Neural Quantum States

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Sep 22, 2026quant-ph

Hyperbolic Restricted Boltzmann Machine Neural Quantum State

We construct the first type of non-Euclidean non-autoregressive neural quantum state (NQS) in the form of the hyperbolic Restricted Boltzmann Machine (HRBM), which is studied in the variational Monte-Carlo (VMC) setting of the Quantum Sherrington-Kirkpatrick (QSK) model whose ground state exhibits volume-law entanglement. Across a 512-fold increase in the Hilbert space dimension corresponding to a system size increase from N=14N=14 to N=24N=24, HRBM NQS robustly outperforms its Euclidean version, the RBM NQS, in terms of better ground state energy optimization as well as lower Renyi-2 S2S_2 and von Neumann SvNS_{vN} absolute entanglement entropy reconstruction errors. More importantly, for all tested QSK system sizes, HRBM NQS demonstrates a superior expressivity in faithfully reproducing the entire entanglement spectrum of the QSK model from the top eigenvalues down to the tail end across 15 orders of magnitude, while RBM NQS consistently overestimates the sub-dominant modes. This work furnishes a proof-of-concept demonstrating that hyperbolic non-autoregressive NQS ansatze, thanks to the exponential volume of the hyperbolic geometry underlying their constructions, might be more natural at representing volume-law quantum systems than conventional Euclidean NQS. Furthermore, an interesting byproduct of this work is the polynomial scaling result of RBM-type NQS ansatze in the QSK volume-law system as the Hilbert space dimension increases exponentially.

H. L. Dao

Neural Quantum StatesQuantum Many-Body PhysicsRestricted Boltzmann MachinesHyperbolic Representation Learning
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Sep 22, 2026quant-ph

When Quantum Meets AI: Quantum Methods for Machine Learning and Machine Learning Methods for Quantum Systems

This thesis studies the intersection of quantum computing and artificial intelligence in two directions: quantum methods for machine learning and machine learning methods for quantum systems. For quantum machine learning, Neural Quantum Embedding learns data representations that increase the trace distance between embedded class ensembles, lowering an embedding-dependent bound on empirical risk and improving classification on noisy quantum hardware. A training objective based on the Hilbert-Schmidt inner product extends this approach to deterministic quantum computation with one qubit (DQC1) and is demonstrated on an NMR quantum processor. A margin-based generalization analysis then connects quantum neural network performance to quantum state discrimination. In the studied benchmarks, margin distributions predict generalization more reliably than parameter-count metrics. For quantum systems, a Mamba-based neural decoder for surface codes matches a reproduced Transformer baseline in memory experiments while reducing inference-cost scaling from quartic to quadratic in code distance. Under an explicit decoder-induced-noise model, it achieves lower logical error rates and a higher effective threshold. For neural quantum states, stochastic reconfiguration is interpreted as tangent-space ridge regression, with its diagonal shift controlling the bias-variance trade-off under finite Monte Carlo sampling. Multi-shift stochastic reconfiguration reduces checkpoint-local validation residuals and update variance relative to fixed-shift SR, at additional computational cost. Together, these contributions show how learned representations, statistical control, and hardware constraints shape the exchange between quantum computing and machine learning.

Tak Hur

Department of Statistics and Data Science Graduate School Yonsei University

Neural Quantum StatesQuantum Error CorrectionQuantum Machine LearningQuantum Neural Networks
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Sep 7, 2026quant-ph

Topology Obstructs Pure Foundation Neural Quantum States

Foundation models for ground states in spin-1/2 systems are a promising method for problems ranging from quantum chemistry to identifying new phase diagrams. Nearly all such models are currently pure-states that condition on the Hamiltonian's parameters, whose Monte Carlo samples give energy estimates according to the variational principle. In this contribution, we show that this representation is topologically obstructed. For any gapped Hamiltonian family whose ground-state bundle is non-trivial, every continuous normalized state-vector model has zero fidelity with the ground state at some parameter value in the Hamiltonian family. For that value, the energy is at least one spectral gap, ΔΔ, with an O(Δ)O(Δ) gap in an open-neighbourhood of that point. We show that this is a sufficient no-go also in the case of degenerate ground-state manifolds, time dynamics, and periodic systems with mixed space-time topology, demonstrating these obstructions on one- and two-qubit systems. We discuss how this causes a spike in the fidelity susceptibility, giving a numerical signature of a phase-transition where there is none. We then show that operator-valued models canonically avoid these obstructions and preserve topological information, implying a structural necessity in representation for foundation neural quantum states.

