多分类局域希尔伯特空间的高效表示：从非线性受限玻尔兹曼机到Kolmogorov-Arnold网络
Efficient Representation of multicategorical local Hilbert spaces: nonlinear Restricted Boltzmann Machines to Kolmogorov-Arnold Networks
AI总结：
本文提出将神经网络量子态的对数波函数作为原始多值自旋变量的非线性函数，以高效表示多分类局域希尔伯特空间，并在RBM和KAN上验证，该方法参数更少、可训练性更好，且能捕捉量子Potts模型的相变。
AI中文摘要：
用于表示自旋-1/2系统的神经网络量子态（NQS）通常由具有二元可见变量的多层感知器（MLP）构建，其中输入自旋首先通过线性仿射映射组合，然后应用更具表达力的非线性变换。表示多分类系统（即自旋-1和自旋-2，或具有两个以上局域自由度的q态量子Potts系统）的标准方法是采用一元或独热编码的MLP架构。遵循几项类似的研究，我观察到，虽然独热构造对于具有两个以上局域态的系统（如自旋S > 1/2的模型或q态Potts模型）在数学上是忠实的编码，但其参数数量随局域态数量增长，且由此产生的优化景观可能阻碍收敛。我在数值上发现，允许log Ψ作为原始多值自旋变量的非线性函数是一种自然的分类推广：它保留了局域基的标记自由度，并以严格更少的参数重现了独热模型，通常还改善了可训练性。我首先在浅层受限玻尔兹曼机（RBM）上对此想法进行基准测试，为其配备几种不同的非线性连接，用于自旋-1、2和3的海森堡链。然后转向Kolmogorov-Arnold网络（KAN），其中每条边携带一个可学习的单变量非线性，并表明它们为同一原理提供了严格更具表达力的实现。最后，我证明了该框架能够捕捉量子Potts哈密顿量的临界行为，恢复其相变。
英文摘要：
Neural network quantum states (NQS) for representing spin-$\frac{1}{2}$ systems are typically built from Multi-Layer Perceptrons (MLPs) with binary visible variables, in which the input spins are first combined through linear affine maps before more expressive nonlinear transformations are applied. The standard way for representing multi-categorical systems, namely spin-1 and spin-2, or the q-state quantum Potts systems with more than two local degrees of freedom, is a unary or one-hot encoded MLP architecture. Following a few similar studies, I observe that while the one-hot construction is the mathematically faithful encoding for systems with more than two local states such as spin-$S$ models with $S > 1/2$ or $q$-state Potts models, its parameter count grows with the number of local states, and the resulting optimization landscape can hinder convergence. I find numerically that allowing $\log Ψ$ to be a nonlinear function of the raw multi-valued spin variable is a natural categorical generalization: it preserves the labelling freedom of the local basis and reproduces the one-hot model with strictly fewer parameters, often with improved trainability. I first benchmark this idea on shallow Restricted Boltzmann Machines (RBMs), equipping them with several distinct nonlinear connections, for spin-1, 2, and 3 Heisenberg chains. I then turn to Kolmogorov-Arnold Networks (KANs), where each edge carries a learnable univariate nonlinearity, and show that they provide a strictly more expressive realization of the same principle. Finally, I demonstrate that this framework captures the critical behaviour of the quantum Potts Hamiltonian, recovering its phase transition.
