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

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.

Quadratic integrate-and-fire neurons exhibit less fragmented loss landscapes and outperform leaky integrate-and-fire neurons in spike-based gradient descent

The ability to train spiking neural networks is essential for modeling biological neural networks as well as for neuromorphic computing. However, for the extensively used leaky integrate-and-fire (LIF) neurons, arbitrarily small parameter changes can induce spike (dis)appearances that disrupt subsequent activity, leading to unstable neural representations and permanently silent neurons during exact spike-based gradient descent. Recent work shows that a class of neuron models, which includes the quadratic integrate-and-fire (QIF) neuron, avoids these discontinuities and enables continuous and even smooth spike-based gradient descent. However, it remains unclear whether these advantages translate into practice. Here, we demonstrate that they do so via a controlled comparison between networks of LIF and QIF neurons on the popular Spiking Heidelberg Digits dataset. Specifically, in a first step, we perform a thorough hyperparameter search to optimize both models, revealing a clear performance advantage of QIF neurons. In a second step, we visualize the loss and gradient landscapes. Consistent with their inferior performance, we find that the loss landscapes of LIF neurons, which are discontinuous, appear more fragmented and the related gradients more erratic. An analysis of the landscapes of single samples indicates that these features arise from changes in the temporal order of spikes, which often cause disruptive spike (dis)appearances. Overall, our results advocate replacing LIF neurons with neuron models exhibiting continuous spiking dynamics, such as QIF neurons, for gradient descent training.