Overcoming Fourier Locking in Quantum Data Re-uploading Classifiers via Spectral Homotopy

Abstract

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 $F_x$ 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 $F_x$ frozen for the entire run, while escaping circuits migrate their frequency content (direct training: $r_{pb}=-0.48$; curriculum: $d=1.34$; both $p<0.001$). The replicated signature is this spectral mobility, not any endpoint value of $F_x$, and trapped circuits retain a fully non-degenerate parameter-space QFIM ($r_{pb}\approx 0$): 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 ($f:1.0\to 3.0$) convexifies the early loss landscape; escaping circuits raise $F_x$ 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.

14 pages, 5 figures, 1 table. Submitted to PRX Quantum. Code and data: https://github.com/moth-quantum/fourier_locking_study. ORCID: 0009-0000-0081-9786

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