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arXiv:2603.28646

Neural Quantum States in Non-Stabilizer Regimes: Benchmarks with Atomic Nuclei

James W. T. Keeble, Alessandro Lovato, Caroline E. P. Robin

nucl-thquant-phSubmitted Mar 30, 2026• Announced Mar 31, 2026• 0 citations
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Abstract

As neural networks are known to efficiently represent classes of tensor-network states as well as volume-law-entangled states, identifying which properties determine the representational capabilities of neural quantum states (NQS) remains an open question. We construct NQS representations of ground states of medium-mass atomic nuclei, which typically exhibit significant entanglement and non-stabilizerness, to study their performance in relation to the quantum complexity of the target state. Leveraging a second-quantized formulation of NQS tailored for nuclear-physics applications, we perform calculations in active orbital spaces using a restricted Boltzmann machine (RBM), a prototypical NQS ansatz. For a fixed number of configurations, we find that states with larger non-stabilizerness are systematically harder to learn, as evidenced by reduced accuracy. This finding suggests that non-stabilizerness is a primary factor governing the compression and representational efficiency of RBMs in entangled regimes, and motivates extending these studies to more sophisticated network architectures.

Comments
13 pages, 5 figures

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As neural networks are known to efficiently represent classes of tensor-network states as well as volume-law-entangled states, identifying which properties determine the representational capabilities of neural quantum states (NQS) remains an open question. We construct NQS representations of ground states of medium-mass atomic nuclei, which typically exhibit significant entanglement and non-stabilizerness, to study their performance in relation to the quantum complexity of the target state. Leveraging a second-quantized formulation of NQS tailored for nuclear-physics applications, we perform calculations in active orbital spaces using a restricted Boltzmann machine (RBM), a prototypical NQS ansatz. For a fixed number of configurations, we find that states with larger non-stabilizerness are systematically harder to learn, as evidenced by reduced accuracy. This finding suggests that non-stabilizerness is a primary factor governing the compression and representational efficiency of RBMs in entangled regimes, and motivates extending these studies to more sophisticated network architectures.
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Despite the huge theoretical potential of neural quantum states, their use in describing generic, highly-correlated quantum many-body systems still often poses practical difficulties. Customized network architectures are under active investigation to address these issues. For a guided search of suited network architectures a deepened understanding of the link between neural network properties and attributes of the physical system one is trying to describe, is imperative. Drawing inspiration from the field of machine learning, in this work we show how information propagation in deep neural networks impacts the physical entanglement properties of deep neural quantum states. In fact, we link a previously identified information propagation phase transition of a neural network to a similar transition of entanglement in neural quantum states. With this bridge we can identify optimal neural quantum state hyperparameter regimes for representing area as well as volume law entangled states. The former are easily accessed by alternative methods, such as tensor network representations, at least in low physical dimensions, while the latter are challenging to describe generally due to their extensive quantum entanglement. This advance of our understanding of network configurations for accurate quantum state representation helps to develop effective representations to deal with volume-law quantum states, and we apply these findings to describe the ground state (area law state) vs. the excited state (volume law state) properties of the prototypical next-nearest neighbor spin-1/2 Heisenberg model.
Modern deep learning has enabled unprecedented achievements in various domains. Nonetheless, employment of machine learning for wave function representations is focused on more traditional architectures such as restricted Boltzmann machines (RBMs) and fully connected neural networks. In this Letter, we establish that contemporary deep learning architectures, in the form of deep convolutional and recurrent networks, can efficiently represent highly entangled quantum systems. By constructing tensor network equivalents of these architectures, we identify an inherent reuse of information in the network operation as a key trait which distinguishes them from standard tensor network-based representations, and which enhances their entanglement capacity. Our results show that such architectures can support volume-law entanglement scaling, polynomially more efficiently than presently employed RBMs. Thus, beyond a quantification of the entanglement capacity of leading deep learning architectures, our analysis formally motivates a shift of trending neural-network-based wave function representations closer to the state-of-the-art in machine learning.
Abstract Neural-network quantum state (NQS) has attracted significant interests as a powerful wave-function ansatz to model quantum phenomena. In particular, a variant of NQS based on the restricted Boltzmann machine (RBM) has been adapted to model the ground state of spin lattices and the electronic structures of small molecules in quantum devices. Despite these progresses, significant challenges remain with the RBM-NQS-based quantum simulations. In this work, we present a state-preparation protocol to generate a specific set of complex-valued RBM-NQS, which we name the unitary-coupled RBM-NQS, in quantum circuits. Our proposal expands the applicability of NQS as prior works deal exclusively with real-valued RBM-NQS for quantum algorithms. With this scheme, we achieve (1) modeling complex-valued wave functions, (2) using as few as one ancilla qubit to simulate M hidden spins in an RBM architecture, and (3) avoiding post-selections to improve scalability.
Abstract Neural quantum state (NQS) ansätze have shown promise in variational Monte Carlo algorithms by their theoretical capability of representing any quantum state. However, the reason behind the practical improvement in their performance with an increase in the number of parameters is not fully understood. In this work, we systematically study the efficiency of restricted Boltzmann Machines (RBMs) to represent the ground states in different phases of the spin-1 bilinear-biquadratic model, as the hidden layer density α increases. We train our ansatz by minimizing two different loss functions: 1) energy, and 2) infidelity of the NQS ansatz w.r.t. that of the exact ground state. We observe that the accuracy of our ansatz saturates with α in both cases. We demonstrate that this can be explained by looking at the spectrum of the quantum geometric tensor (QGT). We find that the rank of the QGT saturates beyond a certain α, and we emphasize that it corresponds to the dimension of the relevant manifold for an optimized NQS. This provides a useful diagnostics for the practical representation power of an NQS ansatz.