To perform machine learning using quantum computers, classical data must first be represented in the quantum realm. This process, known as quantum data encoding, is fundamental to Quantum Machine Learning (QML). How we map classical data points, like a vector $x$, into quantum states, often denoted as $|\phi(x)\rangle$ residing in a Hilbert space, significantly influences the capabilities and performance of subsequent quantum algorithms.

This chapter concentrates on the methods and implications of encoding classical information into quantum systems. We will examine the mathematical structure of quantum feature maps, which define this classical-to-quantum data transformation. You will learn about various encoding strategies, including higher-order polynomial maps and data re-uploading techniques designed to enhance model expressivity. We will also cover methods for analyzing the properties of these maps, such as their ability to generate entanglement and their geometric relationship to kernel methods. Furthermore, we address the practical challenges of encoding high-dimensional data and provide hands-on guidance for implementing custom feature maps using standard quantum computing libraries. By the end of this chapter, you will have a solid grasp of advanced data encoding techniques and their theoretical underpinnings, equipping you to design and select appropriate feature maps for specific QML tasks.