Non-variational supervised quantum kernel methods: a review

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.