Fourier locking & spectral homotopy

What it says

Data re-uploading classifiers — variational QML models where a classical input is re-encoded between trainable layers — exhibit Fourier locking: the trainable Fourier spectrum collapses onto a small integer lattice, capping expressivity even as depth grows. The paper's remedy is spectral homotopy, a continuous deformation of the encoding schedule that unlocks the spectrum. Three experiments (a / b / f) demonstrate the collapse and the fix, with results shipped as three JSON files and five figures.

Why it's next-door to us

Two of our results are Fourier-support arguments in disguise:

• AQFT under noise (G1). Our rule (M) noise sweep and the G1 depolarizing sweep show a truncated AQFT_k1 beating full QFT in 92% of noisy cells. That's a Fourier-support story too: noise removes the high-frequency components; keeping them just wastes gates. Topel's locking is which Fourier components a model can even represent; ours is which survive the channel.
• Rule (M) as a discrete homotopy. Topel deforms encoding angles continuously to unlock the spectrum. Our rule (M) rewrites deform ZX diagrams discretely while preserving the matrix identity — a combinatorial cousin of the same idea: reparameterise until the useful structure appears.

What we took from the artifact

• One JSON per experiment. experiment_a_results.json, experiment_b_results_clean.json, experiment_f_bulletproof.json — externally corroborates the shape we already use under src/data/demos/.
• Split run/render notebooks. Their FourierLocking_Experiments.ipynb writes JSON; FourierLocking_Figures.ipynb reads it. We follow the same split between quantum/pqp_frontier/*.py drivers and the nadarasa.proofs.*.tsx routes.
• Pinned requirements.txt. Now mirrored at quantum/requirements.txt so external readers can reproduce the Selene sweeps without Lovable.