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On September 7, 2026, arXiv cs.LG published the paper “Quantum Kolmogorov–Arnold representation theorem for continuous unitary-valued maps”. This study established two quantum Kolmogorov–Arnold representation theorems for continuous unitary-valued mappings within the open neighborhood of the identity matrix $O_1(\mathbf{I})$. First, it proved the exact additive decomposition of anti-Hermitian-valued mappings in this neighborhood; second, considering the non-commutativity of quantum operators, it derived a finite sequence product form consisting of single-variable matrix exponents. Finally, based on the lifting properties of $\mathcal{SU}(2)$, specific topological counterexamples were provided, indicating that these local representation theorems cannot be extended to the entire unitary group $\mathcal{U}(n)$.
本文在单位矩阵 $O_1(\mathbf{I})$ 的开放邻域内，建立了连续酉值映射的两个量子柯尔莫哥洛夫 - 阿诺德表示定理。首先证明了该邻域内反厄米特值映射的精确加法分解；其次针对量子算符非交换性，导出了由单变量矩阵指数构成的有限序列乘积形式。最后基于 $\mathcal{SU}(2)$ 的提升性质提供了具体拓扑反例，表明这些局部表示定理无法扩展至整个酉群 $\mathcal{U}(n)$。