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arXiv:2512.11632 · SciPost Physics Core

Basis dependence of Neural Quantum States for the Transverse Field Ising Model

Ronald Santiago Cortes, Aravindh S. Shankar, Marcello Dalmonte, Roberto Verdel, Nils Niggemann

quant-phcond-mat.stat-mechSubmitted Dec 12, 2025• Announced Dec 15, 2025• 0 citations
v1 · Dec 12, 2025v2 · Jan 13, 2026v3 · Mar 24, 2026
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Abstract

Neural Quantum States (NQS) are powerful tools used to represent complex quantum many-body states in an increasingly wide range of applications. However, despite their popularity, at present only a rudimentary understanding of their limitations exists. In this work, we investigate the dependence of NQS on the choice of the computational basis, focusing on restricted Boltzmann machines. Considering a family of rotated Hamiltonians corresponding to the paradigmatic transverse-field Ising model, we discuss the properties of ground states responsible for the dependence of NQS performance, namely the presence of ground state degeneracies as well as the uniformity of amplitudes and phases, carefully examining their interplay. We identify that the basis-dependence of the performance is linked to the convergence properties of a cluster or cumulant expansion of multi-spin operators -- providing a framework to directly connect physical, basis-dependent properties, to performance itself. Our results provide insights that may be used to gauge the applicability of NQS to new problems and to identify the optimal basis for numerical computations.

Journal reference
SciPost Phys. Core 9, 027 (2026)
DOI
10.21468/SciPostPhysCore.9.2.027
Comments
23 pages, 10 figures

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Neural Quantum States (NQS) are powerful tools used to represent complex quantum many-body states in an increasingly wide range of applications. However, despite their popularity, at present only a rudimentary understanding of their limitations exists. In this work, we investigate the dependence of NQS on the choice of the computational basis, focusing on restricted Boltzmann machines. Considering a family of rotated Hamiltonians corresponding to the paradigmatic transverse-field Ising model, we discuss the properties of ground states responsible for the dependence of NQS performance, namely the presence of ground state degeneracies as well as the uniformity of amplitudes and phases, carefully examining their interplay. We identify that the basis-dependence of the performance is linked to the convergence properties of a cluster or cumulant expansion of multi-spin operators -- providing a framework to directly connect physical, basis-dependent properties, to performance itself. Our results provide insights that may be used to gauge the applicability of NQS to new problems and to identify the optimal basis for numerical computations.
Neural quantum states are a new family of variational Ans\"atze for quantum-many body wave functions with advantageous properties in the notoriously challenging case of two spatial dimensions. Since their introduction, a wide variety of different network architectures have been employed to study paradigmatic models in quantum many-body physics with a particular focus on quantum spin models. Nonetheless, many questions remain about the effect that the choice of architecture has on the performance on a given task. In this work, we present a unified comparison of a selection of popular network architectures and symmetrization schemes employed for ground-state searches of prototypical spin Hamiltonians, namely, the two-dimensional transverse-field Ising model and the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model. In the presence of a nontrivial sign structure of the ground states, we find that the details of symmetrization crucially influence the performance. We describe this effect in detail and discuss its consequences, especially for autoregressive models, as their direct sampling procedure is not compatible with the symmetrization procedure that we found to be optimal.
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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.