Neural-Network Quantum States (NQS)

• Neural-Network Quantum States (NQS) are variational families of quantum states with amplitudes parameterized by neural networks, enabling flexible many-body representations.
• They employ architectures such as RBMs, autoregressive models, and Transformers to capture complex entanglement patterns including volume-law behavior.
• Applications span spin systems, quantum chemistry, nuclei, and nonlinear waves, though challenges in optimization and phase learning remain.

Neural-Network Quantum States (NQS) are variational many-body wavefunction ansätze in which amplitudes in a chosen computational basis are parameterized by neural networks. In practice, an NQS represents a state $|\psi_\theta\rangle$ through a map from configurations—spin strings, occupation-number vectors, or lattice Fock states—to complex amplitudes, and the parameters $\theta$ are optimized by variational principles such as energy minimization or fidelity maximization. Across the literature, NQS occupy a position complementary to tensor-network methods: they can represent classes of tensor-network states, can capture volume-law entanglement, and have been deployed in spin systems, quantum chemistry, lattice bosons, frustrated magnets, fracton codes, nonlinear wave equations, and mixed spin-fermion lattice models (Keeble et al., 30 Mar 2026).

1. Definition and place among many-body representations

NQS are variational families of quantum states whose basis amplitudes are encoded by a neural network and trained to minimize

$$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$

or, when an exact target is available, to maximize the fidelity

$$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$

In discrete bases, the associated local-energy estimator has the standard form

$$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$

with Monte Carlo expectations taken over $|\psi_\theta(x)|^2$ or related normalized distributions (Keeble et al., 30 Mar 2026).

The term “neural-network quantum state” historically became associated with Restricted Boltzmann Machines (RBMs), but the category is broader. The corpus includes RBMs, symmetric and projected RBMs, group-convolutional networks, multilayer perceptrons, convolutional models, autoregressive recurrent neural networks, Vision Transformer–based feature extractors, tensorized gated RNNs, and Transformer ansätze built for composite local Hilbert spaces (Mezera et al., 2023). In second quantization, NQS are naturally defined on determinants or occupation-number vectors, so fermionic antisymmetry is handled by the basis itself and by the Hamiltonian’s creation–annihilation algebra rather than by an explicit antisymmetrization layer (Keeble et al., 30 Mar 2026).

Within the broader landscape of variational states, NQS are often contrasted with tensor networks. Tensor-network methods are especially effective in low-entanglement, area-law regimes, while NQS have been shown to efficiently represent certain tensor-network classes and volume-law-entangled states, as emphasized in the nuclear benchmarks of medium-mass sd-shell nuclei (Keeble et al., 30 Mar 2026). The tensor-network reinterpretation of RBM-NQS goes further: RBM states can be viewed as correlator product states built from extensively sized GHZ-form correlators, making their relation to CPS explicit and clarifying why they can encode nonlocal constraints, topological structure, and volume-law entanglement with compact parameterizations (Clark, 2017).

2. Core ansätze and mathematical parameterizations

The canonical RBM ansatz writes the amplitude over binary visible variables as a hidden-unit marginalization. In the occupation-number formulation used for nuclei and electronic structure,

$$\langle n|\Psi_\theta\rangle = \sum_{h_i\in\{-1,+1\}} \exp\!\Big( \sum_{j=1}^N a_j n_j + \sum_{i=1}^M b_i h_i + \sum_{i=1}^M\sum_{j=1}^N W_{ij} h_i n_j \Big),$$

which yields after tracing out the hidden layer

$$\ln \langle n|\Psi_\theta\rangle = \sum_{i=1}^N a_i n_i + \sum_{j=1}^M \ln\!\Big[ 2\cosh\!\Big(b_j+\sum_{i=1}^N W_{ji} n_i\Big) \Big].$$

The parameters are generally complex-valued, so amplitude and phase are encoded jointly rather than by separate subnetworks (Keeble et al., 30 Mar 2026).

In electronic-structure applications, the equivalent second-quantized RBM form is often written as

$$w_\theta(o)=\exp\!\Big(\sum_i a_i o_i\Big)\prod_j\Big[1+\exp\!\Big(b_j+\sum_i W_{ij}o_i\Big)\Big],$$

with occupations $o_i\in\{0,1\}$ over spin orbitals and hidden density $\theta$0 determining the hidden-layer width (Li et al., 2023). For spin systems, the usual visible variables are $\theta$1, giving

$\theta$2

again with complex parameters when frustrated sign structure must be represented (Mezera et al., 2023).

