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From Architectures to Applications: A Review of Neural Quantum States

Hannah Lange, Anka Van de Walle, Atiye Abedinnia, Annabelle Bohrdt

arXiv:2402.09402v3cond-mat.dis-nncond-mat.quant-gascond-mat.str-elquant-ph

TL;DR

Simulating quantum many-body states is challenging because exact wave-function representations require exponentially many coefficients. This review synthesizes neural quantum states and their architectures and applications; it concludes that they can represent broad classes of states while compressing coefficients into tractable network parameters. It also notes that optimization can make finding the target state difficult.

  • Problem

    Quantum state tomography methods such as linear inversion and maximum likelihood estimation face exponential scaling in sampling complexity and the number of state parameters.

    Source: §4.1 · p. 29

  • Method

    The review surveys NQS architectures and their use across quantum-state simulation and tomography.

    Source: §5 · p. 32

  • Results

    NQS can represent a broad range of quantum states and compress wave-function coefficients into a tractable number of network parameters.

    Source: §5 · p. 32

  • Takeaways & Limitations

    This versatility makes NQS applicable to numerical quantum-system simulation and quantum state tomography.

    Source: §5 · p. 32

    Stochastic reconfiguration can require many samples for a precise covariance matrix estimate, regularization when the matrix is non-invertible, and costly inversion.

    Source: §3.1.1 · p. 19

Abstract

from arXiv · show

Due to the exponential growth of the Hilbert space dimension with system size, the simulation of quantum many-body systems has remained a persistent challenge until today. Here, we review a relatively new class of variational states for the simulation of such systems, namely neural quantum states (NQS), which overcome the exponential scaling by compressing the state in terms of the network parameters rather than storing all exponentially many coefficients needed for an exact parameterization of the state. We introduce the commonly used NQS architectures and their various applications for the simulation of ground and excited states, finite temperature and open system states as well as NQS approaches to simulate the dynamics of quantum states. Furthermore, we discuss NQS in the context of quantum state tomography.

1. A Short Introduction to Neural Quantum States (NQS)

Neural quantum states represent wave-function coefficients with trainable neural networks, offering a parameter-based alternative to storing the full state. The review introduces their sampling and training, early simulation results, and expressive scope alongside optimization challenges.

1. A Short Introduction to Neural Quantum States (NQS)

  • NQS map configurations σ to wave-function coefficients ψθ(σ), with amplitude and phase often modeled separately.Network parameters are trained so that ψθ approximates the target wave function.Source: §1 · p. 3
  • The original NQS work achieved competitive ground-state energies against state-of-the-art 2D methods and dynamics agreeing well with t-DMRG.The examples were an antiferromagnetic Heisenberg model on a 10 × 10 lattice and an 80-site spin chain.Source: §1 · p. 3, Figure 1 · p. 4
  • NQS can represent some volume-law entangled states without exponential cost, but the Sachdev-Ye-Kitaev ground state is an example that cannot be represented efficiently [29].Some architectures also match or exceed the expressive power of practically usable variational tensor networks.Source: §1 · p. 3, Figure 1 · p. 4
  • NQS training has a non-convex landscape, so finding the ground state can be challenging even when the ansatz is sufficiently expressive.The review points to advanced training strategies as a response to this optimization challenge.Source: §1 · pp. 3–4
  • NQS training can proceed without external data, but performance depends on architecture, wave-function parameterization, optimization, and task.The state may use separate amplitude and phase networks or complex parameters in one network.Source: §1 · p. 4
  • NQS observables are estimated from samples generated according to the NQS amplitudes, with computational cost depending on normalization and operator type.Unnormalized states may require approaches such as Metropolis sampling; local operators can be evaluated efficiently.Source: §1 · pp. 4–5

2. NQS Architectures

The review surveys neural quantum state architectures, their applications, and their trade-offs. It covers feed-forward networks and restricted Boltzmann machines alongside the field’s shifting mix of architectures.

