Quantum Bernstein--Kolmogorov--Arnold approximation

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.

The paper asks whether function approximation from copies of coordinate-encoded qubits admits a Kolmogorov–Arnold representation with target-independent inner observables. Its input is $n$ copies of each of $d$ coordinate states, with all dependence on $x\in[0,1]^d$ confined to those states; the subsequent unitary depends on the target $f:[0,1]^d\to[0,1]$ and $n$, but not on $x$. For $d>1$, it constructs $2d+1$ commuting Hermitian observables $H_q^{(n)}$ and a single target-dependent scalar function $g$, independent of $n$, such that $U_n=\prod_{q=0}^{2d}\exp[-i g(H_q^{(n)})\otimes Y]$ produces an ancilla measurement probability exactly equal to the multivariate Bernstein polynomial $B_nf(x)$. When $f$ is continuous, $g$ can be chosen continuous, and the full output converges uniformly in norm to the original input tensored with $\sqrt{1-f(x)}|0\rangle+\sqrt{f(x)}|1\rangle$. An exact squared-norm error formula is given in terms of Bernstein approximations to $\sqrt f$ and $\sqrt{1-f}$. The construction applies a classical Sprecher-type superposition theorem to $\arcsin\sqrt f$, substitutes commuting normalized zero-count operators for the coordinates, and uses their binomial statistics to establish the probability and coherent-convergence statements. The theorem also gives exact Bernstein probabilities for discontinuous targets, but does not guarantee approximation in that case. No elementary-gate complexity bound is asserted. The discussion describes compression to $d\lceil\log_2(n+1)\rceil$ qubits using the symmetric-subspace structure and notes that Lipschitz regularity permits quantitative convergence estimates.