This paper establishes the Quantum Voronovskaya-Damasclin (QVD) Theorem, a complete asymptotic expansion for Quantum Neural Network Operators (QNNOs) in the approximation of arbitrary quantum channels. This result generalizes the classical Voronovskaya theorem from scalar functions to the non-commutative, multi-dimensional framework of quantum information. By introducing rigorous quantum analogs of Sobolev and H\"older spaces (\ (C^m, (H) \) ) based on Fr\'echet derivatives in the Liouville representation and the completely bounded (diamond) norm, we provide a precise characterization of the approximation error. The expansion reveals a rich structure, explicitly isolating contributions from polynomial terms, fractional corrections arising from H\"older continuity, and non-commutative commutator effects inherent to the operator algebraic setting. For a channel in \ (C^m, (H) \), we derive an explicit bound for the remainder term in the diamond norm of order \ (O (n^- (m+) (n) ^3m/2) \), with a dimension-dependent constant. The theorem's power is demonstrated through several key applications: a quantum central limit theorem elucidating the Gaussian fluctuations of QNNOs, an optimal interpolation scheme for quantum channels using the Kubo--Ando geometric mean, and a quantum Richardson extrapolation method for accelerating convergence. This work provides a foundational bridge between classical approximation theory, operator algebras, and quantum information science, offering a rigorous framework for the asymptotic analysis of quantum machine learning models.
Rômulo Damasclin Santos (Tue,) studied this question.