Autoregressive NQS replace the hidden-variable marginalization by an exact factorization of the Born distribution. In the 2D tensorized gated RNN used for bosonic Hofstadter models,

$\theta$3

with

$\theta$4

This gives exact, uncorrelated sampling from the model distribution without MCMC, even with higher on-site occupations and fixed-$\theta$5 constraints implemented directly in the sampling rule (Döschl et al., 2024).

Transformer-based NQS extend this logic to heterogeneous local Hilbert spaces. In the Ancilla Layer Model, each site carries both fermionic occupations and ancillary spins, the local state is tokenized, a Transformer produces context-aware features, and the final amplitude is a determinant of spin-dependent backflow orbitals,

$\theta$6

thereby combining attention-based long-range conditioning with exact fermionic antisymmetry (Rende et al., 2 Mar 2026). A related feature-space construction freezes a deep Vision Transformer representation and trains only a shallow complex RBM readout,

$\theta$7

so that representation learning and variational readout are cleanly separated (Rende et al., 2024).

The architecture space has also expanded in more specialized directions. KAN-based “SineKAN” NQS represent amplitudes as nested univariate functions with learnable sinusoidal activations, while spin-1 RBMs augment the visible layer with quadratic single-site terms to retain product-state exactness, labeling freedom, and tensor-network gauge equivalence for local Hilbert space dimension three (Shamim et al., 2 Jun 2025). In all of these cases, the ansatz design is tied directly to the structure of the local basis and the sign pattern of the target state.

3. Optimization, sampling, and variational estimators

The dominant training paradigm remains Variational Monte Carlo. For configurations sampled from

$\theta$8

the energy is estimated as

$\theta$9

A widely used optimization scheme is stochastic reconfiguration (SR), or quantum natural gradient, with update

$$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$0

where

$$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$1

In the nuclear sd-shell study, this is regularized by a diagonal shift $$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$2 combined with an RMSProp-inspired scheme to stabilize small-eigenvalue directions of the QGT (Keeble et al., 30 Mar 2026).

Sampling mechanics vary sharply by architecture. RBM-based NQS typically use Metropolis–Hastings proposals, often constrained to symmetry sectors such as fixed particle number or magnetization. In the sd-shell nuclear calculations, proposals were 90% single-orbital swaps and 10% double-orbital swaps, with 4096 parallel chains and 40,960 samples per energy estimator (Keeble et al., 30 Mar 2026). In frustrated spin models such as the Shastry–Sutherland lattice, exchange updates enforce fixed $$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$3 sectors at zero field, while single-spin updates are used at finite field (Mezera et al., 2023). For deep autoregressive models, by contrast, exact ancestral sampling replaces Markov chains entirely, eliminating autocorrelation and burn-in (Döschl et al., 2024).

An important methodological branch replaces stochastic sampling by deterministic configuration selection. In quantum chemistry, a non-stochastic optimization algorithm constructs a selected determinant set on the fly, inspired by selected CI. Expectations and SR matrices are then evaluated by normalized sums over that selected set rather than by MCMC, with the selection criterion

$$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$4

This bypasses Markov-chain sampling, yields smoother convergence, and achieved comparable or superior accuracy to stochastic VMC on strongly correlated molecules (Li et al., 2023).

Another line of work replaces QGT-based optimization by supervised regression toward an imaginary-time target,

$$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$5

together with an overlap loss

$$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$6

This “fixed target until energy decreases” strategy avoids explicit SR inversion and was shown on the 2D $$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$7–$$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$8 Heisenberg model to improve stability over direct first-order energy minimization while remaining competitive with SR and DMRG on moderate sizes (Ledinauskas et al., 2023).

Training reuse has also become a methodological theme. Fine-tuning schemes pretrain a deep feature extractor near a highly expressive point of a phase diagram and then optimize only a shallow output layer elsewhere, reducing the trainable fraction to about 6.6% and giving roughly tenfold savings in cost and memory in the reported systems (Rende et al., 2024). Adaptive RNN schemes similarly grow model width during optimization, reusing smaller networks to initialize larger ones and reducing runtime while improving stability (McNaughton et al., 24 Jul 2025).

4. Expressivity, exact representations, and state complexity

A central theoretical question is what determines NQS representational efficiency. A common misconception is that entanglement alone is the relevant control parameter. The sd-shell nuclear benchmarks directly challenge that view: for a fixed number of configurations, states with larger non-stabilizerness are systematically harder for RBMs to learn, even when they share comparable basis sizes and substantial entanglement (Keeble et al., 30 Mar 2026). The paper quantifies non-stabilizerness through the stabilizer Rényi entropy

$$E(\theta)=\frac{\langle \Psi_\theta|\hat H|\Psi_\theta\rangle}{\langle \Psi_\theta|\Psi_\theta\rangle},$$9

and finds that fidelity degrades systematically with increasing $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$0 once the configuration count exceeds or approaches the RBM parameter count. This suggests that “magic,” not entanglement magnitude per se, is the primary compression bottleneck for RBMs in the entangled regime (Keeble et al., 30 Mar 2026).

