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arXiv:2610.09947

On the Expressive Power and Capacity of Quantum Data Reuploaders

Lord Sen, Shyamapada Mukherjee

quant-phSubmitted Oct 7, 2026• Announced Oct 8, 2026• 0 citations
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Abstract

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.

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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, RRR-reupload architecture (QDR). (i) We make explicit that the accessible frequencies form the set KR=S(R)−S(R)K_R=S^{(R)}-S^{(R)}KR​=S(R)−S(R), which equals the RRR-fold sumset T(R)T^{(R)}T(R) of the difference set T=S−ST=S-ST=S−S of the encoding spectrum, and we characterise it exactly: KR={k∈L: ℓT(k)≤R}K_R=\{k\in\mathcal L:\ \ell_T(k)\le R\}KR​={k∈L: ℓT​(k)≤R}, where L=spanZ(T)\mathcal L=\mathrm{span}_{\mathbb Z}(T)L=spanZ​(T) is an integer lattice and ℓT\ell_TℓT​ is word length in TTT. (ii) We show that the \emph{directions} of the spectrum stabilise already at R=1R=1R=1 (the real span of KRK_RKR​ equals that of L\mathcal LL), whereas the spectrum itself never saturates: the maximal frequency grows linearly in RRR and the number of frequencies grows polynomially, as Θ(Rm)Θ(R^{m})Θ(Rm) with m=rank Lm=\mathrm{rank}\,\mathcal Lm=rankL. (iii) Using the spectrum size, we give a Rademacher-complexity bound on the capacity of QDRs of order ∣KR∣/n\sqrt{|K_R|/n}∣KR​∣/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≥4d\ge4d≥4), where it is highly sensitive to the input-encoding scale; it beats the MLP only on equations with d≤3d\le3d≤3 inputs.
Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. We in particular establish the exact correlations between the parameters of the trainable gates and the Fourier coefficients, resolving an open problem on the universal approximation property of QNN. Second, we discuss the limitations of single-qubit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.
For quantum machine learning, the exact boundary between classical and quantum advantage is still poorly understood. Direct comparison between quantum neural networks (QNNs) and existing classical models, which encompass fundamentally different function classes, often fails to provide broader insight into the difference between the two. Inspired by the techniques of Neural Quantum States and Random Fourier Features, this work introduces Neural Fourier Surrogates (NFS), a stochastic classical neural network architecture for efficiently learning coefficients over the same finite Fourier series support as quantum neural networks. Testing on a selection of tabular benchmark datasets, we find that NFS is an effective classifier architecture broadly competitive with established classical baselines, including a comparable Random Fourier Features model, and possessing comparable performance to data-reuploading QNNs; combined with additional analysis comparing the learned Fourier spectra of QNNs and NFS on synthetic data, these results establish NFS as a natural classical baseline for evaluating QNN performance.
Quantum neural networks (QNNs), represented by parameterized quantum circuits, can be trained in the paradigm of supervised learning to map input data to predictions. Much work has focused on theoretically analyzing the expressive power of QNNs. However, in almost all literature, QNNs' expressive power is numerically validated using only simple univariate functions. We surprisingly discover that state-of-the-art QNNs with strong expressive power can have poor performance in approximating even just a simple sinusoidal function. To fill the gap, we propose four expressibility-enhancing strategies for QNNs: Sinusoidal-friendly embedding, redundant measurement, post-measurement function, and random training data. We analyze the effectiveness of these strategies via mathematical analysis and/or numerical studies including learning complex sinusoidal-based functions. Our results from comparative experiments validate that the four strategies can significantly increase the QNNs' performance in approximating complex multivariable functions and reduce the quantum circuit depth and qubits required.
Data re-uploading parameterized quantum circuits (DRU-PQCs) are universal function approximators, yet their expressivity produces oscillatory, non-convex loss landscapes that resist gradient-based optimization. We show that the primary optimization bottleneck in DRU-PQCs is not insufficient capacity but a structural failure mode we term Fourier locking (FL): because encoding weights and entangling layers are nonlinearly coupled, random initialization on high-frequency targets collapses the encoding parameters into spurious local minima. Two Fisher diagnostics characterize FL. The input-space quantum Fisher information FxF_xFx​ measures the effective frequency content of the encoded state; the Fisher discriminant ratio of the measured features measures their alignment with the class labels. In two independent 50-seed experiments, the locking is literal: trapped circuits hold FxF_xFx​ frozen for the entire run, while escaping circuits migrate their frequency content (direct training: rpb=−0.48r_{pb} = -0.48rpb​=−0.48; curriculum: d=1.34d = 1.34d=1.34; both p<0.001p < 0.001p<0.001). The replicated signature is this spectral mobility, not any endpoint value of FxF_xFx​, and trapped circuits retain a fully non-degenerate parameter-space QFIM (rpb≈0r_{pb} \approx 0rpb​≈0): the failure is spectral misalignment of a responsive state, not a loss of geometric sensitivity. A frequency-staged homotopy protocol that paces the target frequency (f:1.0→3.0f: 1.0 \to 3.0f:1.0→3.0) convexifies the early loss landscape; escaping circuits raise FxF_xFx​ in step with the curriculum, and the escape rate triples (18% vs. 6%). Fourier locking is a frequency-alignment problem, and its remedy is frequency pacing.
Commutation failure controls quantum generalization. When data encoding gates fail to commute with trainable parameter gates across consecutive re-uploading layers, the effective dynamical Lie algebra becomes data-dependent; this dependency invalidates the parameter-isolation assumption underpinning existing Lie-algebraic frameworks (Caro et al., 2022; Ohno, 2025). We resolve this structural gap by proving that the statistical generalization capacity of L-layer data re-uploading circuits is tightly bounded by the effective hypothesis dimension: d_eff(L) = min(L * p_layer, |Ω_L|) <= min(L * p_layer, (2L + 1)^d), where d is the input data dimension, p_layer the number of trainable parameters per layer, and |Ω_L| the cardinality of accessible Fourier frequencies generated through non-commuting Lie brackets in the joint dynamical Lie algebra g_joint = Lie_R{i H_j, i G_k}. Harmonic analysis combined with metric entropy chaining establishes matching upper and lower bounds; together, they settle the exact minimax estimation rate R*(F_L, M) = \tilde{\Theta}(\sqrt{d_eff(L)/M}). Two distinct orbit concepts govern learning. The point-wise adjoint orbit dimension dim(O_O) <= dim(g_joint) at fixed x quantifies single-point state variation, whereas the global function-space Fourier capacity |Ω_L| governs sample complexity across the full data domain; conflating these quantities yields vacuous bounds. An initial basis rotation unlocks harmonic modes up to k_max = L; without this rotation, eigenstate initialization constrains the cutoff to k_max = L - 1. Bandwidth dictates depth. For target functions with harmonic bandlimit k_target, the network underfits when L < k_target and achieves full expressivity convergence once L >= k_target. Permutation-equivariant data encoding compresses the joint DLA to an su(2) subalgebra of dimension 3. This algebraic confinement suppresses the 4^n - 1 dimensional explosion across arbitrary qubit counts and shields the circuit from the curse of dimensionality. Numerical benchmarks in PennyLane across L in {1, 2, 4, 8} and sample sizes M in [10, 200] corroborate this harmonic-algebraic scaling and its associated uniform convergence rates.
Quantum computers can be used for supervised learning by treating parametrized quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate the practical implications of this approach, many important theoretical properties of these models remain unknown. Here, we investigate how the strategy with which data are encoded into the model influences the expressive power of parametrized quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data-encoding gates in the circuit. By repeating simple data-encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realize all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.