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arXiv:2309.11534 · Physical Review Letters

Comment on “Can Neural Quantum States Learn Volume-Law Ground States?”

Zakari Denis, Alessandro Sinibaldi, Giuseppe Carleo

Published Feb 18, 2025• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.134.079701

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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 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.
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.
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.
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.
Scaling laws, the power-law relations between loss, architecture size, and compute observed in modern neural networks, offer a quantitative way to characterize the complexity of a learning problem, with the exponent governing the decay of the loss reflecting how rapidly additional resources translate into improved accuracy, and thus how hard the target is to learn. Whether an analogous framework can characterize the complexity of physical problems remains open. We address this question for Neural-Network Quantum States, a leading variational approach for strongly correlated quantum many-body systems. Using transformer wave functions to approximate ground states of the J1J_1J1​-J2J_2J2​ Heisenberg model on triangular and square lattices with up to 20×2020\times 2020×20 sites, we find that the VVV-score, a measure of accuracy of a variational state, decays as a power law in training compute. Under an appropriate rescaling of compute, results for different system sizes collapse onto a single curve, analogous to scaling collapse in critical phenomena. The resulting power law is, to a good approximation, independent of the number of sites, showing that the transformer Ansatz is size-consistent for the systems considered. The exponent decreases systematically with frustration, identifying it as a quantitative measure of representational difficulty of the ground state and establishing scaling laws as a general framework for benchmarking variational ansätze.
Neural quantum states (NQS) attract a lot of attention due to their potential to serve as a very expressive variational ansatz for quantum many-body systems. Here we study the main factors governing the applicability of NQS to frustrated magnets by training neural networks to approximate ground states of several moderately-sized Hamiltonians using the corresponding wave function structure on a small subset of the Hilbert space basis as training dataset. We notice that generalization quality, i.e. the ability to learn from a limited number of samples and correctly approximate the target state on the rest of the space, drops abruptly when frustration is increased. We also show that learning the sign structure is considerably more difficult than learning amplitudes. Finally, we conclude that the main issue to be addressed at this stage, in order to use the method of NQS for simulating realistic models, is that of generalization rather than expressibility.