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 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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BibTeX
@misc{paul2026walshness,
title = {{Walshness: an intrinsic neural-network representability metric for quantum states}},
author = {Nisarga Paul and Kehuang Chen and Jessica K. Jiang and Haimeng Zhao and Di Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00505},
eprint = {2610.00505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00505},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Walshness: an intrinsic neural-network representability metric for quantum states
AU - Nisarga Paul
AU - Kehuang Chen
AU - Jessica K. Jiang
AU - Haimeng Zhao
AU - Di Luo
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.00505
UR - https://arxiv.org/abs/2610.00505
AB - 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.
ER -