Timothy Heightman, Elena Orlova, Philip Mantrov +1

1ICFO–Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain · 2Simulacra Research Inc., London, UK and Chicago, USA

Neural Quantum StatesQuantum Many-Body Physics
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Aug 12, 2026quant-ph

Hamilton-Zero: A Neural Tensor-Network Foundation Model for Ground States of Arbitrary Quadratic Qubit Hamiltonians

A central promise of useful quantum advantage is the ability to compute ground states of Hamiltonian systems beyond the reach of classical simulation methods. Here we demonstrate that this problem can be effectively amortized across an arbitrary and universal set of Hamiltonians by a foundation model with ∼0.5\sim0.5B variational parameters, trained with contemporary techniques from large language models and deep reinforcement learning. To do this, we formulate spin-1/2\text{spin-}1/2 quantum ground-state learning as manifold variational optimisation over centrally odd scalar functions on SU(2)N\mathrm{SU}(2)^N. This replaces explicit Hilbert-space vector amplitudes with manifold functions on which the Hamiltonian acts through Lie derivatives, evaluated by custom automatic differentiation primitives. We prove that the resulting variational principle on this manifold preserves the spin-1/2\text{spin-}1/2 sector's ground-state upper bound using the Peter-Weyl theorem, then pre-train our foundation model on a dataset of hundreds of thousands of different Hamiltonian systems, varying the connection topology, system size, interaction types and strengths, bringing together a century of many-body literature. Using a novel SU(2)\mathrm{SU}(2) replica-exchange Langevin sampler and sharded natural-gradient optimisation, we train our model with our own extension of the Kronecker-Factored Approximate Curvature (KFAC) optimiser on system sizes up to 64 qubits. On a held-out generalisation dataset, we fine-tune our model on system sizes of up to 1024 qubits, and evaluate on systems up to 8100 qubits.

Timothy Heightman, Elena Orlova, Philip Mantrov +1

1ICFO – Institut de Ci`encies Fot`oniques, The Barcelona Institute of Science and Technology · Castelldefels, Barcelona, Spain · 2Simulacra Research Inc., London, UK & Chicago, USA

Tensor NetworksNeural Quantum StatesQuantum Many-Body Physics
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Jul 21, 2026quant-ph

Enhanced Neural Quantum State via Annealed Gradient Descent

Neural quantum states offer expressive representations of quantum many-body wave functions, yet their practical accuracy can be limited by stochastic optimization rather than representational capacity. Here we identify a finite-sample instability, termed subspace trapping, in which physically important configurations become strongly underestimated, remain absent from successive sampling batches and receive insufficient gradient feedback. This self-reinforcing loss of sampled support can confine optimization to an effective subspace and produce apparently stationary states above the true ground state energy. To address this problem, we introduce annealed gradient descent (AGD), a sampling-aware update with annealing factor that temporarily increases the relative contribution of sampled low-probability configurations while limiting the dominance of high-probability ones. We establish the connection between finite-sample support loss and effective subspace optimization, and then evaluate the method across molecular systems, one and two-dimensional J1J_1-J2J_2 models. Annealed gradient descent suppresses metastable trapping, preserves physically relevant configurations and enables compact neural quantum states to attain chemical accuracy and competitive state-of-the-art performance. These results establish AGD as a lightweight complement to expressive neural architectures, improved sampling strategies for scalable quantum many-body optimization.

Shiwei Zhou, Yiming Huang, Xiao Yuan +1

Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China · China Center of Advanced Science and Technology, Beijing 100190, China · Center on Frontiers of Computing Studies, School of Computer Science, Peking University, Beijing 100871, China

Neural Quantum StatesGradient DescentQuantum Many-Body Physics
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Jul 2, 2026cs.LG

One More Time: Revisiting Neural Quantum States from a Reinforcement Learning Perspective

