Expressivity and generalization error of projected fidelity quantum kernels

Kernel methods solve nonlinear problems using linear models by mapping data into a higher dimensional feature space. While quantum computers can efficiently perform high dimensional feature maps, it has been shown that models based on global fidelity quantum kernels typically do not generalize well (i.e., perform poorly on unseen data) as the number of qubits increases. Projected quantum kernels have been proposed to resolve this generalization issue [1], but their properties remain largely unknown. Here, we study key properties of the projected quantum kernels including their expressivity and generalization error using kernel eigen-decomposition. We show that projected fidelity quantum kernels share a subset of their eigen-spectrum with global fidelity quantum kernels and how projected kernels impose bias on quantum models [2]. We also study how the acts of projection into smaller subspaces and composing these subspaces will affect the performance of quantum models. Finally, we analytically bound their generalization error by the difference between mean purities and mean embedding purities. Our work provides a deeper understanding of the properties of projected fidelity quantum kernels.

[1] Huang, H. Y. et al (2021). Nat. Commun., 12(1), 1-9.

[2] Kübler, J. et al. (2021). Adv. Neural Inf. Process. Syst., 34, 12661-12673.