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

Laura Sáez-Ortuño, Santiago Forgas-Coll, Massimiliano Ferrara

TL;DR

This paper investigates the feasibility of quantum kernel methods for real-world consumer classification in the NISQ era. It introduces a hybrid Q-SVM pipeline with a quantum feature extraction module (QFE) and proves convergence of variational quantum kernels with rate $O(1/\sqrt{T})$ and tight quantum feature extraction separation bounds $\gamma_{\text{quantum}} \ge \gamma_{\text{classical}} \cdot \sqrt{\frac{2^L}{d \cdot \mathrm{poly}(\log d)}}$, along with Nyström-based complexity results. Empirically, the Q-SVM achieves ROC AUC $0.83$ and recall $0.8609$ on a real marketing dataset, showing robust performance under shallow quantum embeddings. The results offer a principled link between near-term quantum hardware capabilities and practical marketing analytics tasks such as customer segmentation and churn prediction, enabling ROC-guided decision thresholds without retraining.

Abstract

This work studies the feasibility of applying quantum kernel methods to a real consumer classification task in the NISQ regime. We present a hybrid pipeline that combines a quantum-kernel Support Vector Machine (Q-SVM) with a quantum feature extraction module (QFE), and benchmark it against classical and quantum baselines in simulation and with limited shallow-depth hardware runs. With fixed hyperparameters, the proposed Q-SVM attains 0.7790 accuracy, 0.7647 precision, 0.8609 recall, 0.8100 F1, and 0.83 ROC AUC, exhibiting higher sensitivity while maintaining competitive precision relative to classical SVM. We interpret these results as an initial indicator and a concrete starting point for NISQ-era workflows and hardware integration, rather than a definitive benchmark. Methodologically, our design aligns with recent work that formalizes quantum-classical separations and verifies resources via XEB-style approaches, motivating shallow yet expressive quantum embeddings to achieve robust separability despite hardware noise constraints.

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

TL;DR

This paper investigates the feasibility of quantum kernel methods for real-world consumer classification in the NISQ era. It introduces a hybrid Q-SVM pipeline with a quantum feature extraction module (QFE) and proves convergence of variational quantum kernels with rate and tight quantum feature extraction separation bounds , along with Nyström-based complexity results. Empirically, the Q-SVM achieves ROC AUC and recall on a real marketing dataset, showing robust performance under shallow quantum embeddings. The results offer a principled link between near-term quantum hardware capabilities and practical marketing analytics tasks such as customer segmentation and churn prediction, enabling ROC-guided decision thresholds without retraining.

Abstract

This work studies the feasibility of applying quantum kernel methods to a real consumer classification task in the NISQ regime. We present a hybrid pipeline that combines a quantum-kernel Support Vector Machine (Q-SVM) with a quantum feature extraction module (QFE), and benchmark it against classical and quantum baselines in simulation and with limited shallow-depth hardware runs. With fixed hyperparameters, the proposed Q-SVM attains 0.7790 accuracy, 0.7647 precision, 0.8609 recall, 0.8100 F1, and 0.83 ROC AUC, exhibiting higher sensitivity while maintaining competitive precision relative to classical SVM. We interpret these results as an initial indicator and a concrete starting point for NISQ-era workflows and hardware integration, rather than a definitive benchmark. Methodologically, our design aligns with recent work that formalizes quantum-classical separations and verifies resources via XEB-style approaches, motivating shallow yet expressive quantum embeddings to achieve robust separability despite hardware noise constraints.

Paper Structure

This paper contains 29 sections, 3 theorems, 36 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Marketing Analytics Applications and Business Impact
  4. Related Work
  5. Preliminaries
  6. Quantum Feature Maps and Kernels
  7. Circuit Ansatz
  8. Main Theoretical Results
  9. Convergence of Variational Quantum Kernels
  10. Quantum Feature Extraction Separation Bounds
  11. Complexity of Approximate Quantum Feature Extraction
  12. Methodology
  13. Task, Data, and Metrics
  14. Model Suite
  15. Training, Evaluation, and Infrastructure
  16. ...and 14 more sections

Key Result

Theorem 3.1

Consider the variational quantum kernel optimization problem: where $\mathcal{L}_{\text{SVM}}(k_{\theta})$ is the SVM hinge loss with quantum kernel $k_{\theta}$, and $R(\theta)$ is an $\ell_2$ regularizer with strength $\lambda > 0$. Assume: Then gradient descent with learning rate $\eta \leq 1/\beta$ achieves: where $\theta^*$ is the global minimum. For $\eta = \Theta(1/\sqrt{T})$, this yield

Figures (3)

  • Figure 1: End-to-end workflow for quantum-enhanced classification.
  • Figure 2: ROC curve for Q-SVM with no skill diagonal.
  • Figure 3: Performance comparison across different methods.

Theorems & Definitions (9)

  • Theorem 3.1: Convergence of Variational Quantum Kernels
  • proof
  • Remark 3.2
  • Theorem 3.3: Quantum Feature Extraction Separation Bounds
  • proof
  • Remark 3.4
  • Proposition 3.5: Complexity of Nyström-Approximated QFE
  • proof
  • Remark 3.6