发表机构
National Institute of Technology, Rourkela（鲁尔克拉国家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究刻画了量子数据重上传器的频谱结构，证明其方向在R=1时稳定而频率数量随R多项式增长，并给出容量界；实验表明单量子比特QDR在低维任务上优于MLP，但在高维回归上表现不佳。
AI 中文摘要
数据重上传——在参数化量子电路的多个层中重复编码经典输入——是增强量子神经网络（QNN）表达能力的关键机制。基于Schuld等人~\n\cite{schuld2021fourier}的傅里叶理论框架，我们研究了一般多变量、$R$次重上传架构（QDR）的频谱。（i）我们明确指出，可访问频率构成集合$K_R=S^{(R)}-S^{(R)}$，该集合等于编码频谱的差集$T=S-S$的$R$折和集$T^{(R)}$，并精确刻画了它：$K_R=\{k\in\mathcal L:\\ \ell_T(k)\le R\}$，其中$\mathcal L=\mathrm{span}_{\mathbb Z}(T)$是一个整数格，$\ell_T$是$T$中的字长。（ii）我们证明，频谱的\emph{方向}在$R=1$时已经稳定（$K_R$的实张成等于$\mathcal L$的实张成），而频谱本身从未饱和：最大频率随$R$线性增长，频率数量以$\Theta(R^{m})$多项式增长，其中$m=\mathrm{rank}\\,\mathcal L$。（iii）利用频谱大小，我们给出了QDR容量的Rademacher复杂度界，阶为$\sqrt{|K_R|/n}$。（iv）我们报告了在函数逼近和分类任务上的实验，包括训练/测试分割、多个随机种子以及参数匹配的经典基线。模拟频谱与理论完全吻合。单量子比特QDR在多个平滑或周期性的单变量目标以及螺旋任务上优于匹配的MLP，但在多变量回归（$d\ge4$）上明显更差，且对输入编码尺度高度敏感；仅在输入维度$d\le3$的方程上优于MLP。
英文摘要
Data reuploading---repeatedly encoding classical inputs at multiple layers of a parameterized quantum circuit---is a central mechanism for enhancing the expressivity of quantum neural networks (QNNs). Building on the Fourier-theoretic framework of Schuld et al.~\cite{schuld2021fourier}, we study the frequency spectrum of the general multivariate, $R$-reupload architecture (QDR). (i) We make explicit that the accessible frequencies form the set $K_R=S^{(R)}-S^{(R)}$, which equals the $R$-fold sumset $T^{(R)}$ of the difference set $T=S-S$ of the encoding spectrum, and we characterise it exactly: $K_R=\{k\in\mathcal L:\ \ell_T(k)\le R\}$, where $\mathcal L=\mathrm{span}_{\mathbb Z}(T)$ is an integer lattice and $\ell_T$ is word length in $T$. (ii) We show that the \emph{directions} of the spectrum stabilise already at $R=1$ (the real span of $K_R$ equals that of $\mathcal L$), whereas the spectrum itself never saturates: the maximal frequency grows linearly in $R$ and the number of frequencies grows polynomially, as $Θ(R^{m})$ with $m=\mathrm{rank}\,\mathcal L$. (iii) Using the spectrum size, we give a Rademacher-complexity bound on the capacity of QDRs of order $\sqrt{|K_R|/n}$. (iv) We report experiments on function approximation and classification with train/test splits, several seeds and parameter-matched classical baselines. The simulated spectra match the theory exactly. A single-qubit QDR beats a matched MLP on several smooth or periodic univariate targets and on the spiral task, but it is clearly worse on multivariate regression ($d\ge4$), where it is highly sensitive to the input-encoding scale; it beats the MLP only on equations with $d\le3$ inputs.
