When Do Quantum Kernels Help? An Empirical and Diagnostic Study of the ZZ Feature Map

Abstract

Quantum kernel methods promise richer feature spaces than classical kernels by encoding data into the exponentially large Hilbert space of a quantum circuit, but it is unclear how often this promise translates into an actual advantage on realistic data. We build a from-scratch, exactly-simulated implementation of the Havlíček et al. (2019) ZZ feature-map fidelity kernel — without any quantum computing library — validate it against an independent brute-force construction, and use it to run a controlled comparison against classical (RBF, polynomial, linear) kernels under an identical model-selection protocol. On a synthetic dataset engineered so its labels are a function of the same feature map (a positive control), the quantum kernel reaches 98.3% test accuracy versus 48–53% (chance level) for every classical kernel. On the MiniBooNE particle-identification benchmark — a real high-energy-physics classification task (Fermilab electron-neutrino vs. muon-neutrino events) with no engineered relationship to any quantum feature map — the result reverses sharply: classical kernels reach 81–87% accuracy while the quantum kernel collapses to 71.7% at both PCA dimensionalities we test, exactly the majority-class baseline in both cases. We diagnose this gap using kernel-target alignment and kernel-value statistics, and connect it to existing theory on the inductive bias and concentration of quantum kernels. Our results are an honest negative result for quantum kernels as a drop-in replacement for classical kernels on generic data, and a clean positive control demonstrating our simulator and evaluation protocol are correct and can detect an advantage when one is present by construction.