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This theoretical paper proposes a quantum analogue of the Kolmogorov-Arnold representation theorem to address expressivity limits in quantum neural networks. It is a proof-based framework, not an experimental result, and its claims that the measurement probabilities reproduce multivariate Bernstein polynomials and converge uniformly to target functions still need scrutiny of the construction and its scaling. For quantum ML readers, it is a formal link between classical approximation theory and quantum circuit design.
Quantum Bernstein--Kolmogorov--Arnold approximation
arXiv:2610.41608v1 Announce Type: new Abstract: This paper addresses the expressivity and representation gap in Quantum Neural Networks by establishing a quantum analogue of the classical Kolmogorov--Arnold representation theorem. We propose a constructive approximation framework over fixed input coherent states, where the dependence of the function argument is embedded entirely in coordinate qubits, while the target function dictates a single outer unitary evolution. By applying a Sprecher-type representation to the rotation angle of the state, we demonstrate that the multidimensional operator factorises into \(2d+1\) commuting universal inner observables. We prove that the resulting measurement probability precisely reproduces the multidimensional Bernstein polynomial of the target function. For any continuous multivariate function, the joint quantum state converges uniformly to the target output register, linking classical superposition theory with probabilistic and coherent quantum circuits.
Abstract:This paper addresses the expressivity and representation gap in Quantum Neural Networks by establishing a quantum analogue of the classical Kolmogorov--Arnold representation theorem. We propose a constructive approximation framework over fixed input coherent states, where the dependence of the function argument is embedded entirely in coordinate qubits, while the target function dictates a single outer unitary evolution. By applying a Sprecher-type representation to the rotation angle of the state, we demonstrate that the multidimensional operator factorises into \(2d+1\) commuting universal inner observables. We prove that the resulting measurement probability precisely reproduces the multidimensional Bernstein polynomial of the target function. For any continuous multivariate function, the joint quantum state converges uniformly to the target output register, linking classical superposition theory with probabilistic and coherent quantum circuits.
Quantum Physics (quant-ph); Functional Analysis (math.FA)
arXiv:2610.41608 [quant-ph]
(or arXiv:2610.41608v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.41608 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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