Neural Quantum States in Many-Body Physics

• Neural Quantum States (NQS) are variational wave functions that use neural networks to parameterize both amplitude and phase in many-body quantum systems, enabling precise modeling of entanglement and correlations.
• Advanced optimization and sampling techniques, such as stochastic reconfiguration and autoregressive methods, enhance NQS scalability and efficiency across spin, fermionic, and bosonic systems.
• Their versatility extends to applications in quantum magnetism, quantum chemistry, and topological phases, driving innovations in quantum simulation and complexity analysis.

Neural Quantum States (NQS) are a class of variational wave functions in which the amplitude and phase for many-body quantum systems are parameterized by neural networks. By leveraging the universal approximation capabilities of artificial neural architectures, NQS provide an alternative to traditional variational Ansätze and have become an influential framework for the paper, simulation, and analysis of quantum many-body problems. NQS approaches have enabled precise modeling of complex quantum systems—including those with high entanglement and rich quantum correlations—beyond the reach of conventional methods. Their flexibility permits application to spin, fermionic, and bosonic systems, quantum chemistry, quantum dynamics, and topologically ordered phases, and their underlying principles are driving developments in quantum complexity theory, algorithmic resource assessment, and scalable quantum simulation.

1. Fundamental Principles and Mathematical Formulation

Neural Quantum States employ neural networks to encode the wave function $\psi(\mathbf{n})$ defined on the Hilbert space of a quantum system, where $\mathbf{n}$ labels a basis configuration. A key example is the RBM (Restricted Boltzmann Machine) NQS, given by

$$\psi_\theta(\mathbf{n}) = \exp\left[\sum_{j} a_j n_j\right] \prod_{l} 2 \cosh\left(b_l + \sum_{j} W_{lj} n_j\right),$$

where $a_j$, $b_l$, and $W_{lj}$ are variational network parameters. More general architectures, such as multilayer perceptrons (MLPs), convolutional neural networks (CNNs), graph neural networks, and Transformers, allow for modeling more complex correlation and entanglement structures.

The variational principle is central: the NQS wave function is optimized to minimize the expectation value of the Hamiltonian,

$$E_\theta = \frac{\langle \psi_\theta | \hat{H} | \psi_\theta \rangle}{\langle \psi_\theta | \psi_\theta \rangle},$$

where observables are calculated using Markov chain Monte Carlo (MCMC) or autoregressive sampling to generate configurations distributed according to $|\psi_\theta(\mathbf{n})|^2$.

For fermionic systems, antisymmetry is enforced either by embedding Slater determinants (combined with neural backflow or Jastrow factors) or through Jordan-Wigner strings. In representing quantum states, NQS allow both amplitude and phase to be treated, bestowing the expressibility necessary for sign- or phase-problematic systems (Lange et al., 14 Feb 2024).

2. Algorithmic Advances: Optimization and Sampling

The non-convexity of the NQS optimization landscape has spurred the development of specialized algorithms. Stochastic reconfiguration (SR), which approximates imaginary-time evolution using the quantum geometric tensor, is widely adopted despite its scaling as $O(N_\theta^2)$ (Ledinauskas et al., 2023). Recent work bypasses SR by using first-order gradient descent with adaptive time stepping, overlap-based losses, and Euclidean metrics, enabling use of much larger networks and more stable convergence.

Sampling is a computational bottleneck, as generating independent samples from complicated $|\psi_\theta|^2$ landscapes is challenging. Kinetic sampling protocols based on continuous-time Monte Carlo (the Zanella process) provide improved autocorrelation properties and ergodic exploration compared to traditional Metropolis-Hastings, especially in rugged probability landscapes (Bagrov et al., 2020). In autoregressive architectures, the probability can be perfectly factorized—$P(\mathbf{n}) = \prod_i P(n_i | n_{<i})$—allowing for uncorrelated direct sampling, as leveraged in recurrent neural network and gated RNN NQS (Döschl et al., 7 May 2024). Further advancements in large-scale parallelism and hybrid sampling algorithms enable NQS frameworks to scale to thousands of processing nodes and tackle ab initio quantum chemistry for sizable molecular systems (Xu et al., 30 Jun 2025).

3. Expressivity, Correlations, and Internal Mechanisms

NQS provide a framework capable of capturing both area-law and volume-law entanglement, as well as complex correlation structures. Detailed analysis via correlation-based interpretable neural network architectures reveals that—at least in the computational basis—faithful representation of a general quantum state requires contributions from correlations of all possible orders. The Fourier (Boolean) expansion

$$\psi(\sigma) = \sum_{S \subseteq [L]} \bar{\psi}(S) \prod_{i\in S} \sigma_i$$

shows that to reconstruct the wave function exactly, the network must be capable of representing all (up to $L$-body) spin products. This need for high-order correlations persists even for “simple” product or antiferromagnetic states. The network’s architecture, activation function parity, and depth critically influence its capacity to represent these high-order terms (Döschl et al., 19 Aug 2025).