Neural quantum states (NQS) provide a flexible and scalable framework for approximating quantum many-body wavefunctions. Among NQS parameterizations, autoregressive models are especially attractive because they enable exact, independent sampling from the Born distribution, avoiding the autocorrelation and mixing issues of Markov chain methods. Yet their optimization remains comparatively underexplored: Adam is a scalable method but ignores function space geometry, while stochastic reconfiguration is principled but costly and numerically fragile in large models. To address this gap, we show that variational energy minimization can be viewed as an advantage policy-gradient problem over the Born distribution, motivating trust-region optimization for NQS training. We introduce Proximal Wavefunction Optimization (PWO), a principled trust-region algorithm that clips probability-ratio changes in the amplitude channel and phase increments in the phase channel. PWO avoids explicit matrix inversion, reuses samples across multiple updates, and combines the scalability of first-order optimization with theoretical guarantees. Across Ising and frustrated J1J_1-J2J_2 one- and two-dimensional spin systems, PWO improves stability and wall-clock convergence over Adam, minSR, and SPRING. Finally, we fine-tune a 1.51.5B-parameter RWKV-7 model, demonstrating NQS optimization at a scale over three orders of magnitude beyond prior work.

Juan Agustín Duque, Sergio García Heredia, Vinicius Hernandes +4

Mila Quebec AI Institute, Université de Montréal, Montréal, Canada · Applied Quantum Algorithms ⟨aQaL⟩, LIACS & LION, Leiden University, Leiden, Netherlands · QuTech and Kavli Institute of Nanoscience, Delft University of Technology, Delft, Netherlands +1

Neural Quantum StatesSecond-Order OptimizationPolicy Gradient MethodsTrust-Region OptimizationQuantum Many-Body Physics
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Jul 1, 2026quant-ph

Mechanistic Interpretability and Causal Feature Steering of Neural Quantum States via Sparse Autoencoders

Neural Quantum States (NQS) are a remarkably expressive class of variational ansätze for quantum many-body wavefunctions, yet little is understood about their internal mechanisms: trained on variational objectives alone, how do NQS accurately capture physical observables that they have never been explicitly optimized for? In this work, we present a systematic approach to analyze the internal activations of NQS using sparse autoencoders. We extract features from the residual stream and demonstrate that these features strongly correlate with physical observables such as order parameters, staggered magnetization, and half-chain correlators, across both ground state representation and real-time dynamics. Remarkably, the discovery of these features is entirely unsupervised, with no physical labels provided. We further establish that such features causally affect the corresponding observables predicted by NQS, by showing that targeted, post-training intervention on a \textit{single} feature smoothly and monotonically steers the corresponding observable, while leaving the variational energy nearly unchanged. These results demonstrate that NQS are not merely functional approximators, but encode rich, interpretable internal representations of physical information. Our approach provides both a diagnostic and an intervention tool for NQS, and serves as a foundation for using mechanistic interpretability towards more reliable, transparent NQS.

Zihao Qi, Christopher Earls

Department of Physics, Cornell University, Ithaca, NY 14853, USA. · Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA.

Neural Quantum StatesMechanistic InterpretabilityQuantum Many-Body Physics
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Jul 1, 2026cond-mat.str-el

Holographic Quantum Transformer: A Generalist Neuro-Symbolic Architecture for Solving Frustrated Systems via Generative Attention

Simulating two-dimensional frustrated quantum matter is a grand challenge due to the sign problem and exponential Hilbert space complexity. In this work, we introduce the Holographic Quantum Transformer (HQT), a physics-inspired generative architecture that leverages global self-attention to resolve non-local entanglement patterns. We validate HQT on the square lattice J1−J2J_1-J_2 Heisenberg model. On the heavily frustrated 8×88 \times 8 lattice at the quantum critical point (J2=0.5J_2=0.5), HQT reaches a ground-state energy per site (E/NE/N) of −0.5001(1)\mathbf{-0.5001(1)}, consistent with the expected finite-size scaling trend. Beyond numerical accuracy, HQT exhibits intrinsic physical awareness, autonomously recovering the underlying J2J_2 interaction geometry through interpretable attention maps. Our central contribution is ``Holographic Transfer", a zero-shot size-extrapolation protocol with rapid alignment: a model trained on 8×88 \times 8 systems is directly projected onto larger 10×1010 \times 10 lattices via continuous positional-embedding interpolation and head re-initialization, achieving high-fidelity initialization and rapid convergence. This zero-shot protocol yields an energy of E/N=−0.49782(3)E/N = \mathbf{-0.49782(3)}, statistically consistent with the variational state of the art while requiring no from-scratch training on the target lattice. Our results establish generative attention as a scalable paradigm for transferable quantum simulation.

