Persistence of architecture-induced coefficient couplings during training

Determine to what extent the coefficient–coefficient coupling structure predicted from the circuit harmonic matrix C under uniform parameter sampling persists during actual training of parametrized quantum circuits, and ascertain how such couplings augment or restrict learned correlations in the data.

Background

Section 4 derives that second-order statistics of trainable Fourier coefficients factor through the circuit harmonic matrix C, yielding architecture-induced covariance and correlation structures prior to training. The authors note that these structures provide diagnostics that are computable without data or training. The outstanding issue is whether, and how strongly, these architecture-induced couplings manifest during actual optimization on data.

References

How much of these couplings survive into actual training and how much they augment or restrict learned correlations in data is left as future work.

— Circuit Harmonic Matrices: A Spectral Framework for Quantum Machine Learning  (2604.04292 - Campbell et al., 5 Apr 2026) in Section 4 (Coefficient statistics from the interaction matrix)