Distribution of QNTKs around the parameter-averaged mean and higher-order kernel extensions

Characterize the distribution of quantum neural tangent kernels (QNTKs) around their parameter-averaged mean for parametrized quantum circuits, determine how this distribution is shaped by the circuit architecture through the circuit harmonic matrix C, and develop extensions to higher‑order kernel objects.

Background

Section 6 shows that parameter-averaged harmonic QNTKs can be reconstructed as C diag(||k||2) C†. Numerics indicate structural agreement with correlation matrices, but only average behavior is analyzed. The authors identify as future work the need to go beyond averages to distributions and to extend kernel-based descriptions beyond second order.

References

Characterising the distribution of QNTKs around this parameter-averaged mean, and understanding how that distribution is shaped by the architecture through $C$, alongside the extension to higher-order kernel objects, are is left for future work.

— Circuit Harmonic Matrices: A Spectral Framework for Quantum Machine Learning  (2604.04292 - Campbell et al., 5 Apr 2026) in Section 6.3 (Parameter Averaged Kernels from C: Results)