2. NQS Architectures

  • By June 2024, cited works used a range of architectures in similar fractions, with no clear architecture choice for a given physical problem.The field moved from early RBM work toward CNNs and FFNNs, and more recently toward autoregressive networks, transformers, and fermionic systems.Source: §2 · p. 6

2.1. Feed Forward Neural Networks (FFNNs)

  • FFNNs use input, hidden, and output layers; NQS applications include discrete and continuous systems and direct targeting of ground or excited states.RBF networks use distance-dependent activations and have been applied to a quantum harmonic oscillator with an applied field.Source: §2.1 · pp. 6–7

2.2. Restricted Boltzmann Machines (RBMs)

  • RBMs add hidden nodes to increase expressivity and restrict connections to visible-hidden pairs, while shallow RBMs have limits that deep RBMs can overcome.Their connectivity supports volume-law entanglement and applications across varied physical systems, including higher-dimensional systems.Source: §2.2 · pp. 8–9

2.3. Convolutional Neural Networks (CNNs) and Group CNNs

CNNs use shared convolutional filters to capture local patterns, while group CNNs build in transformation symmetries. The review describes their use across lattice geometries and reports applications to frustrated models and volume-law states.

2.3. Convolutional Neural Networks (CNNs) and Group CNNs

  • CNN filters share weights across input locations to detect local patterns, reducing parameters relative to fully connected networks; convolution produces feature maps passed to later layers.CNNs are presented as a viable approach for two-dimensional systems, including the frustrated J1−J2 antiferromagnetic Heisenberg model.Source: §2.3 · p. 9
  • Deep CNNs encode 2D volume-law entangled states with O(N) parameters, versus O(N) for RBMs and O(N^2) for fully connected networks.CNNs have also been applied to ground-state calculations and dynamics; skip connections, layer normalization, and a large-parameter stochastic reconfiguration variant address training challenges in deep networks.Source: §2.3 · p. 10
  • CNN methods adapt to triangular, honeycomb, and Kagome lattices using regular filters and, for some geometries, strategically placed virtual vertices.These adaptations are described as preserving lattice properties while expanding information exchange and receptive fields.Source: §2.3 · p. 10
  • Group convolutions make feature maps equivariant to transformations such as rotations, translations, and reflections, unlike standard CNNs, which lack some of these symmetries.Group CNNs generate equivariant feature maps indexed by group elements; the wave-function coefficient uses the character of the symmetry operation.Source: §2.3 · p. 10
  • G-CNNs have been used for J1−J2 Heisenberg ground states on square and triangular lattices, and deeper models calculate low-lying excited states by changing symmetry-operation characters.Applications also include a reported spinon pair density wave on the Kagome lattice and dimer-state paramagnetic and canted magnetic order phases on the Maple-Leaf lattice at different Jd/J.Source: §2.3 · pp. 10–11

2.4. Graph Neural Networks

The review covers graph-based message passing and autoregressive networks, then details recurrent neural networks (RNNs) as an autoregressive architecture. It describes lattice applications, sampling and expressivity trade-offs, and RNN links to tensor-network representations.

2.4. Graph Neural Networks

  • Graph neural networks retain the Hamiltonian lattice as a graph and update node values through typically pairwise messages from immediate neighbors.A sublattice encoding can supply each site’s position within the unit cell as additional input.Source: §2.4 · p. 11
  • Graph structure and message passing allow transfer across system sizes; reviewed studies consider hard-core bosonic t−V and J1−J2 Heisenberg models on multiple lattice geometries.The Heisenberg study compares complex weights with separate amplitude and phase networks; the hard-core bosonic Hamiltonian is stoquastic and requires no sign structure to be learned.Source: §2.4 · p. 11

2.5. Autoregressive Networks

  • Autoregressive networks normalize amplitudes through conditional probabilities, enabling direct sampling instead of Markov-chain sampling, but this sampling can reduce expressivity in some cases.The review also notes that stochastic reconfiguration can cause optimization problems for autoregressive NQS.Source: §2.5 · pp. 11–12

2.5.1. Recurrent Neural Networks (RNNs)

  • RNN cells pass hidden states between sites and produce local conditional probabilities and phases; shared weights can model bulk properties and support retraining on larger systems.GRUs provide long-term memory for modeling long-range correlations; RNNs are applied in one- and two-dimensional systems, with conservation laws, spatial symmetries, and Marshall-sign implementation reported to improve results in one study.Source: §2.5.1 · pp. 12–13
  • A 2D tensorized MPS-RNN encodes area-law entanglement and supports perfect sampling and efficient wave-function evaluation, unlike PEPS.The review also reports that receiving hidden states from two directions improves state compression relative to tensor-network classes supporting polynomial-time wave-function evaluation.Source: §2.5.1 · p. 13

2.6. Transformers

Transformers use attention to model dependencies among system sites, with NQS decoders commonly masking future inputs for autoregressive sampling. The review describes ground-state, dynamics, and generalization applications, alongside efficiency and output-size trade-offs in patched models.

