A Quantum Voronovskaya Theorem: Asymptotic Expansions for Quantum Neural Network Operators

We prove the Quantum Voronovskaya--Damasclin Theorem, establishing the first complete asymptotic expansion for Quantum Neural Network Operators (QNNOs) when approximating arbitrary quantum channels. This result generalizes the classical Voronovskaya theorem for Bernstein polynomials to the non‑commutative, multi‑dimensional setting of quantum information theory. Within a rigorous functional analytic framework, we introduce quantum analogues of Sobolev and Hölder spaces $\mathcal{C}^{m,\gamma}(\mathcal{H})$ based on Fréchet differentiability in the Liouville representation, and we measure approximation errors using the diamond (completely bounded trace) norm. Our expansion, \[ \Psi_n(\Phi)(\rho) = \Phi(\rho) + \sum_{j=1}^{m} \frac{a_j(\Phi,\rho)}{n^j} + \sum_{j=1}^{\lfloor m/2\rfloor} \frac{b_j(\Phi,\rho)}{n^{j+\gamma}} + \sum_{j=1}^{\lfloor m/3\rfloor} \frac{c_j(\Phi,\rho)}{n^{j+2\gamma}} + O\!\left(n^{-(m+\gamma)}(\log n)^{3m/2}\right), \]explicitly separates three distinct contributions: integer‑order polynomial terms involving Fréchet derivatives and even kernel moments; fractional corrections governed by Marchaud fractional derivatives that capture Hölder regularity of order $\gamma$; and intrinsically non‑commutative commutator terms that vanish in classical settings. The remainder estimate is sharp, with an explicit constant depending on $m$, $\gamma$, and the Hilbert space dimension $d$. As applications, we derive a quantum central limit theorem for QNNOs, construct optimal interpolation geodesics between channels using Kubo--Ando means, and develop a quantum Richardson extrapolation method that reveals fundamental acceleration limits imposed by fractional smoothness. These results establish a rigorous bridge between classical approximation theory, fractional calculus, and quantum machine learning, providing a powerful tool for the design and analysis of quantum neural networks.