Entanglement diagnostics remain useful but more nuanced. In the same work, simple single-orbital entropies did not correlate robustly with NQS accuracy, while multipartite proton–neutron 8-tangles did, in line with the correlation between accuracy and non-stabilizerness. The RBM systematically underestimated $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$1 of exact states and could even overestimate the bipartite proton–neutron entropy $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$2, reinforcing the point that distinct complexity diagnostics probe different structural bottlenecks (Keeble et al., 30 Mar 2026).

Exact representability results make the geometry of these bottlenecks concrete. RBM-NQS can be recast as correlator product states built from GHZ-form correlators, and this tensor-network view yields exact representations for weighted-graph states, lattice Laughlin states, toric-code states, and resonating-valence-bond states (Clark, 2017). Compact exact RBM constructions have since been derived for all Jastrow states and all stabilizer states with at most $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$3 hidden units and system-extensive connectivity, showing that low hidden density $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$4 can already be highly expressive when the architecture matches the target structure (Pei et al., 2021).

Basis dependence is another major factor. For spin-1 systems, the AKLT state requires $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$5 hidden units in the $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$6 basis but admits an analytic NQS construction with only $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$7 hidden units in the $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$8 basis, with strong numerical evidence for exact compactness already at $$F(\theta)=\frac{|\langle \Psi_\theta|\Psi_{\mathrm{ex}}\rangle|^2}{\langle \Psi_\theta|\Psi_\theta\rangle}.$$9 (Pei et al., 2021). This does not imply a universal preferred basis, but it does show that NQS compactness can depend dramatically on local basis choice and parameterization.

A further subtlety is that not every formally specified RBM-like object defines a valid normalized wavefunction. The amplitude-ratio analysis of NQS formalizes the fact that NQS naturally provide sampling and ratio access, not direct normalization access, and constructs a three-node example whose unnormalized amplitude vanishes for all configurations, so it cannot be normalized or sampled from at all (Havlicek, 2022). This result is best read as a warning about consistency and normalizability rather than as a pathology of practical training: amplitude ratios are powerful, but they do not by themselves guarantee a well-formed quantum state.

5. Applications across spins, fermions, bosons, nuclei, chemistry, and nonlinear waves

NQS are now used across a remarkably broad range of physical settings. In nuclear many-body theory, a second-quantized RBM on sd-shell active spaces was used to model medium-mass nuclei such as $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$0, $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$1, and $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$2, achieving accurate energies with substantial compression relative to the basis dimension while revealing the role of non-stabilizerness in limiting representational efficiency (Keeble et al., 30 Mar 2026).

In quantum chemistry, NQS have been developed both as algorithmic tools and as large-scale HPC workloads. Deterministic selected-configuration optimization improved stability in strongly correlated molecules (Li et al., 2023), while large-scale transformer-based NQS training on Fugaku introduced multistage sampling parallelism, cache-centric optimization, and SIMD-accelerated local-energy evaluation, yielding up to $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$3 speedup and parallel efficiency up to $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$4 on 1,536 nodes while reaching FCI-level accuracy on benchmark molecules such as $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$5 and $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$6 (Xu et al., 30 Jun 2025). Hybrid tensor-network/NQS ansätze have also been introduced for chemistry, including bounded-degree graph RNNs and RBM-inspired correlators, with chemical accuracy reported for H50, $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$7, and a $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$8 hydrogen cluster (Wu et al., 25 Jul 2025).

In frustrated and strongly correlated lattice models, shallow complex RBMs have reproduced the phase structure and magnetization plateaus of the Shastry–Sutherland model (Mezera et al., 2023), while Transformer-based determinant NQS have accurately described the Ancilla Layer Model, including LL, LL*, and Luther–Emery phases under both OBC and PBC (Rende et al., 2 Mar 2026). KAN-based SineKAN NQS achieved high-precision energies and correlation functions for the TFIM, anisotropic Heisenberg model, and the antiferromagnetic $$E_{\mathrm{loc}}(x)=\sum_{x'} H_{x x'}\frac{\psi_\theta(x')}{\psi_\theta(x)},$$9–$|\psi_\theta(x)|^2$0 chain, outperforming RBM, LSTM, and MLP baselines on the $|\psi_\theta(x)|^2$1 $|\psi_\theta(x)|^2$2–$|\psi_\theta(x)|^2$3 benchmark considered (Shamim et al., 2 Jun 2025).

