AI 中文总结
研究量子神经网络算子逼近任意量子信道时的渐近展开，证明其误差分三部分，取得三项进展，包括量子中心极限定理等，还通过数值测试验证理论速率，为量子神经网络设计分析提供理论与工具。
AI 中文摘要
我们证明了量子神经网络算子在逼近任意量子信道时的完整渐近展开式，这是经典Voronovskaya定理的非交换类似物。展开式表明逼近误差分为三个根本不同的部分：涉及普通Fréchet导数的\(1/n\)的整数次幂；由Marchaud分数导数控制的分数次幂，它捕捉了信道的Hölder光滑性；以及没有经典对应物的纯量子交换项。余项由一个显式常数严格界定。我们对一个经典类似物进行了数值测试，证实了预测的收敛速度和对数修正。基于此展开式，我们取得了三项主要进展：量子神经网络算子波动的量子中心极限定理；通过Kubo-Ando均值在量子信道之间构造最优插值测地线的方法；以及对分数光滑性如何限制量子神经网络逼近加速的系统理解。数值测试进一步表明理论速率是精确的，对数增强是不可避免的。我们的工作在经典逼近理论、分数微积分和量子机器学习之间建立了一座严谨的桥梁，为有限维量子神经网络的设计和分析提供了理论见解和实用工具。
英文摘要
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.
Comments24 pages
