Quantum Kernel Methods: Convergence Theory, Separation Bounds and Applications to Marketing Analytics

Quantum machine learning (QML) has emerged as one of the most promising near-term applications of quantum computing, with quantum kernel methods representing a particularly elegant bridge between classical machine learning theory and quantum computational advantages [17, 8, 18]. The fundamental insight underlying quantum kernels is that quantum circuits can efficiently compute inner products in exponentially large Hilbert spaces, potentially capturing data relationships that are intractable for classical methods [12]. This study presents an end-to-end feasibility test of a quantum-enhanced method for supervised classification on a real consumer dataset, evaluated with ROC analysis. We examine whether shallow, NISQ-compatible quantum embeddings and kernels can improve class separability and enable threshold-based operation along the ROC curve without retraining. We report 0.7790 accuracy, 0.7647 precision, 0.8609 recall, 0.8100 F1, and 0.83 AUC. Recent work has demonstrated empirical success of quantum support vector machines (Q-SVM) in various domains, from particle physics [21] to bioinformatics [19]. However, the theoretical foundations explaining when and why quantum advantage emerges remain incomplete. While pioneering studies by Schuld and Killoran [17] established the mathematical equivalence between variational quantum circuits and kernel methods, and Liu et al. [12] proved unconditional quantum advantage for specifically constructed problems, several fundamental questions remain open: 1. What are the convergence guarantees for variational quantum kernel optimization? 1