We present an overview of the method of Neural Quantum States applied to the many-body problem of atomic nuclei. Through the lens of group representation theory, we focus on the problem of constructing neural-network ansätze that respect physical symmetries. We explicitly prove that determinants, which are among the most common methods to build antisymmetric neural-network wave functions, can be understood as the result of a group convolution. We also identify the reason why this construction is so efficient in practice compared to other group convolutional operations. We conclude that group representation theory is a promising avenue to incorporate explicitly symmetries in Neural Quantum States.
We present an overview of the method of Neural Quantum States applied to the many-body problem of atomic nuclei. Through the lens of group representation theory, we focus on the problem of constructing neural-network ansätze that respect physical symmetries. We explicitly prove that determinants, which are among the most common methods to build antisymmetric neural-network wave functions, can be understood as the result of a group convolution. We also identify the reason why this construction is so efficient in practice compared to other group convolutional operations. We conclude that group representation theory is a promising avenue to incorporate explicitly symmetries in Neural Quantum States.
Neural network wave functions have shown promise as a way to achieve high accuracy in solving the many-body quantum problem. These wave functions most commonly use a determinant or a sum of determinants to antisymmetrize many-body orbitals, which are described by a neural network. In many cases, the wave function is projected onto a fixed-spin state. Such a treatment is allowed for spin-independent operators; however, it cannot be applied to spin-dependent problems, such as Hamiltonians containing spin–orbit interactions. We show that for spin-independent Hamiltonians, a strict upper bound property is obeyed between a traditional Hartree–Fock-like determinant, full spinor wave function, the full determinant wave function, and a generalized spinor wave function. The relationship between a spinor wave function and the full determinant arises because the full determinant wave function is the spinor wave function projected onto a fixed-spin, after which antisymmetry is implicitly restored in the spin-independent case. For spin-dependent Hamiltonians, the full determinant wave function is not applicable, because it is not antisymmetric. Numerical experiments on the H 3 molecule and two-dimensional homogeneous electron gas confirm these bounds.
A long-standing goal of nuclear theory is to explain how the structure and dynamics of atomic nuclei and neutron-star matter emerge from the underlying interactions among protons and neutrons. Achieving this goal requires solving the nuclear quantum many-body problem with high accuracy across a wide range of length scales and density regimes. In this review, we discuss how artificial neural network representations of the nuclear many-body wave function have significantly extended the capabilities of continuum quantum Monte Carlo methods. In particular, neural network quantum states enable calculations of larger systems than were previously accessible and provide a flexible framework for capturing phenomena that challenge conventional approaches, including the emergence of nuclear clusters and superfluid phases in dense matter. We highlight recent applications to finite nuclei, infinite nuclear and neutron matter, and dynamical processes relevant to lepton-nucleus and nucleus-nucleus scattering. We also discuss conceptual and methodological connections with condensed matter physics, emphasizing developments in neural network quantum states that bridge strongly correlated systems across disciplines. Together, these developments demonstrate how neural-network methods open new avenues toward unified and accurate descriptions of nuclear structure, matter, and reactions.
Neural quantum states provide flexible variational representations of quantum many-body wave functions by combining neural-network parametrizations with Monte Carlo sampling. In this perspective, we review recent advances in their application to condensed-matter systems, focusing on frustrated quantum magnets, interacting lattice fermions, and non-equilibrium dynamics. We discuss the architectures, symmetry constraints, optimization methods, and sampling strategies underlying state-of-the-art calculations, and summarize practical guidelines for reliable simulations. We also examine the principal remaining challenges, including learning non-trivial sign and phase structures, controlling variational bias, enforcing physical symmetries, scaling optimization to large networks, and achieving stable real-time evolution. Finally, we outline promising directions in which neural quantum states may extend the reach of classical simulations of strongly correlated quantum matter.
Spin plays a fundamental role in understanding electronic structure, yet many real-space wavefunction methods fail to adequately consider it. We introduce the spin-adapted antisymmetrization method, a general procedure that enforces exact total spin symmetry for antisymmetric many-electron wavefunctions in real space. In the context of neural network-based quantum Monte Carlo, our method leverages the expressiveness of deep neural networks to capture electron correlation while enforcing exact spin adaptation via group representation theory. This framework provides a principled route to embed physical priors into otherwise black-box neural network wavefunctions, yielding a compact representation of correlated systems with neural network orbitals. To demonstrate its effectiveness, we study the spin ladder of iron-sulfur clusters, a long-standing challenge for many-body methods due to their dense spectrum of nearly degenerate spin states. Our results reveal accurate resolution of low-lying spin states and spin gaps in [Fe2S2] and [Fe4S4] clusters, offering insights into their electronic structures. In sum, these findings establish our framework as a practical route toward spin-adapted neural network-based quantum Monte Carlo for strongly correlated systems. The authors introduce a general framework that embeds exact spin symmetry directly into the neural network wavefunction, guaranteeing spin-pure states by construction. This design enables accurate resolution of near-degenerate spin states in challenging chemical systems.
Neural network quantum states emerge as a promising tool for solving quantum many-body problems. However, its successes and limitations are still not well-understood in particular for Fermions with complex sign structures. Based on our recent work [J. Chem. Theory Comput. 21, 10252-10262 (2025)], we generalizes the restricted Boltzmann machine to a more general class of states for Fermions, formed by product of `neurons' and hence will be referred to as neuron product states (NPS). NPS builds correlation in a very different way, compared with the closely related correlator product states (CPS) [H. J. Changlani, et al. Phys. Rev. B, 80, 245116 (2009)], which use full-rank local correlators. In constrast, each correlator in NPS contains long-range correlations across all the sites, with its representational power constrained by the simple function form. We prove that products of such simple nonlocal correlators can approximate any wavefunction arbitrarily well under certain mild conditions on the form of activation functions. In addition, we also provide elementary proofs for the universal approximation capabilities of feedforward neural network (FNN) and neural network backflow (NNBF) in second quantization. Together, these results provide a deeper insight into the neural network representation of many-body wavefunctions in second quantization.
Quantum neural network architectures that have little to no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks the action of which commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNNs) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNNs) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a S U ( 2 ) -equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity. Published by the American Physical Society 2024