Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.
Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.
Abstract Since the entry of kernel theory in the field of quantum machine learning, quantum kernel methods (QKMs) have gained increasing attention with regard to both probing promising applications and delivering intriguing research insights. Benchmarking these methods is crucial to gain robust insights and to understand their practical utility. In this work, we present a comprehensive large-scale study examining QKMs based on fidelity quantum kernels (FQKs) and projected quantum kernels (PQKs) across a manifold of design choices. Our investigation encompasses both classification and regression tasks for five dataset families and 64 datasets, systematically comparing the use of FQKs and PQKs quantum support vector machines and kernel ridge regression. This resulted in over 20,000 models that were trained and optimized using a state-of-the-art hyperparameter search to ensure robust and comprehensive insights. We delve into the importance of hyperparameters on model performance scores and support our findings through rigorous correlation analyses. Additionally, we provide an in-depth analysis addressing the design freedom of PQKs and explore the underlying principles responsible for learning. Our goal is not to identify the best-performing model for a specific task but to uncover the mechanisms that lead to effective QKMs and reveal universal patterns.
Quantum machine learning (QML) is an emerging interdisciplinary field which incorporates some features of quantum computing with machine learning. Quantum hardware development has sparked research interest for AI scientists because it gives capabilities in high-dimensional data processing and simultaneous operation execution. The investigation within this paper traces the main features of QML by reviewing variational quantum classifiers (VQCs) as well as quantum kernels and hybrid quantum-classical models. VQCs serve as quantum parameterized circuits whose optimization process with classical feedback achieves quantum superiority for classification applications. Quantum kernels expand Hilbert space features to improve traditional kernel methods, thus proving their functionality in quantum feature space. Hybrid approaches unite NISQ hardware with classical systems, which makes QML applicable to real-world applications right now. (Rietsche et al., 2022; Tychola et al., 2023) An analysis explores separate functions and combined effects of these components, which enhance model performance, extend generalization, and boost computational efficiency. The paper discusses necessary knowledge first, along with existing applications, before analyzing them against traditional methods. The paper reviews current QML tools while exploring their operational readiness as well as practical issues and deployment barriers for broader adoption. The paper conducts an in-depth investigation of significant limitations that include hardware noise alongside questions regarding scalability and interpretability. This paper shows how QML will transform machine learning applications through its review of obstacles that must be resolved to achieve its complete potential development. The study presents researchers and practitioners with an extensive comprehension of QML developments and emerging paths for this revolutionary field.
Quantum and classical machine learning have been naturally connected through kernel methods, which have also served as proof-of-concept for quantum advantage. Quantum embeddings encode classical data into quantum feature states, enabling the construction of embedding quantum kernels (EQKs) by measuring vector similarities and projected quantum kernels (PQKs) through projections of these states. However, in both approaches, the model is influenced by the choice of the embedding. In this work, we propose using the training of a quantum neural network (QNN) to construct neural quantum kernels, specifically neural EQKs and neural PQKs—problem-inspired kernel functions. Unlike previous approaches, our method requires the kernel matrix to be constructed only once, significantly reducing computational overhead. To achieve this, we introduce a scalable training method for an n -qubit data reuploading QNN. Furthermore, we demonstrate neural quantum kernels can alleviate exponential concentration and enhance generalization capabilities compared to problem-agnostic kernels, positioning them as a scalable and robust solution for quantum machine learning applications.
Anomaly detection (AD) underpins fraud prevention, network security, industrial condition monitoring, medical diagnosis, and searches for new physics. However, it remains challenging due to rare anomalies, scarce labeled data, and the high dimensionality and strong correlation of sensor data. Many classical machine learning (ML) algorithms are designed to address AD in various settings, but the growth of quantum computing opens a new path for these problems. Quantum machine learning (QML) offers new tools, from exploring exponentially large Hilbert spaces and promising speed-ups to the native processing of quantum data. Thus, this review connects the two fields. We first present the basics of quantum computing, quantum algorithms, and QML with the most influential literature, and formalize the classical anomaly scores that quantum methods inherit. We then systematically review 30 primary studies of quantum anomaly detection (QAD) published between 2018 and 2026, express each method family through its anomaly score, and organize the field into a taxonomy of six families: quantum kernel methods, quantum autoencoders, quantum generative models, variational quantum classifiers, quantum linear-algebra methods, and detectors for quantum data. A cross-study analysis shows that the field has moved from asymptotic speed-up proposals to small hybrid experiments, that kernels and autoencoders dominate, that only nine studies run on gate-based hardware, and that none yet demonstrates a robust, practically relevant advantage over well-tuned classical detectors. We close with the main obstacles, such as data embedding, barren plateaus, kernel concentration, and weak baselines, and a research roadmap that prioritizes quantum data, fair benchmarks against classical baselines, and robust detectors for sequential and distributed data.
Quantum kernel methods, i.e., kernel methods with quantum kernels, offer distinct advantages as a hybrid quantum-classical approach to quantum machine learning (QML), including applicability to Noisy Intermediate-Scale Quantum (NISQ) devices and usage for solving all types of machine learning problems. Kernel methods rely on the notion of similarity between points in a higher (possibly infinite) dimensional feature space. For machine learning, the notion of similarity assumes that points close in the feature space should be close in the machine learning task space. In this paper, we discuss the use of variational quantum kernels with task-specific quantum metric learning to generate optimal quantum embeddings (a.k.a. quantum feature encodings) that are specific to machine learning tasks. Such task-specific optimal quantum embeddings, implicitly supporting feature selection, are valuable not only to quantum kernel methods in improving the latter's performance, but they can also be valuable to non-kernel QML methods based on parameterized quantum circuits (PQCs) as pretrained embeddings and for transfer learning. This further demonstrates the quantum utility, and quantum advantage (with classically-intractable quantum embeddings), of quantum kernel methods.
The quantum support vector machines (QSVMs) offer a paradigm change that may leverage quantum mechanical processes to outperform traditional support vector machines (SVMs) in high-dimensional and nonlinear classification tasks. These machines are located at the crossroads of supervised machine learning and quantum computing. The purpose of this chapter is to provide a comprehensive technical analysis of QSVMs, with a particular emphasis on the use of variation quantum circuits for implicit feature space transformations and the development of these circuits within the context of quantum kernel estimation. Through the use of quantum state preparation and interference, we provide a comprehensive explanation of the possible fabrication of quantum kernels that are based on inner product evaluations in exponentially huge Hilbert spaces. When it comes to boosting model expressivity beyond classical limits, the importance that entanglement and quantum feature mapping play is specifically stressed. In addition to this, we investigate algorithmic implementations on noisy intermediate-scale quantum (NISQ) hardware, focusing on problems associated with quantum decoherence, gate fidelity, and loop depth optimization. Several fields, including quantum-enhanced drug discovery, anomaly detection in quantum sensor networks, fault classification in quantum error correction, and portfolio optimization, are now the focus of research that is being conducted in order to develop real-world applications. This chapter comes to a close with an analysis of the existing theoretical limits, resource needs, and future perspectives for the use of QSVMs in machine learning applications in order to achieve a quantum advantage. In order to facilitate the development of scalable quantum machine learning models that are founded on strong mathematical and computational notions, this synthesis is geared at academics and practitioners.