Asymptotic Expansions for Quantum Neural Network Operators: A Non-Commutative Voronovskaya Theorem

We establish a complete asymptotic expansion for Quantum Neural Network Operators (QNNOs) approximating arbitrary quantum channels, providing a non-commutative analogue of the classical Voronovskaya theorem. Within a rigorous functional analytic framework, we introduce quantum Sobolev and Hölder spaces Cm,γ(H) based on Fréchet differentiability in the Liouville representation, and we measure approximation errors using the diamond norm. Our main result, the Quantum Voronovskaya–Damasclin Theorem,