CRC Press eBooks

Quantum Feature Spaces and Kernels

Hamid D. Ismail

Published Nov 4, 2026• 0 citations

Abstract

This chapter develops the theory and practice of quantum feature maps and quantum kernels as a central framework in quantum machine learning (QML). Building on the data-encoding methods introduced in the previous chapter, it explains how classical inputs are embedded into quantum states and how similarity between data points is then defined through overlaps in Hilbert space. This chapter introduces the quantum kernel as the squared overlap between encoded states, establishes its fundamental algebraic properties, and shows how it satisfies the conditions required for direct use in classical kernel-learning methods such as support vector machines, kernel ridge regression, and Gaussian process models. A major focus is placed on the circuit-level realization and estimation of quantum kernels. This chapter presents kernel evaluation through encoding unitaries and examines three principal estimation methods: the SWAP test, the Hadamard test, and the direct return-probability method, with attention to their trade-offs in qubit overhead, circuit depth, and suitability for NISQ hardware. It also develops the geometric interpretation of quantum kernels in Hilbert space, relating kernel values to angular separation, decision boundaries, expressivity, and generalization. Together, these ideas provide a unified foundation for understanding how quantum circuits, feature maps, and classical learning algorithms interact in hybrid quantum-classical kernel models.

DOI: 10.1201/9781042002818-6

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