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

2026-07-03

AI Summary

This work extends the classical Kolmogorov–Arnold representation theorem into the quantum domain by investigating the structure of multivariate continuous unitary-valued mappings in a neighborhood of the identity matrix. By integrating local parametrizations of Lie groups, the matrix exponential map, and analysis of skew-Hermitian operators, the paper introduces two novel local quantum Kolmogorov–Arnold representations: one based on an additive decomposition within the exponent of skew-Hermitian maps, and another constructed as a sequential product of finitely many single-variable matrix exponentials leveraging non-commutativity. The study rigorously establishes the exact forms of both representations and demonstrates, via a topological counterexample on SU(2), that they cannot be globally extended—thereby highlighting the inherent optimality and limitations of such local constructions.

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.

Problem

• Kolmogorov–Arnold representation
• unitary-valued maps
• quantum analogues
• continuous functions
• matrix exponentials

Innovation

• Quantum Kolmogorov–Arnold theorem
• unitary-valued maps
• matrix exponential decomposition
• non-commutative factorization
• topological obstruction