Walshness: an intrinsic neural-network representability metric for quantum states

Abstract: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 and only if it has bounded Walshness 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 recovers 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.

Quantum Physics (quant-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn)

arXiv:2610.00505 [quant-ph]

(or arXiv:2610.00505v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.00505 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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