2.6. Transformers

  • Self-attention creates all-to-all interactions between sites; NQS commonly use a decoder whose mask allows connections to previous sequence elements but not subsequent ones, enabling exact sampling.The decoder applies softmax to masked query-key dot products and combines multi-head attention with a feed-forward network.Source: §2.6 · pp. 13–14
  • Transformer quantum states obtain results comparable to DMRG for a 1D frustrated spin model with relatively few parameters and have also been applied to dynamics, steady states, and quantum chemistry.Reported ground-state applications include the 1D transverse-field Ising, Heisenberg J1−J2, and XYZ models.Source: §2.6 · p. 14
  • Large patched transformers improve efficiency but increase output dimension exponentially with patch size; passing their outputs to a patched RNN breaks inputs into smaller sub-patches.The patched model is reported to capture ground-state properties and phase transitions of large Rydberg systems, competing with quantum Monte Carlo results.Source: §2.6 · p. 14
  • Transformer states trained with Hamiltonian parameters as inputs can generate ground states for unseen parameter values without additional training, though with slightly larger error.The review also describes generalization to different unseen systems and to systems outside the training set.Source: §2.6 · pp. 14–15
  • Factored attention in a transformer has been mapped to solving the inverse Potts problem with pseudo-likelihood; combined with a patched transformer, it yields high-quality J1−J2 Heisenberg ground-state energies.The review also describes a transformer-plus-FFNN architecture for ground-state search in the Shastry-Sutherland model.Source: §2.6 · pp. 14–15

2.7. NQS for Fermionic Systems

Fermionic NQS either build antisymmetry into the ansatz or use Jordan–Wigner transformations to map bosonic NQS to fermionic wave functions. The review describes the trade-offs and common constructions of both approaches.

2.7. NQS for Fermionic Systems

  • Fermionic NQS use either ansätze with built-in fermionic statistics or bosonic NQS antisymmetrized through a Jordan–Wigner transformation.The latter avoids requiring a specialized fermionic architecture, while Jordan–Wigner strings are non-local and may cause problems for some architectures.Source: §2.7 · p. 15, §2.7.2 · p. 16

2.7.1. Fermionic Architectures

  • Slater–Jastrow ansätze combine an antisymmetric component, typically a Slater determinant, with a Jastrow factor that captures correlations.In second quantization, neural networks can parameterize the Jastrow factor; examples include RBM+PP and neural backflow approaches [157, 139, 151, 152, 161–163].Source: §2.7.1 · pp. 15–16
  • First-quantized FermiNet and PauliNet achieve high accuracies in ab initio molecular simulations, while neural antisymmetric constructions can reduce determinant costs but may reduce accuracy.The review notes that evaluating Slater determinants is costly in first quantization, and that an alternative deep-network construction can reduce accuracy.Source: §2.7.1 · p. 15

2.7.2. Bosonic Architectures with Jordan Wigner Strings

  • Jordan–Wigner strings supply the fermionic exchange sign in expectation-value calculations, and the resulting approach has been applied to molecules, lattice models, and solid-state systems.The exchange sign comes from the exponential string; it need not be implemented in the NQS architecture itself.Source: §2.7.2 · p. 16

2.8. Other Design Choices

NQS performance depends on choices such as phase-and-amplitude representation, symmetry handling, activation functions, and model size. The review also identifies open-source toolboxes for NQS simulation and tomography.

