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arXiv:2604.03294

Expressibility of neural quantum states: a Walsh-complexity perspective

Taige Wang

cond-mat.str-elcond-mat.dis-nncs.LGquant-phSubmitted Mar 27, 2026• Announced Mar 31, 2026• 0 citations
v1 · Mar 27, 2026v2 · Apr 7, 2026
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Abstract

Neural quantum states are powerful variational wavefunctions, but it remains unclear which many-body states can be represented efficiently by modern additive architectures. We introduce Walsh complexity, a basis-dependent measure of how broadly a wavefunction is spread over parity patterns. States with an almost uniform Walsh spectrum require exponentially large Walsh complexity from any good approximant. We show that shallow additive feed-forward networks cannot generate such complexity in the tame regime, e.g. polynomial activations with subexponential parameter scaling. As a concrete example, we construct a simple dimerized state prepared by a single layer of disjoint controlled-$Z$ gates. Although it has only short-range entanglement and a simple tensor-network description, its Walsh complexity is maximal. Full-cube fits across system size and depth are consistent with the complexity bound: for polynomial activations, successful fitting appears only once depth reaches a logarithmic scale in $N$, whereas activation saturation in $\tanh$ produces a sharp threshold-like jump already at depth $3$. Walsh complexity therefore provides an expressibility axis complementary to entanglement and clarifies when depth becomes an essential resource for additive neural quantum states.

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8 pages, 2 figures

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Neural quantum states are powerful variational wavefunctions, but it remains unclear which many-body states can be represented efficiently by modern additive architectures. We introduce Walsh complexity, a basis-dependent measure of how broadly a wavefunction is spread over parity patterns. States with an almost uniform Walsh spectrum require exponentially large Walsh complexity from any good approximant. We show that shallow additive feed-forward networks cannot generate such complexity in the tame regime, e.g. polynomial activations with subexponential parameter scaling. As a concrete example, we construct a simple dimerized state prepared by a single layer of disjoint controlled-ZZZ gates. Although it has only short-range entanglement and a simple tensor-network description, its Walsh complexity is maximal. Full-cube fits across system size and depth are consistent with the complexity bound: for polynomial activations, successful fitting appears only once depth reaches a logarithmic scale in NNN, whereas activation saturation in tanh⁡\tanhtanh produces a sharp threshold-like jump already at depth 333. Walsh complexity therefore provides an expressibility axis complementary to entanglement and clarifies when depth becomes an essential resource for additive neural quantum states.
Neural quantum states (NQS) have emerged as powerful representations of quantum states with rapidly expanding applications across quantum many-body physics. Yet our understanding of when neural networks can efficiently represent physical quantum states remains limited, in part due to the nonlinear parameterization of NQS, the intricate sign structure of quantum states, and the sensitive dependence on basis choice. We introduce Walshness, a complexity metric of a quantum state which quantifies whether it admits a compact neural-network representation. We rigorously prove that a general quantum many-body state admits an efficient NQS representation if it has bounded Walshness, and vice versa, for various NQS architectures. We provide an algorithm for finding the optimal basis by minimizing the energy of a corresponding classical spin model. This allows us to empirically improve the learnability of various physical quantum states, including ground states of the transverse-field Ising model and mixed-field toric code, often reducing infidelity by orders of magnitude, and generalizes the basis choice given by the well-known Marshall sign rule in frustrated antiferromagnets. Our results provide an intrinsic complexity metric for quantum states with provable connections to their neural-network representability.
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Part of the challenge for quantum many-body problems comes from the difficulty of representing large-scale quantum states, which in general requires an exponentially large number of parameters. Neural networks provide a powerful tool to represent quantum many-body states. An important open question is what characterizes the representational power of deep and shallow neural networks, which is of fundamental interest due to the popularity of deep learning methods. Here, we give a proof that, assuming a widely believed computational complexity conjecture, a deep neural network can efficiently represent most physical states, including the ground states of many-body Hamiltonians and states generated by quantum dynamics, while a shallow network representation with a restricted Boltzmann machine cannot efficiently represent some of those states.One of the challenges in studies of quantum many-body physics is finding an efficient way to record the large system wavefunctions. Here the authors present an analysis of the capabilities of recently-proposed neural network representations for storing physically accessible quantum states.
An approximate quantum state preparation method is introduced, called the Walsh series loader (WSL). The WSL approximates quantum states defined by real-value functions of single real variables with a depth independent of the number nnn of qubits. Two approaches are presented. The first one approximates the target quantum state by a Walsh series truncated at $O(1/\sqrt{\ensuremath{\epsilon}})$, where $\ensuremath{\epsilon}$ is the precision of the approximation in terms of infidelity. The circuit depth is also $O(1/\sqrt{\ensuremath{\epsilon}})$, the size is $O(n+1/\sqrt{\ensuremath{\epsilon}})$, and only one ancilla qubit is needed. The second method accurately represents quantum states with sparse Walsh series. The WSL loads sss-sparse Walsh series into nnn qubits with a depth doubly sparse in sss and kkk, the maximum number of bits with value 1 in the binary decomposition of the Walsh function indices. The associated quantum circuit approximates the sparse Walsh series up to an error $\ensuremath{\epsilon}$ with a depth O(sk)O(sk)O(sk), a size O(n+sk)O(n+sk)O(n+sk), and one ancilla qubit. In both cases, the protocol is a repeat-until-success procedure with a probability of success $P=\mathrm{\ensuremath{\Theta}}(\ensuremath{\epsilon})$, giving an averaged total time of $O(1/{\ensuremath{\epsilon}}^{3/2})$ for the WSL and $O(sk/\ensuremath{\epsilon})$ for the sparse WSL. Amplitude amplification can be used to reach a probability of success $P=\mathrm{\ensuremath{\Theta}}(1)$, modifying the quantum circuit size to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}((n+1/\sqrt{\ensuremath{\epsilon}})/\sqrt{\ensuremath{\epsilon}})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}((n+sk)/\sqrt{\ensuremath{\epsilon}})$ and the depth to $O([log{(n)}^{3}+1/\sqrt{\ensuremath{\epsilon}}]/\sqrt{\ensuremath{\epsilon}})$ and $O([log{(n)}^{3}+sk]/\sqrt{\ensuremath{\epsilon}})$, respectively. Amplitude amplification reduces by a factor $O(1/\sqrt{\ensuremath{\epsilon}})$ the total time dependence on $\ensuremath{\epsilon}$ but increases the size and depth of the associated quantum circuits, making them linearly dependent on nnn. These protocols give overall efficient algorithms with no exponential scaling in any parameter. They can be generalized to any complex-value, multivariate, almost-everywhere-differentiable function. The repeat-until-success Walsh series loader is so far the only method that prepares a quantum state with a circuit depth and an averaged total time independent of the number of qubits.
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.