This chapter examines quantum data encoding as a foundational component of quantum machine learning (QML), emphasizing that the way classical data are embedded into quantum states strongly shapes model expressivity, trainability, and practical feasibility. It first develops a general framework in which encoding is viewed as a data-dependent unitary transformation and from a machine-learning perspective as a quantum feature map that defines the geometry of learning in Hilbert space. This chapter then compares the principal encoding strategies: basis encoding, valued for simplicity and interpretability but limited in expressive power; angle encoding, which is hardware-friendly and central to variational models; amplitude encoding, which offers exponential qubit compression but faces significant state-preparation challenges; and hybrid encoding, which combines multiple schemes to balance compression, expressivity, and implementability. A major theme of this chapter is that encoding should be understood as the construction of a quantum feature map whose induced quantum kernel determines similarity, decision boundaries, and generalization behavior. The discussion further highlights the role of encoding in variational quantum algorithms, the relationship between feature-map complexity and potential quantum advantage, and the practical constraints imposed by qubit count, circuit depth, noise, and measurement overhead.
This chapter examines quantum data encoding as a foundational component of quantum machine learning (QML), emphasizing that the way classical data are embedded into quantum states strongly shapes model expressivity, trainability, and practical feasibility. It first develops a general framework in which encoding is viewed as a data-dependent unitary transformation and from a machine-learning perspective as a quantum feature map that defines the geometry of learning in Hilbert space. This chapter then compares the principal encoding strategies: basis encoding, valued for simplicity and interpretability but limited in expressive power; angle encoding, which is hardware-friendly and central to variational models; amplitude encoding, which offers exponential qubit compression but faces significant state-preparation challenges; and hybrid encoding, which combines multiple schemes to balance compression, expressivity, and implementability. A major theme of this chapter is that encoding should be understood as the construction of a quantum feature map whose induced quantum kernel determines similarity, decision boundaries, and generalization behavior. The discussion further highlights the role of encoding in variational quantum algorithms, the relationship between feature-map complexity and potential quantum advantage, and the practical constraints imposed by qubit count, circuit depth, noise, and measurement overhead.
The development of quantum computers has been the stimulus that enables the realization of Quantum Machine Learning (QML), an area that integrates the calculational framework of quantum mechanics with the adaptive properties of classical machine learning. This article suggests a broad architecture that allows the connection between classical data pipelines and quantum algorithms, hybrid quantum-classical models emerge as a promising route to scalable and near-term quantum benefit. At the core of this paradigm lies the Classical-Quantum (CQ) paradigm, in which the qubit states of high-dimensional classical data are encoded using sophisticated classical encoding strategies which encode the data in terms of amplitude and angle of rotation, along with superposition mapping. These techniques allow compression of information exponentially into Hilbert space representations, which, together with reduced sample complexity, allows greater feature expressivity. We also examine variational quantum circuits, quantum gates expressed as trainable variables that run with classical optimizers to overcome decoherence, noise, and gate-depth constraints of the existing Noisy Intermediate-Scale Quantum (NISQ) devices. Experimental comparisons with a Quantum Naive Bayes classifier prove that even small quantum circuits can approximate probabilistic inference with competitive accuracy compared to classical benchmarks, and have much better robustness to noisy data distributionsThis model does not only explain the algorithmic and architectural design of QML, it also offers a roadmap to the implementation of quantum kernels, variational algorithms, and hybrid feedback loops into practice, including optimization, computer vision, and medical diagnostics. The results support the idea that hybrid architectures with strong data encoding and adaptive error protection are key to moving QML out of theory to practice.
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.
The encoding of classical data into quantum states constitutes the primary performance bottleneck in Quantum Machine Learning (qml) on Noisy Intermediate-Scale Quantum (nisq) devices. No existing framework jointly characterises resource cost, expressivity, and noise robustness, nor provides actionable selection guidelines for practitioners. This survey addresses that gap through a systematic review of 66 primary works (2017-2026) assembled via a PRISMA-adapted protocol across five academic databases. Four principal contributions are made. First, a three-axis cost-expressivity-robustness taxonomy classifies all major encoding families - basis, angle, dense-angle, amplitude, data re-uploading, and IQP - along independently measurable axes. Second, closed-form depth-fidelity bounds under nisq decoherence channels identify the critical gate-error rate p* ~ 10^-3 below which amplitude encoding is viable. Third, a unified treatment of Fourier expressivity, barren-plateau onset, and quantum kernel concentration as functions of the encoding circuit provides the first joint trainability analysis. Fourth, a five-regime decision framework maps (D, n, p, tau) - feature dimension, qubit budget, error rate, and task type - to a hardware-grounded encoding recommendation. The central finding is that for p >= 10^-3, shallow angle-based encodings consistently outperform amplitude encoding in practice, despite the latter's exponential qubit advantage.
This chapter introduces the central role of information encoding in quantum machine learning, examining how classical and quantum data can be mapped into quantum states for processing by quantum algorithms. It surveys key encoding strategies—basis, angle, amplitude, and Hamiltonian-based—and discusses their theoretical foundations, resource requirements, and suitability for current quantum hardware. The chapter also explores quantum autoencoders, the nature of quantum data, and practical considerations such as noise, scalability, and reversibility. Together, these topics provide a comprehensive framework for understanding how data is embedded in quantum systems to enable meaningful learning and inference.
Quantum machine learning is often motivated by the idea that quantum systems can expose useful high-dimensional structure that is difficult to access with classical models. We isolate one central component of this claim: the fixed data-encoding map. Amplitude, angle, and basis encoding are evaluated as deterministic feature maps for classical supervised learning under matched output dimensionality and strong classical controls. The benchmark compares these encodings against raw linear models, random Fourier features, polynomial features, PCA, RBF SVMs, and shallow neural networks across diverse classical datasets. Rather than treating performance as a single endpoint, we analyze the geometry of each representation through effective rank, condition number, centered kernel alignment, predictive performance, and practical overhead. The resulting picture is mechanistic: amplitude encoding can remove magnitude information through unit-sphere normalization, angle encoding can become geometrically redundant with raw linear features, and basis encoding can impose a binary Hamming geometry that is poorly aligned with smooth decision structure. These findings do not argue against quantum computation, however, they show that fixed quantum-inspired encoding geometry alone is not a reliable source of machine-learning advantage on classical data.
This chapter develops variational quantum models as the dominant framework for near-term quantum machine learning (QML), emphasizing their role as trainable quantum circuits embedded within hybrid quantum-classical optimization loops. Building on earlier chapters on encoding and quantum feature geometry, it formulates parameterized quantum circuits (PQCs) as families of unitary transformations acting on reference states, with model outputs obtained through expectation values of observables. This measurement-based structure makes variational learning inherently stochastic and closely ties model behavior to circuit architecture, observable choice, and optimization strategy. This chapter examines variational quantum circuits (VQCs) as learning models, introduces quantum neural networks (QNNs) as structured variational architectures, and analyzes layered, data-reuploading, recurrent, and hybrid classical-quantum designs. It highlights the central tension between expressivity and trainability, showing how universality and approximation power are counterbalanced by optimization barriers such as barren plateaus. The discussion further develops ansatz design strategies, supervised, fidelity-based, and energy-based losses, regularization under noise, and practical methods for gradient estimation and optimization, including the parameter-shift rule, Simultaneous Perturbation Stochastic Approximation (SPSA), and layer-wise training. Together, these ideas establish variational quantum models as a flexible yet hardware-constrained paradigm whose practical success depends on careful coordination among architecture, measurement, and optimization.