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 O1(I) โ ๐ฐ(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 ๐ฎ๐ฐ(2) to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group ๐ฐ(n) without encountering fundamental structural obstructions.
Related
- Algebraic Kolmogorov--Arnold representation theorem for quantum measurement
- Modular Heights of Unitary Shimura Varieties III: Proof of the Main Theorem
- The Lq-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths
- A compactness theorem for twisted-unitary elements of integral group rings, with a certified route to the theta-unitary Case A at window B(4) of the Promislow group
- A Representation Theorem for Fractional Viscoelastic Networks
BibTeX
@misc{dzhenzher2026quantum,
title = {{Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps}},
author = {Sviatoslav V. Dzhenzher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.03187},
eprint = {2607.03187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.03187},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Quantum Kolmogorov--Arnold representation theorem for continuous unitary-valued maps
AU - Sviatoslav V. Dzhenzher
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2607.03187
UR - https://arxiv.org/abs/2607.03187
AB - 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(I) โ ๐ฐ(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 ๐ฎ๐ฐ(2) to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group ๐ฐ(n) without encountering fundamental structural obstructions.
ER -