Characterization of reachable coefficient sets in quantum data reuploaders

Characterize the sets of Fourier coefficient vectors that can actually be realized by the trainable blocks of multivariate quantum data reuploading models, beyond the frequencies permitted by the spectrum K_R and subject to the architecture's coefficient constraints and parameter count.

Background

The paper establishes that an R-reupload quantum data reuploading model has an accessible frequency set K_R determined by the encoding spectrum, but emphasizes that spectral accessibility alone does not imply that every corresponding Fourier coefficient can be realized. The coefficients are constrained by the trainable quantum blocks, the observable, the parameterization, and the number of trainable parameters.

The experiments and theoretical discussion identify these coefficient constraints as a separate limitation from the growth of the frequency spectrum. A general characterization of the reachable coefficient sets would clarify the actual finite-depth expressivity and capacity of quantum data reuploaders, and would help distinguish expressivity limitations caused by coefficient constraints from those caused by optimization.