On the Expressive Power and Capacity of Quantum Data Reuploaders

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∈ℒ: ℓ_T(k)≤ R}, where ℒ=span_ℤ(T) is an integer lattice and ℓ_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 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=rank ℒ. (iii) Using the spectrum size, we give a Rademacher-complexity bound on the capacity of QDRs of order √(|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≥4), where it is highly sensitive to the input-encoding scale; it beats the MLP only on equations with d≤3 inputs.

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Cite

BibTeX RIS

BibTeX

@misc{sen2026expressive,
  title = {{On the Expressive Power and Capacity of Quantum Data Reuploaders}},
  author = {Lord Sen and Shyamapada Mukherjee},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09947},
  eprint = {2610.09947},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09947},
  note = {CC BY-SA 4.0},
}

RIS

TY  - UNPB
TI  - On the Expressive Power and Capacity of Quantum Data Reuploaders
AU  - Lord Sen
AU  - Shyamapada Mukherjee
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09947
UR  - https://arxiv.org/abs/2610.09947
AB  - 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∈ℒ: ℓ_T(k)≤ R}, where ℒ=span_ℤ(T) is an integer lattice and ℓ_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 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=rank ℒ. (iii) Using the spectrum size, we give a Rademacher-complexity bound on the capacity of QDRs of order √(|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≥4), where it is highly sensitive to the input-encoding scale; it beats the MLP only on equations with d≤3 inputs.
ER  -

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