量子版柯尔莫哥洛夫-阿诺德定理，实现酉矩阵函数的精确分解
Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps
- 在单位矩阵邻域内，将多变量酉映射分解为单变量指数形式的加法组合
- 基于反对易矩阵指数的加法分解与非交换性导致的乘积分解两种形式
- 揭示了全局推广受限于拓扑障碍，适用于量子计算与量子神经网络研究者
经典柯尔莫哥洛夫-阿诺德表示定理指出，任意连续多元函数可精确分解为有限个一元连续函数与加法运算的复合。该结果近期启发了经典机器学习中的柯尔莫哥洛夫-阿诺德网络（KANs）及其量子扩展（QKANs）。本文在单位酉群 \/mathcal{U}(n) 中单位矩阵的开1-邻域 \(O_1(\mathbf{I})\) 内，建立了两个关于连续酉值多变量映射的量子版本表示定理。首先，证明了目标酉映射的矩阵指数可被精确表示为反对易值映射的加法组合；其次，由于量子算子不满足交换性，推导出以有限序列乘积形式表达目标酉映射的因式分解版本。最后，通过基于 \mathcal{SU}(2) 的提升性质构造具体拓扑反例，证明这些局部表示定理无法在不遭遇根本结构性障碍的情况下全局推广至整个酉群 \mathcal{U}(n)。
原文摘要 · Abstract (English)
The classical Kolmogorov--Arnold representation theorem states that any continuous multivariate function can be exactly decomposed into a finite composition of univariate continuous functions and addition operations. This foundational result has recently inspired the development of Kolmogorov--Arnold Networks (KANs) in classical machine learning, as well as their extensions into the quantum domain (QKANs). In this paper, we establish two quantum analogues of the Kolmogorov--Arnold representation theorem for continuous unitary-valued maps of several variables within an open $1$-neighbourhood of the identity matrix \(O_1(\mathbf{I}) \subset \mathcal{U}(n)\). First, we prove a representation theorem that yields an exact additive decomposition inside the matrix exponent of anti-Hermitian-valued maps. Second, due to the non-commutative nature of quantum operators, we derive a factorised version expressing the target unitary map as a finite sequential product of univariate matrix exponentials. Finally, we provide a concrete topological counterexample based on the lifting property of \(\mathcal{SU}(2)\) to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group \(\mathcal{U}(n)\) without encountering fundamental structural obstructions.
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