The ZZ feature map induces a signless Laplacian metric: a closed-form classical surrogate for quantum kernel regression

Abstract

Bandwidth-tuned quantum kernels have been shown to lose their advantage over classical kernels and to resemble radial basis function kernels, but the analytical support for that observation rests on separable encoding circuits and captures entangling circuits only qualitatively. We close this gap for the ZZ feature map. We prove that in the small-bandwidth regime the induced kernel has, to leading order, the anisotropic Mahalanobis geometry defined by M = I + pi^2 Q, where Q is the signless Laplacian of the entanglement graph, and that it admits a parameter-free Gaussian surrogate matching it through second order. The quadratic structure persists at any fixed depth as a pullback of the Fubini-Study metric. The anisotropy depends entirely on a phase convention: under the unshifted convention the metric is the identity irrespective of entanglement, which accounts analytically for the isotropic resemblance previously reported. The derivation is verified against direct simulation for path, cycle and complete graphs, with relative error below 10^-3. The corresponding classical kernel requires no fitted parameters and no quantum simulation. Compared with the quantum kernel on two near-infrared spectroscopic benchmarks, across four targets, five preprocessing pipelines and 100 resampled splits per cell, the paired 95% bootstrap interval contains zero in 18 of 20 cells, the typical relative difference in test error is 2.7%, and the two families select the same preprocessing pipeline in 96% of splits. The regime in which the reduction fails adds no robust predictive value: restricting the grid to the classical regime improves mean test error by 3.8%, a gain that an equally large random restriction does not reproduce.