连续酉值映射的量子柯尔莫哥洛夫 - 阿诺德表示定理
Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps
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中文总结 AI 辅助
研究连续酉值映射的量子柯尔莫哥洛夫 - 阿诺德表示定理，通过反厄米值映射矩阵指数内的精确加法分解及酉映射的有限序列积等方法，给出两个定理并以反例表明局部定理不能全局扩展。
中文摘要 AI 辅助
经典柯尔莫哥洛夫 - 阿诺德表示定理启发了机器学习中相关网络发展。本文为单位矩阵\(1\)邻域内多变量连续酉值映射建立两个量子类似定理，一是反厄米值映射矩阵指数内的加法分解定理，二是酉映射的因式分解版本，还给出反例说明局部定理不能全局扩展。
英文摘要
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.
发表机构
- Moscow Institute of Physics and Technology（莫斯科物理技术研究所）
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