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

arXiv: https://arxiv.org/abs/2607.11013 · HTML: https://mlsanity.com/paper/2607.11013/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 18 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Builds on (most cited first, 4 of 4)

- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Predictive Performance of Deep Quantum Data Re-uploading Models](https://mlsanity.com/paper/2505.20337/) (2025)

## Abstract (arXiv)

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 Fx 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 Fx frozen for the entire run, while escaping circuits migrate their frequency content (direct training: rpb = -0.48; curriculum: d = 1.34; both p < 0.001). The replicated signature is this spectral mobility, not any endpoint value of Fx, and trapped circuits retain a fully non-degenerate parameter-space QFIM (rpb ≈ 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 → 3.0) convexifies the early loss landscape; escaping circuits raise Fx 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.