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arXiv:2510.11797 · Physical Review Letters

Bound on Entanglement in Neural Quantum States

Nisarga Paul

Published Mar 10, 2026• 1 citation
arXivHTMLPDF

Abstract

Variational wave functions offer a practical route around the exponential complexity of many-body Hilbert spaces, but their expressive power is often sharply constrained. Matrix product states, for instance, are efficient but limited to area law entangled states. Neural quantum states (NQS) are widely believed to overcome such limitations, yet little is known about their fundamental constraints. Here we prove that feed-forward neural quantum states acting on n spins with k scalar nonlinearities, under certain analyticity assumptions, obey a bound on entanglement entropy for any subregion: S≤cklog n, for a constant c. This establishes an NQS analog of the area law constraint for matrix product states and rules out volume law entanglement for NQS with O(1) nonlinearities. We demonstrate analytically and numerically that the scaling with n is tight for a wide variety of NQS. Our Letter establishes a fundamental constraint on NQS that applies broadly across different network designs, while reinforcing their substantial expressive power.

DOI
10.1103/rpj5-cns6

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Abstract A study of the artificial neural network representation of quantum many-body states is presented. The locality and entanglement properties of states for shallow and deep quantum neural networks are investigated in detail. By introducing the notion of local quasi-product states, for which the locally connected shallow feed-forward neural network states and restricted Boltzmann machine states are special cases, we show that Rényi entanglement entropies of all these states obey the entanglement area law. Besides, we also investigate the entanglement features of deep Boltzmann machine states and show that locality constraints imposed on the neural networks make the states obey the entanglement area law. Finally, as an application, we apply the notion of Rényi entanglement entropy to understand the power of neural networks, and show that image classification problems can be efficiently solved must obey the area law.
Passetti et al. [Physical Review Letters 131, 036502 (2023)] recently assessed the potential of neural quantum states (NQS) in learning ground-state wave functions with volume-law entanglement scaling. They focused on NQS using feedforward neural networks, specifically applied to the complex SYK Hamiltonian for fermions. Their numerical results hint at an exponential increase in the required variational parameters as the system size grows, apparently tied to the entanglement growth within the SYK ground state. This challenges the general utility of NQS for highly entangled wavefunctions, contrasting with established analytical and numerical findings. Based on our experiments, we show that suitably chosen NQS can learn ground states with volume-law entanglement both for spin and fermionic problems. We argue that the setup utilized in the aforementioned letter reveals the inefficiency of non-fermionic NQS to learn fermionic states, rather than a general connection between entanglement content and learnability hardness.
We establish a direct connection between general tensor networks and deep feed-forward artificial neural networks. The core of our results is the construction of neural-network layers that efficiently perform tensor contractions and that use commonly adopted nonlinear activation functions. The resulting deep networks feature a number of edges that closely match the contraction complexity of the tensor networks to be approximated. In the context of many-body quantum states, this result establishes that neural-network states have strictly the same or higher expressive power than practically usable variational tensor networks. As an example, we show that all matrix product states can be efficiently written as neural-network states with a number of edges polynomial in the bond dimension and depth that is logarithmic in the system size. The opposite instead does not hold true, and our results imply that there exist quantum states that are not efficiently expressible in terms of matrix product states or projected entangled pair states but that are instead efficiently expressible with neural network states.
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We study whether neural quantum states based on multilayer feed-forward networks can find ground states which exhibit volume-law entanglement entropy. As a testbed, we employ the paradigmatic Sachdev-Ye-Kitaev model. We find that both shallow and deep feed-forward networks require an exponential number of parameters in order to represent the ground state of this model. This demonstrates that sufficiently complicated quantum states, although being physical solutions to relevant models and not pathological cases, can still be difficult to learn to the point of intractability at larger system sizes. Hence, the variational neural network approach offers no benefits over exact diagonalization methods in this case. This highlights the importance of further investigations into the physical properties of quantum states amenable to an efficient neural representation.
Neural quantum states (NQS) have emerged as a powerful variational ansatz for representing quantum many-body wave functions. Their internal mechanisms, however, remain poorly understood. We investigate the role of correlations in NQS-like quantum state representation by employing a correlation-based interpretable neural network architecture and then proving our observations using Boolean function theory. The correlator neural network demonstrates that, even for simple product states, correlations up to the systemsize in the chosen computational basis are required to represent a quantum state faithfully. We explain these observations using Fourier expansion, which reveals the correlator basis as the effective basis of the internal NQS structure, the resulting necessity for high-order correlations as supported by an entanglement bound that scales with the correlation order, the consequences of linear dependencies in constrained Hilbert spaces for correlation requirements, and the connections between spin basis rotations and the correlator basis. Furthermore, we analyze how neural networks achieve high correlation orders by increasing the magnitude of the network weights, which can be compensated by increasing the network depth. Lastly, we discuss how activation functions, network architectures, and the choice of reference basis influence correlation requirements. Our results provide insights and a better understanding of the internal structure and requirements of NQS, enabling a more systematic use of NQS in future research.