QUBIT-REUPLOAD FOURIER-RESOLUTION BOUNDS FOR COLLECTIVE-PAULI QUANTUM PINNs

Abstract

Finite-dimensional data-reuploading circuits with fixed linear-angle encodings have finite Fourier spectra determined by their encoding generators. We translate this established representation into sharp continuum obstructions for a quantum physics-informed neural network targeting Dirichlet Schrödinger modes. A model with \(q\) qubits and \(L\) copies of the fixed collective Pauli encoder has bandwidth at most \(K=q L\). After an antisymmetrizing boundary wrapper that enforces both boundary conditions exactly, its output lies in the first \(K\) sine modes. If the target index satisfies \(n>K\), then every normalized output has zero overlap with the target, phase-free \(L^{2}\) error \(\sqrt{2}\), and fixed-energy residual at least \(\pi^{2}\left(n^{2}-K^{2}\right) / 2\); the residual bound is sharp in the ambient sine space. We also compute the exact infimum of residual plus a squared-mass normalization penalty, including its collapse-to-zero transition, and prove a shift-invariant extension for real bounded potentials. Thus \(q L \geq n\) is necessary to remove this spectral obstruction, though not sufficient for representability or training. Deterministic numerical checks accompany the analytic proofs.