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

Rômulo Damasclin Chaves dos Santos

Published May 21, 2026• 0 citations

Abstract

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, reveals a multiscale decomposition of the error into three distinct contributions: integer-order terms involving Fréchet derivatives and even kernel moments, fractional corrections governed by Marchaud fractional derivatives that capture Hölder regularity of order γ, and intrinsically non-commutative commutator terms that vanish in classical settings. The remainder is sharply bounded by O(n−(m+γ)(log n)3m/2) with an explicit constant depending on m, γ, and the Hilbert space dimension d. As applications, we derive a quantum central limit theorem for QNNO fluctuations, construct optimal interpolation geodesics between quantum channels using Kubo–Ando means, and develop a quantum Richardson extrapolation method that reveals fundamental acceleration limits imposed by fractional smoothness. Our 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 in finite-dimensional settings.

DOI: 10.20944/preprints202605.1369.v1

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

Rômulo Damasclin Chaves dos Santos

Published May 21, 2026• 0 citations

Publisher

Abstract

DOI: 10.20944/preprints202605.1369.v1

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\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.

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: \[ \norm{R_{m,n}(Φ,\bullet)}_\diamond \le C_{m,γ,d} \|Φ\|_{\cC^{m,γ}} \, n^{-(m+γ)} (\log n)^{3m/2}. \] 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.

\mathcal{C}^{m,\gamma}(\mathcal{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

\mathcal{C}^{m,\gamma}(\mathcal{H})

, we derive an explicit bound for the remainder term in the diamond norm of order

O(n^{-(m+\gamma)}(\log 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.

L^2

L^p

C^0

, and Sobolev

H^k

, along with a classical-quantum comparative analysis (as Supplemental Materials).