Spectral Quantum Memory for Implicit Neural Representations

Implicit neural representations (INRs) model signals as continuous functions from coordinates to values, but their performance depends strongly on how spectral capacity is allocated. Yet most Quantum INR (QINR) architectures treat spectral components as independent outputs of separate quantum blocks. We introduce the Spectral Quantum Memory Model (SQMM), a memory coupled quantum INR in which sequential hybrid data reuploading layers (bands) communicate through a shared coherent quantum memory register. Each band reads the memory, computes on its own coordinate upload, and writes part of its state back, so later bands can build on the computation of earlier ones. When memory read and write layers are input independent, we prove that the memory state forms an operator valued Fourier series whose support is contained in the Minkowski sum of the initial memory spectrum and the local band spectra. Downstream bands then condition their Fourier coefficients on the spectral content stored by earlier bands, and the memory coupling creates no new primitive frequencies. This theory motivates a spectral memory alignment loss that encourages memory to encode the spectrum expressed by the deployed prefix of the model. Across audio and image reconstruction benchmarks, SQMM improves over compact QINR and INR baselines.