On the Expressive Power and Capacity of Quantum Data Reuploaders

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: $K_R=\{k\in \mathcal{L}:{\mathrm{\ell }}_T(k)\le R\}$, where $\mathcal{L}={\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}_{\mathbb{Z}}(T)$ is an integer lattice and ${\mathrm{\ell }}_T$ 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 $\mathrm{\Theta }(R^m)$ with $m=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\mathcal{L}$. (iii) Using the spectrum size, we give a Rademacher-complexity bound on the capacity of QDRs of order $\sqrt{\mathrm{\mid }{K}_R\mathrm{\mid }\mathrm{/}n}$. (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 ($d\ge 4$), where it is highly sensitive to the input-encoding scale; it beats the MLP only on equations with $d\le 3$ inputs.