Expressivity of IQP Feature-Map Quantum Kernels via Reproducing Kernel Hilbert Spaces

Similarity in many decision systems is governed not by distance alone but by interactions among variables. In fraud and anomaly detection, small local perturbations can cross interaction-sensitive decision boundaries while leaving ambient distance almost unchanged. Motivated by this setting, we introduce a thin-slab interaction model and an interaction-driven quantum kernel constructed from entangled Pauli-string feature maps. The feature map explicitly encodes sparse high-order block interactions. We show that the resulting fidelity kernel is positive semidefinite, admits an exact block-factorized formulation, and induces a geometry sensitive to changes in interaction regime. Across balanced and imbalanced synthetic experiments spanning third-, fourth-, sixth-, and eighth-order interactions, the proposed kernel consistently outperforms linear, radial basis function, Laplacian, and polynomial kernels, as well as an engineered-interaction linear baseline supplied with the planted block products. On real fraud-detection benchmarks, it achieves the highest mean accuracy and F1 on Credit Card Fraud Detection and ranks second on IEEE-CIS Fraud Detection. Executed on a 156-qubit IBM Quantum processor in a fourth-order setting, the hardware-estimated kernel matches the noise-free simulator within seed-to-seed variability and retains its advantage over the baselines. These findings show that quantum-kernel performance depends on alignment between feature-map geometry and the underlying predictive structure, rather than on Hilbert-space dimension alone. Because the prescribed block-factorized kernel can also be evaluated exactly on a classical computer, the results establish predictive and representational value rather than computational quantum speedup.

Projected quantum kernels (PQKs) have emerged as a notable approach to circumvent the hardness of machine learning tasks arising from the exponential dimensionality of the quantum Hilbert space. However, recent PQK studies often rely heavily on extensive classical hyperparameter tuning of the radial basis function (RBF) kernel parameter and the support vector machine regularization parameter. The extensive tuning can mask the true representational power of the underlying PQKs induced by quantum feature maps, which has not been explored in detail. Thus, this work aims to expose the structural capabilities of PQKs by isolating and evaluating their native inductive bias through comparisons between untuned and tuned PQKs against the RBF baseline. Furthermore, we propose novel quantum feature maps that leverage two-qubit entanglement to capture multivariate correlations in IoT data. The results show that extensive tuning can lead to competitive performance architectures, while our proposed entanglement topologies exhibit a significantly stronger native inductive bias. This reveals that superior quantum architectural design can yield structurally better PQKs with improvements arising from quantum-mechanical properties rather than from classical hyperparameter tuning.

Quantum kernel methods have been proposed as a promising approach for leveraging near-term quantum computers for supervised learning, yet rigorous benchmarks against strong classical baselines remain scarce. We present a comprehensive empirical study of quantum kernel support vector machines (QSVMs) across nine binary classification datasets, four quantum feature maps, three classical kernels, and multiple noise models, totalling 970 experiments with strict nested cross-validation. Our analysis spans four phases: (i) statistical significance testing, revealing that none of 29 pairwise quantum-classical comparisons reach significance at $α= 0.05$; (ii) learning curve analysis over six training fractions, showing steeper quantum slopes on six of eight datasets that nonetheless fail to close the gap to the best classical baseline; (iii) hardware validation on IBM ibm_fez (Heron r2), demonstrating kernel fidelity $r \geq 0.976$ across six experiments; and (iv) seed sensitivity analysis confirming reproducibility (mean CV 1.4%). A Kruskal-Wallis factorial analysis reveals that dataset choice dominates performance variance ($\varepsilon^2 = 0.73$), while kernel type accounts for only 9%. Spectral analysis offers a mechanistic explanation: current quantum feature maps produce eigenspectra that are either too flat or too concentrated, missing the intermediate profile of the best classical kernel, the radial basis function (RBF). Quantum kernel training (QKT) via kernel-target alignment yields the single competitive result -- balanced accuracy 0.968 on breast cancer -- but with ~2,000x computational overhead. Our findings provide actionable guidelines for quantum kernel research. The complete benchmark suite is publicly available to facilitate reproduction and extension.