{"ID":5935804,"CreatedAt":"2026-07-07T01:22:02.77346169Z","UpdatedAt":"2026-07-07T02:10:06.972658124Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03187","arxiv_id":"2607.03187","title":"Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2607.03187","url_pdf":"https://arxiv.org/pdf/2607.03187v1","authors":"[\"Sviatoslav V. Dzhenzher\"]","published":"2026-07-03T10:46:32Z","proceeding":"quant-ph","tasks":"[\"quant-ph\",\"cs.LG\",\"math.FA\"]","methods":"[]","has_code":false}