Autoregressive neural architectures can reduce the required maximum correlation order in systems with conditional independence, while non-autoregressive (e.g., correlator transformer) or determinant-based methods expose the full correlation structure more naturally. Fourier analysis enables diagnosis of representational failures in NQS and suggests avenues for improvement via basis rotations or tailored activation architectures.

4. Applications Across Physics and Chemistry

The adaptability of NQS ansätze has led to diverse applications:

• Quantum Magnetism and Spin Models: NQS have captured ground and excited states, phase transitions, and magnetization plateaus in a variety of spin models, including the $J_1$–$J_2$ Heisenberg, Shastry-Sutherland, and Kitaev models. Transformer-based NQS have been used to resolve emergent stripe order and long-range correlations in the 2D Hubbard model, achieving state-of-the-art energies and capturing experimentally relevant phases (Gu et al., 3 Jul 2025).
• Strongly Correlated Electrons and Quantum Chemistry: Hybrid approaches combining tensor network structures (matrix-product states) with neural networks (bounded-degree graph RNNs and RBM-inspired correlators) have reached chemical accuracy in complex clusters and real molecular systems. Efficient semi-stochastic energy evaluation dramatically reduces simulation time, and large-scale simulations scale to thousands of nodes using advanced parallel and cache-centric strategies (Wu et al., 25 Jul 2025, Xu et al., 30 Jun 2025).
• Dynamics and Finite Temperature: NQS, by augmenting variational principles to time-dependent and thermal settings (e.g., via purification in an enlarged Hilbert space), allow simulation of quantum evolution and open-system dynamics. Linear response methods and dynamical spectral function estimations are accessible through NQS frameworks.
• Topological Phases, Fracton Models, and Quantum Memory: NQS architectures specifically designed to incorporate stabilizer constraints as local correlations (e.g., in the checkerboard, X-cube, or Haah’s code) can represent highly entangled topological ground states efficiently, revealing strong first-order phase transitions and probing the stability of fracton orders under quantum fluctuations (Machaczek et al., 17 Jun 2024).
• Resource Theory and Quantum Complexity: NQS methods can quantify "magic"—or non-stabilizerness—of many-body wavefunctions via Monte Carlo estimation of stabilizer Rényi entropies, providing insight into the quantum computational resources intrinsic to a given state, e.g., for quantum advantage proposals or fault-tolerant computing (Sinibaldi et al., 13 Feb 2025).

5. Generalization, Double Descent, and Physically-Informed Design

The generalization capabilities of NQS differ from conventional machine-learning networks. Recent studies have identified the double descent phenomenon: as the network size increases, generalization first degrades, then improves only upon vastly overparameterizing the network (beyond the Hilbert space dimension)—a regime seldom reached in practical NQS implementations due to the exponential scaling of the quantum Hilbert space. Optimal generalization is typically achieved when the number of parameters matches the number of unique training samples. Sampling strategy plays a critical role; importance sampling of high-probability configurations and physics-aware architecture choices (such as enforcing symmetry) significantly improve learning performance (Moss et al., 31 Jul 2025).

This finding suggests that brute-force scaling of network size is not sufficient for improved representation in quantum many-body settings and that architecture must encode physical knowledge—symmetry constraints, basis choices, or known correlation structure—for efficiency and accuracy.

6. Recent Innovations and Future Directions

Advancements include fine-tuning paradigms—pretraining deep networks near phase transitions followed by low-cost output layer adjustment enables rapid, accurate mapping of entire phase diagrams (Rende et al., 12 Mar 2024). Adaptive training, where small neural networks are incrementally grown and parameters reused to initialize larger models, reduces computational cost and stabilizes training trajectories, with impacts on energy variances and resource utilization in large-scale simulations (McNaughton et al., 24 Jul 2025).

Novel NQS architectures, such as Kolmogorov-Arnold Networks with learnable sinusoidal activations, have demonstrated superior performance for large spin chain models, outperforming RBMs, MLPs, and RNNs in energy accuracy and computational cost (Shamim et al., 2 Jun 2025).

Hybrid NQS-tensor network methods and quantum-enabled variational Monte Carlo algorithms—utilizing quantum-assisted sampling—offer polynomial efficiency and pave the way for future quantum-classical hybrid simulation protocols (Sajjan et al., 16 Dec 2024).

Future directions for NQS research include integration with ab initio materials modeling (via density functional perturbation theory), extension to open and finite-temperature systems, development of more expressive yet interpretable architectures, and reduction of sampling and inference bottlenecks (critical for quantum embedding loops and large Hilbert space exploration) (Zhou et al., 15 Sep 2025). Progress in physics-informed architecture, optimization beyond SR, and scalable parallelism will further extend the impact of NQS-based methods in quantum simulation, quantum information, and condensed matter research.