Xingran Guo, Tiaojie Xiao, Jie Liu +1

National University of Defense Technology Changsha, China · State University of New York at New Paltz New Paltz, NY, USA

Neural Quantum StatesZero-Shot LearningTransformer AttentionQuantum Many-Body Physics
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Jun 24, 2026quant-ph

Two-dimensional Hyperbolic RNN Neural Quantum State

In the first part of this work, we construct the first type of two-dimensional (2D) hyperbolic neural quantum state (NQS) in the form of the Lorentz 2DRNN (Recurrent Neural Network) and benchmark its performance against the Euclidean 2DRNN in the paradigmatic N×NN\times N 2D Transverse Field Ising Model (2DTFIM) setting with different lattice sizes up to N=12N=12 and at different transverse magnetic field strengths. We find that hyperbolic Lorentz 2DRNN NQS definitively outperform Euclidean 2DRNN NQS when the system is at the phase transition point when the physics can be described by a conformal field theory (CFT), which is known to be dual to an Anti-de-Sitter (AdS) space whose spatial geometry is hyperbolic. In the second part of this work, we benchmark the performances of the recently introduced one-dimensional (1D) hyperbolic NQS including Poincaré RNN/GRU and Lorentz RNN/GRU against their Euclidean NQS versions in N×NN\times N 2DTFIM, which has to be converted to a one-dimensional setting to allow for the use of 1D NQS. The findings in this case extend our previous results that 1D hyperbolic NQS definitively outperform 1D Euclidean NQS, thanks to the combined effects of the hierarchical structure comprising the first and NthN^{th} neighbor interactions present in the 1D system arising from the 2D lattice and the CFT physics at the critical point. While more studies with larger system sizes are required, our work serves as a proof-of-concept for the utility, effectiveness as well as the superior performances of one- and two-dimensional hyperbolic NQS ansatzes compared to the existing Euclidean NQS in many-body quantum physics systems, especially when these systems exhibit structural hierarchy or when they are at criticality, or a combination of both.

H. L. Dao

Neural Quantum StatesRecurrent Neural NetworksQuantum Many-Body PhysicsHyperbolic Representation Learning
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Jun 11, 2026cond-mat.dis-nn

Low-variance estimators overcome the phase-gradient bottleneck in complex-valued neural quantum states

Complex neural quantum states are difficult to optimize when their wavefunction phase carries gauge, chiral, fermionic, or topological structure. We show that the major failure mode is not only ansatz expressivity, but the Monte Carlo estimator used to learn this phase. For separated amplitude-phase states, differentiating the local energy at fixed samples gives a different unbiased estimator of the same variational Monte Carlo phase force, without changing the objective. We further extend the construction to coupled two-head networks by keeping the amplitude-gradient contribution and applying the direct derivative only to the phase path. An adaptive minimum-variance mixture interpolates between standard and direct estimators during training. Across flux ladders, chiral chains, two-dimensional flux cylinders, an interacting fermion ladder, shared-network controls, and a fractional quantum Hall benchmark, the resulting estimators reduce phase-gradient variance, suppress seed failures, and often move multi-percent standard-gradient plateaus to sub-percent accuracy.

Yi-Ran Xue, Rui Wang, Baigeng Wang +1

*National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China. · Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA. · Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China. +3

Neural Quantum StatesComplex-Valued Neural NetworksQuantum Many-Body PhysicsVariance Reduction
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May 21, 2026cond-mat.str-el

Comment on "Spin-1/2 Kagome Heisenberg Antiferromagnet: Machine Learning Discovery of the Spinon Pair-Density-Wave Ground State"

A recent article [Phys. Rev. X 15, 011047 (2025)] utilizes group-equivariant convolutional neural networks to study the ground state of the kagome Heisenberg antiferromagnet. On the largest finite-size cluster studied to date (N=108N=108), the authors report variational energies significantly lower than other numerical methods, including state-of-the-art density matrix renormalization group (DMRG) calculations. In contrast to previous results suggesting a possible spin-liquid ground state, the authors observe a spinon pair-density-wave ground state. We find that: (i) the reported low energies are artifacts of broken ergodicity in the Metropolis--Hastings sampling, since the single-spin-flip update rule utilized by the authors effectively freezes the Markov chains; and (ii) when ergodic sampling is enforced via spin-exchange updates, the neural network converges to energies significantly higher than existing DMRG results, calling the paper's claims into question.