2.8. Other Design Choices

  • For the J1−J2 ground state on small clusters, a complex-valued RBM outperformed two separate real-valued networks for amplitude and phase.The study compared these alternative representations of the wave-function amplitude and phase.Source: §2.8 · pp. 16–17
  • Restricting training to a target symmetry sector can improve performance, using configuration constraints for global U(1) symmetry or transformed samples for spatial symmetries.Group CNNs and gauge-equivariant networks can preserve certain symmetries explicitly.Source: §2.8 · p. 17
  • Activation-function choice can strongly affect FFNN performance, while RBM accuracy can saturate at high hidden-layer width as the quantum geometric tensor rank saturates.Overparameterization followed by pruning is also discussed as a possible performance-improvement strategy [174].Source: §2.8 · p. 17

2.9. Open-Source Toolboxes

  • NetKet supports ground-state search, dynamics, and tomography, while jVMC targets variational Monte Carlo, FermiNet supports atomic and molecular ground states, and QuCumber provides RBM-based tomography.The listed packages are based on Google’s JAX library, except QuCumber, for which this passage does not state a library.Source: §2.9 · pp. 17–18

3.1. Ground States

For the J1−J2 model on 10×10 lattices, several NQS architectures achieve results competitive with or better than a high-bond-dimension DMRG benchmark. The review also explains VMC and stochastic reconfiguration, optimization challenges, and proposed remedies.

3.1.1. Variational Monte Carlo

  • VMC trains a variational state by minimizing its sampled Hamiltonian expectation value, and stochastic reconfiguration adjusts updates using the parameter-space geometry.SR is motivated as imaginary-time evolution projected onto the variational manifold, producing an update involving the quantum geometric tensor [64, 66, 70].Source: §3.1.1 · pp. 18–19
  • SR can be unstable without precise estimates of its matrix, may require regularization, and has costly inversion scaling with parameter count.Iterative solvers, sequential local optimization, sample-sized inversions, and adaptive learning-rate solvers are among proposed remedies [87] [100].Source: §3.1.1 · p. 19
  • Other optimization strategies include artificial-temperature annealing, symmetry constraints, transfer learning, pruning, external-data pretraining, and a few post-training Lanczos steps.Few Krylov vectors are typically accessible because each requires separate training [145] [204].Source: §3.1.1 · p. 19, §3.1.2 · p. 19

3.1.3. Results on the J1−J2 model

  • Recent CNN, RBM, and transformer results on 10×10 J1−J2 lattices at J2/J1 = 0.5 are competitive with or outperform DMRG using 8192 SU(2) states.Some approaches use more than 10^5 parameters, while some with around 10^3–10^4 parameters obtain competitive energy errors; implementation and optimization choices matter. Figure 10 compares the methods.Source: Figure 10 · p. 20, §3.1.3 · p. 21

3.1.2. Optimization Challenges

  • A stochastic-representation approach trains by regression on projected wave-function samples without requiring amplitude-distributed samples, energy evaluation, or energy gradients.It iterates by using the updated trial function to generate new sample coordinates.Source: §3.1.2 · p. 20
  • NQS optimization faces rugged landscapes, and fixing exact phase or amplitude during training did not systematically improve ground-state representation in the studied models.The findings for the J1−J2 and bosonic and fermionic t−J models suggest that phase–amplitude interplay is important [35].Source: §3.1.1 · p. 19, §3.1.2 · p. 20

3.2. Excited States

NQS excited-state methods target either a different symmetry sector from the ground state or low-energy states in the same sector. The review covers symmetry targeting, orthogonality constraints, and expanded-system approaches.

3.2. Excited States

  • Different-symmetry excited states can be targeted by restricting the wave function or training loss to the desired sector, enabling NQS calculations of quasiparticle dispersions.Methods include quantum-number projections and a momentum-target penalty; the translation operator used in one approach is inexpensive to evaluate [35].Source: §3.2 · pp. 21–22
  • For excited states in the ground state’s symmetry sector, some methods enforce orthogonality to the ground state or lower excited states, but require explicit ground-state representation.This requirement makes calculating higher excited states computationally demanding.Source: §3.2 · p. 22
  • An alternative recasts finding the K lowest excited states as finding the ground state of an expanded system containing all targeted states.VMC on the expanded Hamiltonian yields a representation from which excited-state energies, expectation values, and overlaps can be retrieved [212].Source: §3.2 · p. 22
  • A separate proposal constructs excited states through single-particle excitations on top of the ground state, assuming one-particle excitations dominate the low-lying spectrum..Source: §3.2 · p. 22

3.3. Dynamics

NQS can simulate quantum dynamics, but t-VMC stability and accumulated errors constrain long-time evolution. The review describes p-tVMC and other alternatives, alongside applications to local quenches and spectral functions.

