A Non-Commutative Voronovskaya Theorem for Quantum Neural Network Operators

We prove a complete asymptotic expansion for quantum neural network operators when they approximate arbitrary quantum channels. This is the non-commutative analogue of the classical Voronovskaya theorem. The expansion reveals that the approximation error splits into three fundamentally different parts: integer powers of \(1/n\) involving ordinary Fréchet derivatives; fractional powers governed by Marchaud fractional derivatives, which capture the Hölder smoothness of the channel; and purely quantum commutator terms that have no classical counterpart. The remainder is bounded sharply by an explicit constant:
\[
\norm{R_{m,n}(\Phi,\bullet)}_\diamond \le C_{m,\gamma,d} \|\Phi\|_{\cC^{m,\gamma}} \, n^{-(m+\gamma)} (\log n)^{3m/2}.
\]
We present a numerical test for a classical analogue that confirms the predicted convergence rate and the logarithmic correction, directly validating the asymptotic theory. Based on this expansion, we obtain three major advances: a quantum central limit theorem for the fluctuations of quantum neural network operators, a method to construct optimal interpolation geodesics between quantum channels via Kubo-Ando means, and a systematic understanding of how fractional smoothness limits the acceleration of quantum neural network approximations. The numerical test further demonstrates that the theoretical rates are sharp and that logarithmic enhancements are unavoidable. Altogether, our work builds a rigorous bridge between classical approximation theory, fractional calculus, and quantum machine learning, offering both theoretical insight and practical tools for designing and analyzing quantum neural networks in finite dimensions.