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

arXiv: https://arxiv.org/abs/2607.03187 · HTML: https://mlsanity.com/paper/2607.03187/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 43 references that are in the map.
- First posted on arXiv 3 Jul 2026.

## Builds on (most cited first, 11 of 11)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [Convolutional Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.13155/) (2024)
- [Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2509.14026/) (2025)
- [Addressing common misinterpretations of KART and UAT in neural network literature](https://mlsanity.com/paper/2408.16389/) (2024)
- [Physics-informed time series analysis with Kolmogorov-Arnold Networks under Ehrenfest constraints](https://mlsanity.com/paper/2509.18483/) (2025)
- [Kolmogorov--Arnold stability](https://mlsanity.com/paper/2504.05255/) (2025)
- [QuKAN: A Quantum Circuit Born Machine approach to Quantum Kolmogorov Arnold Networks](https://mlsanity.com/paper/2506.22340/) (2025)
- [Probing Quantum Spin Systems with Kolmogorov-Arnold Neural Network Quantum States](https://mlsanity.com/paper/2506.01891/) (2025)

## Abstract (arXiv)

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) subset 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 (SU(2)) to demonstrate that these local representation theorems cannot be globally extended to the entire unitary group (U(n)) without encountering fundamental structural obstructions.