Walshness：量子态的一种内禀神经网络可表示性度量

中文摘要 AI 辅助

神经量子态（NQS）已成为量子态的强大表示形式，其在量子多体物理中的应用正在迅速扩展。然而，我们对神经网络何时能高效表示物理量子态的理解仍然有限，部分原因在于NQS的非线性参数化、量子态复杂的符号结构以及对基选择的敏感依赖。我们引入了Walshness，一种量子态的复杂度度量，用于量化该量子态是否允许紧凑的神经网络表示。我们严格证明，对于多种NQS架构，一个一般的量子多体态当且仅当具有有界的Walshness时，才允许高效的NQS表示。我们提供了一种通过最小化相应经典自旋模型的能量来寻找最优基的算法。这使我们能够经验性地提高各种物理量子态的可学习性，包括横向场伊辛模型的基态和混合场环面码，通常将不保真度降低数个数量级，并恢复和推广了受挫反铁磁体中著名的Marshall符号规则所给出的基选择。我们的结果为量子态提供了一种内禀复杂度度量，并与其神经网络可表示性建立了可证明的联系。

英文摘要

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.

发表机构

• California Institute of Technology（加州理工学院）
• Tsinghua University（清华大学）

机构由 AI 辅助整理，请以论文原文为准。

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