3.3. Dynamics

  • t-VMC optimizes time-dependent network weights by minimizing variational residuals, but covariance-matrix inversion and accumulated errors complicate time evolution.The Moore–Penrose pseudoinverse requires a cutoff for small singular values, and Krylov methods may fail to converge.Source: §3.3 · p. 23
  • t-VMC has been applied to systems of up to 256 spins and timescales up to 10s, but long-time access remains challenging.Its stability depends on the variational ansatz and can be affected by statistical bias or exponential sample complexity when the wave function contains zeros.Source: §3.3 · p. 24
  • p-tVMC uses an s-order Runge–Kutta approximant and Monte Carlo-estimated infidelity to project the exactly evolved state onto the variational manifold at each time step.Unlike t-VMC, it does not require an expansion of the variational state in its parameters.Source: Figure 12 · p. 23, §3.3 · p. 24
  • In the transverse-field Ising model, an RBM lets p-tVMC compute off-diagonal elements exactly and scale linearly with the number of parameters, though propagator decomposition or expansion can be costly.The method is described as usable when t-VMC fails or is inefficient, without the same biases or vanishing signal-to-noise ratio.Source: §3.3 · p. 24
  • The implicit midpoint alternative optimizes the next-time-step state against the discrete Schrödinger flow and can preserve the symplectic form of Hamiltonian dynamics.The approach avoids complicating network optimization with intermediate quantities.Source: §3.3 · p. 24
  • Other dynamics approaches include supervised local-unitary circuits, training on exactly evolved states, quantum-device hybrid calculations, and variational classical networks.The review also describes local-quench calculations of dynamical structure factors using t-VMC with CNN states or Chebyshev expansion with RBMs.Source: §3.3 · pp. 24–25, §3.3.1 · p. 25

3.3.1. Spectral Functions

  • For spectral functions, the review describes calculating a correction vector through an extension of stochastic reconfiguration, using a ground state obtained beforehand with SR.Source: §3.3.1 · p. 25

3.4. Finite Temperature States

The review describes NQS approaches to finite-temperature states using purification, METTS, quantum typicality, and direct density-matrix representations. These methods use imaginary-time evolution or free-energy optimization, with differing constraints on the represented state.

3.4. Finite Temperature States

  • Purification represents a thermal state using a pure state with ancilla sites, evolves it in imaginary time, then traces out the ancillas to obtain the density matrix.A modified RBM is one NQS architecture used for purification.Source: §3.4 · p. 25, Figure 13 · p. 26
  • METTS estimates thermal averages from an ensemble of pure states generated by imaginary-time evolution and repeated projective-measurement collapse.RBMs have been used to prepare the initial product states.Source: §3.4 · pp. 25–26, Figure 13 · p. 26
  • In purification and METTS, keeping each new wave function within the variational manifold during imaginary-time evolution can modify β.Source: §3.4 · p. 26
  • Quantum typicality approximates an infinite-temperature state with a pair-product wave function and an NQS component, then evolves the typical state in imaginary time.Pair-product wave functions can model interactions and impose no double occupancy through a Gutzwiller projection.Source: §3.4 · p. 26
  • A CNN representation of a mixed-state density matrix can be evolved from infinite temperature, but does not guarantee hermiticity or positive definiteness.Source: §3.4 · p. 27
  • An alternative to imaginary-time evolution minimizes a modified free energy, replacing von Neumann entropy with the more efficiently evaluated second Rényi entropy.Source: §3.4 · p. 27

3.5. Open Systems

Open-system NQS represent density operators to address the larger parameter burden of mixed states and simulate Lindblad dynamics. The review covers POVM and purification representations, time-dependent solutions, and steady-state optimization.