Helia Kamal, Dominik Kufel, DinhDuy Vu +2

Department of Physics, Harvard University, Cambridge, MA 02138, USA · Harvard Quantum Initiative, Cambridge, MA 02138, USA · Department of Physics, Boston University, Boston, MA 02215, USA

Neural Quantum StatesMarkov Chain Monte CarloQuantum Many-Body Physics
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May 13, 2026cond-mat.str-el

Parallel Scan Recurrent Neural Quantum States for Scalable Variational Monte Carlo

Neural-network quantum states have emerged as a powerful variational framework for quantum many-body systems, with recent progress often driven by massively parallel architectures such as transformers. Recurrent neural network quantum states, however, are frequently regarded as intrinsically sequential and therefore less scalable. Here we revisit this view by showing that modern recurrent architectures can support fast, accurate, and computationally accessible neural quantum state simulations. Using autoregressive recurrent wave functions together with recent advances in parallelizable recurrence, we develop variational ansätze, called parallel scan recurrent neural quantum states (PSR-NQS), which can be trained efficiently within variational Monte Carlo in one and two spatial dimensions. We demonstrate accurate benchmark results and show that, with iterative retraining, our approach reaches two-dimensional spin lattices as large as 52×5252\times52 while remaining in agreement with available quantum Monte Carlo data. Our results establish recurrent architectures as a practical and promising route toward scalable neural quantum state simulations with modest computational resources.

Ejaaz Merali, Mohamed Hibat-Allah, Mohammad Kohandel +2

Department of Physics and Astronomy, University of California, Davis, California 95616, USA · Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada · Vector Institute, Toronto, Ontario, M5G 0C6, Canada +1

Neural Quantum StatesRecurrent Neural NetworksQuantum Many-Body Physics
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Apr 28, 2026cond-mat.str-el

QERNEL: a Scalable Large Electron Model

We introduce QERNEL, a foundational neural wavefunction that variationally solves families of parameterized many-electron Hamiltonians and captures their ground states throughout parameter space within a single model. QERNEL combines FiLM-based parameter conditioning with scale-efficient architectural elements -- mixture of experts and grouped-query attention, substantially improving expressivity at low computational cost. We apply QERNEL to interacting electrons in semiconductor moiré heterobilayers, training a single weight-shared model for systems of up to 150 electrons. By solving the many-electron Schrödinger equation conditioned on moiré potential depth, QERNEL captures both quantum liquid and crystal states and discovers the sharp phase transition between them, marked by abrupt changes in interaction energy and charge density. Our work establishes a foundation model for moiré quantum materials and a scalable architecture toward a Large Electron Model for solids.

Khachatur Nazaryan, Liang Fu

Department of Physics, Massachusetts Institute of Technology, Cambridge, MA-02139,USA

Neural Quantum StatesMaterials ScienceQuantum Many-Body PhysicsQuantum Neural Networks
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Apr 27, 2026quant-ph

New non-Euclidean neural quantum states from additional types of hyperbolic recurrent neural networks

In this work, we extend the class of previously introduced non-Euclidean neural quantum states (NQS) which consists only of Poincare hyperbolic GRU, to new variants including Poincare RNN as well as Lorentz RNN and Lorentz GRU. In addition to constructing the new non-Euclidean hyperbolic NQS ansatzes, we generalize the results of our earlier work regarding the definitive outperformances delivered by hyperbolic Poincare GRU NQS when benchmarked against their Euclidean counterparts in the Variational Monte Carlo (VMC) experiments involving the Heisenberg J1J2J_1J_2 and J1J2J3J_1J_2J_3 models. Here, using larger systems consisting of 100 spins, we find that all four hyperbolic RNN/GRU NQS variants always outperform their respective Euclidean counterpart with the same architecture. In our experiments, among the four hyperbolic NQS, Lorentz RNN stands out in particular because despite having almost three times fewer parameters, it is capable of surpassing the more complex Poincare GRU and Lorentz GRU to emerge as the best overall hyperbolic NQS ansatz on many instances involving different J2J_2 and (J2,J3J_2, J_3) couplings. Given the findings from this work showing that the four newly constructed hyperbolic RNN/GRU NQS ansatzes are able to outperform the well-established Euclidean RNN/GRU NQS in Heisenberg spin models, we establish the utility and efficiency of the hyperbolic Poincare RNN/GRU and Lorentz RNN/GRU NQS for future variational studies of quantum many-body systems, especially those exhibiting a hierarchical structure in the form of the different degrees of nearest-neighbor interactions.

H. L. Dao

Neural Quantum StatesRecurrent Neural NetworksQuantum Many-Body Physics
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