AI 中文总结
研究Motzkin自旋链基态的非常规纠缠行为，通过因果前缀和模块等核心设计，利用四种主流神经网络架构为无色和彩色Motzkin态构建精确表示，展现神经网络捕捉非平凡纠缠特征的能力，提供基准示例与设计框架。
AI 中文摘要
Motzkin自旋链是典型的无挫折一维量子系统，其基态具有可精确求解的组合结构和违反面积律的奇异纠缠标度。无色Motzkin态表现出与系统大小\(N\)成临界对数纠缠发散\(\log N\)，而其彩色对应态具有超临界亚线性\(\sqrt{N}\)纠缠增长。这种非常规纠缠行为超出了标准矩阵乘积态的表达能力。本文系统地为无色和彩色Motzkin态构建了精确的、无需训练的神经网络表示，涵盖四种主流架构。核心设计利用因果前缀和模块及位置选择性整流线性门。对于彩色态，还引入了专用因果堆栈模块。结果表明神经网络能准确捕捉传统张量网络无法企及的非平凡纠缠特征，为基准测试提供原型示例及未来神经网络量子态发展的建设性设计框架。
英文摘要
Motzkin spin chains are paradigmatic frustration-free one-dimensional quantum systems whose ground states feature exactly solvable combinatorial structures and exotic, area-law-violating entanglement scaling. Specifically, colorless Motzkin states exhibit critical logarithmic entanglement divergence \(\log N\) with system size \(N\), while their colorful counterparts host supercritical sublinear \(\sqrt{N}\) entanglement growth. Such unconventional entanglement behaviors place these states well beyond the expressive capability of standard matrix product states, which are fundamentally constrained by the entanglement area law. Here, we systematically construct exact, training-free neural-network representations for both colorless and colorful Motzkin states across four mainstream architectures, including recurrent, feedforward, convolutional, and transformer networks. Our core design leverages a causal prefix-sum module, implementable via recurrent updates, feedforward mappings, or masked attention layers, combined with position-selective rectified linear gates that enforce the Motzkin height constraints. For the colorful states, we further introduce a dedicated causal stack module that explicitly encodes the last-in-first-out color-matching rule. Our results demonstrate that neural architectures can accurately capture highly non-trivial entanglement features inaccessible to conventional tensor networks, providing prototypic examples for benchmarking and a constructive design framework for future neural-network quantum state developments targeting strongly entangled quantum systems.
