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

quant-pharXiv:2607.11907Jul 2, 20260 citations

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

Rômulo Damasclin Chaves dos Santos

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.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.

May 21, 20260 citations

Asymptotic Expansions for Neural Network Approximations of Quantum Channels

Rômulo Damasclin Chaves dos Santos

This paper establishes the Quantum Voronovskaya--Damasclin (QVD) Theorem, providing a complete asymptotic characterization of Quantum Neural Network Operators in the approximation of arbitrary quantum channels. The result extends the classical Voronovskaya theorem from scalar approximation to the non-commutative operator framework of quantum information theory. We introduce rigorous quantum analogues of Sobolev and Hölder spaces defined through Fréchet differentiability in the Liouville representation and measured using the completely bounded (diamond) norm. Within this framework, we derive an explicit asymptotic expansion of the approximation error and identify the fundamental mechanisms governing convergence. The expansion separates integer-order differential contributions, fractional corrections associated with limited regularity, and intrinsically non-commutative effects arising from operator algebra structure. We also establish a sharp remainder estimate with explicit dependence on the regularity of the channel and the dimension of the underlying Hilbert space. Several applications demonstrate the scope of the theory. These include a quantum central limit theorem describing the fluctuation regime of quantum neural network operators, an optimal interpolation method based on operator geometric means, and a convergence acceleration procedure inspired by Richardson extrapolation. The results provide a rigorous mathematical foundation for the asymptotic analysis of quantum neural network models and establish a direct connection between classical approximation theory, operator algebras, and quantum information science, with implications for quantum algorithms and quantum machine learning.This paper establishes the Quantum Voronovskaya--Damasclin (QVD) Theorem, providing a complete asymptotic characterization of Quantum Neural Network Operators in the approximation of arbitrary quantum channels. The result extends the classical Voronovskaya theorem from scalar approximation to the non-commutative operator framework of quantum information theory. We introduce rigorous quantum analogues of Sobolev and Hölder spaces defined through Fréchet differentiability in the Liouville representation and measured using the completely bounded (diamond) norm. Within this framework, we derive an explicit asymptotic expansion of the approximation error and identify the fundamental mechanisms governing convergence. The expansion separates integer-order differential contributions, fractional corrections associated with limited regularity, and intrinsically non-commutative effects arising from operator algebra structure. We also establish a sharp remainder estimate with explicit dependence on the regularity of the channel and the dimension of the underlying Hilbert space. Several applications demonstrate the scope of the theory. These include a quantum central limit theorem describing the fluctuation regime of quantum neural network operators, an optimal interpolation method based on operator geometric means, and a convergence acceleration procedure inspired by Richardson extrapolation. The results provide a rigorous mathematical foundation for the asymptotic analysis of quantum neural network models and establish a direct connection between classical approximation theory, operator algebras, and quantum information science, with implications for quantum algorithms and quantum machine learning.

quant-pharXiv:2603.18033Mar 11, 20260 citations

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

Rômulo Damasclin Chaves dos Santos

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.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.

May 16, 20260 citations

Asymptotic Expansions for Neural Network Approximations of Quantum Channels

Rômulo Damasclin Chaves dos Santos

Feb 26, 20260 citations

Asymptotic Approximation of Quantum Channels by Quantum Neural Network Operators

Rômulo Damasclin Santos

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

Feb 24, 20260 citations

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 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.

\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.