Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps

What the math gives to ML

The paper supplies a local coordinate principle for unitary-valued functions: near the identity, a unitary map can be represented through anti-Hermitian generators and scalar univariate functions, while noncommutativity leads naturally to an ordered product of matrix exponentials. This suggests a structured unitary neural layer whose dependence on a high-dimensional input is mediated by a small number of one-dimensional splines or small MLPs, rather than a dense unconstrained matrix-valued map. The transferable asset is the combination of local Lie-algebra coordinates, additive generator decomposition, and noncommutative factorization. A practical first test is to replace a dense input-conditioned unitary layer with a product of low-rank or fixed-generator exponentials whose scalar coefficients depend on individual input coordinates.