Building upon the concept of quantum feature maps which encode classical data $x$ into quantum states $|\phi(x)\rangle$, this chapter centers on quantum kernel methods. These methods compute a similarity measure between data points by evaluating functions related to the inner product $\langle\phi(x)|\phi(x')\rangle$ in the quantum feature space $\mathcal{H}$. This calculation forms the basis of the quantum kernel $k(x, x')$, allowing classical kernel-based machine learning algorithms to operate implicitly within potentially high-dimensional quantum Hilbert spaces.

You will learn the formalism defining quantum kernels and the procedures for estimating kernel matrix entries using quantum circuits, both on simulators and hardware. We will examine the mathematical properties that characterize these kernels, such as positive semi-definiteness, and explore the geometric differences compared to classical kernels. Important practical considerations, including the phenomenon of kernel concentration in high dimensions and strategies to address it, will be analyzed. A significant portion of the chapter is dedicated to the Quantum Support Vector Machine (QSVM), detailing its operation and practical implementation for classification problems. Hands-on sections will guide you through implementing and comparing various quantum kernel approaches.