiltonians, and study the homogeneous electron gas. FermiNet calculations of the ground-state energies of small electron gas systems are in excellent agreement with previous initiator full configuration interaction quantum Monte Carlo and diffusion Monte Carlo calculations. We investigate the spin-polarized homogeneous electron gas and demonstrate that the same neural network architecture is capable of accurately representing both the delocalized Fermi liquid state and the localized Wigner crystal state. The network converges on the translationally invariant ground state at high density and spontaneously breaks the symmetry to produce the crystalline ground state at low density, despite being given no a priori knowledge that a phase transition exists.
Neural-network variational Monte Carlo (NNVMC) has emerged as a powerful tool for solving quantum many-body problems, yet systematic pathways for improving its accuracy remain largely heuristic. Here, we introduce a physically motivated basis transformation for NNVMC that enhances variational expressivity without increasing the complexity of the neural-network ansatz itself. By formulating the many-body wave function in a Gaussian basis, we introduce a single learnable locality parameter, ααα, that reshapes the target ground state into a more learnable representation. This approach introduces minimal computational overhead and can be readily combined with existing neural-network architectures. Using the three-dimensional homogeneous electron gas as a benchmark, we show that the optimized basis transformation consistently lowers the variational energy for both FermiNet and message-passing neural-network architectures. Notably, for the latter, it enables a more precise determination of the Fermi liquid to Wigner crystal phase transition. More broadly, our results highlight basis transformation as a new route to improving NNVMC in continuous space, showing that accuracy can be enhanced not only by refining the ansatz but also by making the target ground state easier to represent.
We introduce Fermi Sets, a universal and physically interpretable neural architecture for fermionic many-body wavefunctions. Building on a ``parity-graded'' representation [1], we prove that any continuous fermionic wavefunction on a compact domain can be approximated to arbitrary accuracy by a linear combination of K antisymmetric basis functions--such as pairwise products or Slater determinants--multiplied by symmetric functions. A key result is that the number of required bases is provably small: K=1 suffices in one-dimensional continua (and on lattices in any dimension), K=2 suffices in two dimensions, and in higher dimensions K grows at most linearly with particle number. The antisymmetric bases can be learned by small neural networks, while the symmetric factors are implemented by permutation-invariant networks whose width scales only linearly with particle number. Thus, Fermi Sets achieve universal approximation of fermionic wavefunctions with minimal overhead while retaining clear physical interpretability. As a numerical illustration, a single Fermi Sets model applied to metallic solid hydrogen in three dimensions, trained simultaneously across multiple nuclear geometries, surpasses all diffusion Monte Carlo benchmarks.