Impact of encoder-induced redundancy on spectral bias in training

Investigate how encoder-side redundancy, which scales coefficient variances via path multiplicity in the circuit harmonic analysis, affects spectral bias during training of parametrized quantum circuits.

Background

The analysis links encoder redundancy (multiple paths contributing to the same input frequency) to coefficient variances via C, suggesting a mechanism for spectral bias where low-frequency components may receive larger variance. The open task is to connect this architectural effect to actual training dynamics and spectral bias behavior.

References

Exploring how this affects spectral bias in training is left to future work, however, currently spectral bias analysis in variational quantum circuits can be found in .

— Circuit Harmonic Matrices: A Spectral Framework for Quantum Machine Learning  (2604.04292 - Campbell et al., 5 Apr 2026) in Section 4.3 (Expressivity, redundancy, and architectural coupling)

Although Theorem~\ref{theorem:tight_approx_exp_variance} is restricted to $L=1$, this observation suggests that the same variance gain may persist for practical ansatzes with $L>1$ and continue to mitigate vanishing expressivity. A formal analysis of the $\varepsilon$-approximate 2-design setting for $L>1$ remains open.

— Fourier Symmetrization for Geometric Quantum Machine Learning  (2610.01874 - Wang et al., 1 Oct 2026) in Section 2.2.2, “ε-approximate 2-designs”

We do not show that phase-locking causes the training cost; since it is a property of the two-dimensional state space, its effect cannot be separated from that of the dimension of the state space.

— Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits  (2610.21778 - Poppel et al., 15 Oct 2026) in Section 5, Discussion and conclusion