3.5. Open Systems

  • For N spin-1/2 particles, a density operator requires 4^N coefficients, making open-system simulation more challenging than representing a wave function.Open-system dynamics are governed by the Lindblad master equation.Source: §3.5 · p. 27
  • NQS density operators can use informationally complete POVM outcome probabilities or additional nodes to represent environmental mixing.The POVM representation models positive amplitudes but does not generally guarantee a positive-definite density matrix; purification does.Source: §3.5 · pp. 27–28
  • Time-dependent Lindblad solutions can minimize a distance between the evolved and target representations, including Fubini–Study, trace, or Kullback–Leibler distances.For one approach, expanding distance metrics around small times yields a TDVP update that reduces sampling cost and makes optimization convex.Source: Time dependent solution of the… · p. 28
  • For steady states, neural density operators are trained to satisfy the stationary-state condition by minimizing the expectation of the Liouvillian or an L2 norm.Applying the adjoint Liouvillian leads to an optimization involving L†L, which is Hermitian and has a real spectrum.Source: Steady states · pp. 28–29

4. Learning from Data

The review surveys NQS for reconstructing quantum states from measurements, compressing data through latent representations, and combining experimental or simulated data with variational training. These approaches address incomplete measurements and can support state estimates and observables beyond direct experimental access.

4.1. Quantum State Tomography

  • NQS tomography can learn state structure and generalize from incomplete data, while conventional inversion-based methods scale exponentially in system size.Full QST assumes identical copies and informationally complete measurements; these conditions often do not hold, motivating approximate schemes.Source: §4.1 · p. 29
  • NQS tomography has reconstructed finite-temperature Ising states and highly entangled states with more than a hundred qubits; training commonly minimizes KL divergence between measured and model distributions.Some work instead uses classical shadows to approximate infidelity.Source: §4.1 · p. 29
  • Pure-state reconstruction is often insufficient for experimental states, so mixed-state methods include purification, neural density operators, POVMs, and constraints such as positivity or symmetries.Noise layers can mitigate measurement-error effects in some settings, and filtering or density-matrix constraints can improve reconstruction performance.Source: §4.1 · p. 29, Figure 15 · p. 30

4.1.1. Latent Space Representations

  • A variational autoencoder compresses input into a latent bottleneck and decodes it to an output; after training, random latent inputs can generate new uncorrelated samples.For the variational form, the encoder supplies Gaussian means and variances, and latent values are sampled for the decoder.Source: §4.1.1 · p. 30, Figure 16 · p. 31
  • Latent representations have been used to reconstruct positive wave functions and ground states, extract features from finite-temperature samples, and measure local complexity via latent-space size.One reconstruction conditions the decoder on the magnetic field, while another quantifies compression by the ratio of network parameters to Hilbert-space dimension.Source: §4.1.1 · p. 31

4.2. Hybrid Training

  • Hybrid training reconstructs a state from existing numerical or experimental data before variational energy minimization, and reported examples improve ground-state energy results.For LiH, H2, and the lattice Schwinger model, measurements outside the computational basis are needed; reported hybrid results also allow complex-observable estimates such as entanglement entropy without additional quantum resources.Source: §4.2 · pp. 31–32

5. Summary and Outlook

NQS combine broad expressive power and efficient parameter compression, but optimization remains sensitive to architecture and training choices. The review highlights training advances, hybrid initialization, and generalization as approaches for addressing that challenge and extending NQS to phase-diagram exploration.

5. Summary and Outlook

  • NQS are variational wave functions represented by neural networks, with expressivity spanning frustrated spin, bosonic, and fermionic many-body states in one to three dimensions.The limits of this expressive power remain under study.Source: §5 · p. 32
  • NQS compress exponentially many wave-function coefficients into tractable network parameters, competitive with state-of-the-art numerical methods.Source: §5 · p. 32
  • Optimization remains challenging because the landscape has many local minima and depends sensitively on architecture, hyperparameters, and optimization strategies.Source: §5 · p. 32
  • Modified stochastic reconfiguration incorporates loss-landscape geometry, with some spin-system ground-state energies reaching numerical precision [87].Source: §5 · p. 32
  • Hybrid starts from experimental or numerical data, or tensor-network initialization, combine method advantages and are promising for addressing optimization [118].Source: §5 · p. 32
  • Generalization from training data supports exploring toolboxes for entire phase diagrams rather than individual states [98].Source: §5 · p. 32
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