For bosonic lattice systems in magnetic fields, a 2D tensorized gated RNN NQS was used to study Hofstadter–Bose–Hubbard and long-range Rydberg-dressed models up to $|\psi_\theta(x)|^2$4 sites, identifying Hofstadter phases, Wigner crystals, and bubble crystals with exact autoregressive sampling and explicit amplitude–phase outputs (Döschl et al., 2024). In fracton physics, correlation-enhanced RBMs gave exact representations of the checkerboard, X-cube, and Haah codes and were then used variationally to study the field-driven first-order transition of the checkerboard model on lattices up to 512 qubits (Machaczek et al., 2024).

NQS ideas have also migrated beyond standard lattice many-body settings. A continuous-space “NNQS” for the nonlinear Schrödinger equation parameterized stationary states of a harmonically trapped Gross–Pitaevskii equation, including excited states that are difficult for imaginary-time evolution, and then used those states to study spatiotemporal chaos via Lyapunov exponents and extended self-similarity (Zhao et al., 11 Jun 2025). This suggests that the NQS framework is best understood not as a lattice-specific technology but as a general variational paradigm for wavefunction learning.

6. Limitations, misconceptions, and active directions

Several limitations recur across the literature. First, optimization remains a major bottleneck. High-fidelity representation is often not the issue; rather, flat or rugged energy landscapes, small excitation gaps, and sampling noise impede convergence. In the Hofstadter RNN study, fidelity-optimization tests suggested that representational power was not the main obstacle in the topological regime, whereas optimization was (Döschl et al., 2024). In the sd-shell nuclei, odd systems with small gaps often converged to admixtures of low-lying states, and adding a total-$|\psi_\theta(x)|^2$5 constraint helped in some cases (Keeble et al., 30 Mar 2026).

Second, sign and phase structure remain architecture-sensitive. Complex RBMs can encode phases, but non-stabilizer or non-stoquastic regimes often expose their limitations. Fracton simulations under $|\psi_\theta(x)|^2$6 perturbations showed that learning complex phases in simple RBMs is substantially harder than learning stoquastic amplitudes, with large variance and unstable gradients at larger sizes (Machaczek et al., 2024). This has motivated architectures with explicit amplitude–phase decomposition, attention-based nonlocal modeling, and determinant or Pfaffian structure.

Third, sampling is not uniformly benign. RBM-based VMC must contend with autocorrelation, proposal design, and noisy QGT estimates. Autoregressive models eliminate MCMC but may introduce ordering sensitivity or large parameter counts. Deterministic or semi-stochastic estimators reduce variance but can lose efficiency when the wavefunction support is not sparse (Li et al., 2023). Quantum-assisted sampling has even been proposed as a way to accelerate Markov-chain convergence for RBM/Hopfield NQS, with the reported method scaling polynomially in storage and requiring constant measurements per iteration in the tested setup (Sajjan et al., 2024).

A recurrent misconception is that NQS are synonymous with black-box flexibility. The strongest recent results often come from physics-informed design rather than generic overparameterization. Examples include symmetry-projected and correlation-enhanced RBMs for fracton codes (Machaczek et al., 2024), bounded-degree graph RNNs for molecular entanglement structure (Wu et al., 25 Jul 2025), and solvable-point-guided spin–charge-separated architectures for valence-bond states, where local-rule convolutions, geometric pooling, and sector separation yielded approximately $|\psi_\theta(x)|^2$7 parameter scaling in a gapless regime compared with approximately $|\psi_\theta(x)|^2$8 for MPS in the same setting (Wu et al., 15 Jun 2026). This suggests that compactness is closely tied to whether the architecture encodes the correct constrained Hilbert space and sign structure.

The current research frontier therefore combines three themes. One is expressivity beyond RBMs: Transformers, Pfaffian and backflow constructions, deep autoregressive models, and hybrid NQS–tensor-network forms. Another is optimization: deterministic selection, adaptive model growth, imaginary-time target regression, and fine-tuning of pretrained feature extractors. The third is state-complexity awareness: diagnostics such as stabilizer Rényi entropies, multipartite entanglement measures, and basis optimization are increasingly used not just to analyze target states but to guide ansatz design itself (Keeble et al., 30 Mar 2026).

Taken together, these developments define NQS less as a single ansatz family than as an evolving program in variational many-body representation: neural architectures are treated as wavefunction parameterizations whose success depends on how well they reflect basis choice, locality or nonlocality, symmetry sector, constrained configuration space, and the specific form of amplitude and phase structure in the target state.