Non-variational supervised quantum kernel methods: a review
Authors: John Tanner, Chon-Fai Kam, Jingbo Wang
Organizations: Centre for Quantum Information, Simulation and Algorithms, The University of Western Australia, 35 Stirling Hwy, Crawley WA, 6009, Australia · Quantum Theory Group, Dipartimento di Fisica e Chimica Emilio Segr`e, Universit`a degli Studi di Palermo, Via Archirafi 36, I-90123 Palermo, Italy
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.
Figures & tables
Figure 1: If a feature map ϕ associated with a kernel K embeds input samples into a high-dimensional feature space F in a suitable manner, then kernel methods based on K can convert a non-linear relationship between inputs and labels into a linear relationship in F . (a) In regression settings, suppose that ϕ maps inputs into F such that samples with similar continuous labels (indicated by their colour) lie on approximately parallel hyperplanes. In this case, the label of an unseen input x∈X can be predicted by projecting its embedded representation ϕ(x) linearly along an appropriate direction in F . (b) In binary classification, suppose instead that ϕ arranges inputs in F so that samples from different classes (represented by red triangles and blue squares) lie on opposite sides of a separating hyperplane. The label of an unseen input x∈X can then be predicted by projecting its embedded representation ϕ(x) onto the direction normal to this hyperplane. The sign of the resulting value, possibly after applying a fixed offset b∈R , determines the predicted class. In both (a) and (b) , the axis along which we project is shown as ϕ(x′) . However in general, this axis is usually given by a linear combination of the form ∑i=1Mαiϕ(xi) , where the xi ’s are the input training data samples (see Eq. 3 ).
Figure 2: The circuit diagram used to estimate KF with the Loschmidt echo test. In this case KF(x,x′) is given by the expected value of the global observable (∣0⟩⟨0∣)⊗n , which can equivalently be interpreted as the probability of measuring every qubit to be in the state ∣0⟩ at the end of the computation.
Figure 3: The circuit diagram used to estimate KF with the swap test. In this case KF(x,x′) is given by the expected value of the 1-qubit Pauli- Z operator measured from the top ancilla qubit at the end of the computation.
Figure 4: The circuit diagram used to estimate KF with the Hadamard test. In this case KF(x,x′) is given by the sum of the squares of the expected value of the 1-qubit Pauli- Z operator measured from the top ancilla qubit for the a=0 and a=1 cases. In other words KF(x,x′)=⟨Z1⟩a=02+⟨Z1⟩a=12 .
Figure 5: Applying 1-qubit state tomography and classical post-processing to estimate the PQK KP(x,x′) .
Figure 6: Applying local swap tests and classical post-processing to estimate the PQK KP(x,x′) .
Figure 7: An illustration of a 4-qubit quantum state represented in terms of an explicit array (left) and a tensor network (right).
Figure 8: An illustration of a 4-qubit MPS with open boundary conditions (left) and a 12-qubit PEPS on a 3×4 rectangular lattice with open boundary conditions (right).
Figure 9: An illustration of the inner-product ⟨ψ(x)∣ψ(x′)⟩ between 4-qubit quantum states ∣ψ(x)⟩ and ∣ψ(x′)⟩ in TN notation used to calculate the FQK. Note that ψ∗(x) represents the complex conjugate of ψ(x) .
Figure 10: An illustration of the overlap tr(ρi(x)ρi(x′)) for i=1 in TN notation used to calculate the PQK with 4-qubit quantum systems. Note that with n -qubit quantum systems, the same computation needs to be performed for all i∈{1,…,n} to compute the PQK.
Figure 11: Reducing the quantum bandwidth β reduces the expressibility of the associated quantum feature map. Above we see the yz -plane of the Bloch sphere, with the x -axis pointing out of the page. The figure shows the states prepared by the 1-qubit feature map ρ(βx)=e−iXβx∣0⟩⟨0∣eiXβx for x∈{0,π} with different values of β . As β gets smaller, the states ρ(0) and ρ(βπ) get closer together, hence reducing the expressibility of the feature map.
Quantum kernel methods are a promising branch of quantum machine learning, but their per- formance across data types and encoding choices remains incompletely understood. To address this gap, we developed an algorithmic framework for variational quantum kernels utilizing resource- efficient ansätze for complex classification tasks and introduced a parameter scaling technique to accelerate convergence. We conducted a comprehensive benchmark of this framework on eight chal- lenging, real-world and high-dimensional datasets covering tabular, image, time series, and graph data. Our results show that the proposed quantum kernel demonstrated competitive classification accuracy compared to standard classical kernels in classical simulation, such as the radial basis function (RBF) kernel. This work demonstrates that properly designed quantum kernels can func- tion as versatile, high-performance tools, laying a foundation for quantum-enhanced applications in machine learning. Further research is needed to fully assess the practical performance of quantum methods.
Jiang Yuhan, Matthew Otten
Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA · Department of Chemistry, University of Wisconsin–Madison, Madison, WI 53706, USA
Quantum kernel methods have been proposed as a promising approach for leveraging near-term quantum computers for supervised learning, yet rigorous benchmarks against strong classical baselines remain scarce. We present a comprehensive empirical study of quantum kernel support vector machines (QSVMs) across nine binary classification datasets, four quantum feature maps, three classical kernels, and multiple noise models, totalling 970 experiments with strict nested cross-validation. Our analysis spans four phases: (i) statistical significance testing, revealing that none of 29 pairwise quantum-classical comparisons reach significance at α=0.05; (ii) learning curve analysis over six training fractions, showing steeper quantum slopes on six of eight datasets that nonetheless fail to close the gap to the best classical baseline; (iii) hardware validation on IBM ibm_fez (Heron r2), demonstrating kernel fidelity r≥0.976 across six experiments; and (iv) seed sensitivity analysis confirming reproducibility (mean CV 1.4%). A Kruskal-Wallis factorial analysis reveals that dataset choice dominates performance variance (ε2=0.73), while kernel type accounts for only 9%. Spectral analysis offers a mechanistic explanation: current quantum feature maps produce eigenspectra that are either too flat or too concentrated, missing the intermediate profile of the best classical kernel, the radial basis function (RBF). Quantum kernel training (QKT) via kernel-target alignment yields the single competitive result -- balanced accuracy 0.968 on breast cancer -- but with ~2,000x computational overhead. Our findings provide actionable guidelines for quantum kernel research. The complete benchmark suite is publicly available to facilitate reproduction and extension.
Siavash Kakavand, Christoph Strohmeyer, Michael Schlotter
SHARE at FAU, Schaeffler Technologies AG & Co. KG, Herzogenaurach 91074, Bavaria, Germany
Kernel methods compare inputs through feature maps. Quantum kernels follow the same principle: input data are encoded into quantum states, which define quantum feature representations in Hilbert spaces. Kernel values are then obtained by estimating inner products between these states using suitable quantum circuit measurements. As a result, quantum kernels may be intractable to compute classically while remaining efficiently computable on quantum hardware, potentially leading to a quantum advantage. However, designing effective quantum kernels remains a major challenge. Many quantum kernels, such as the fidelity kernel, suffer from exponential concentration. This results in near-identity kernel matrices that fail to capture meaningful data correlations and lead to overfitting and poor generalization. In this paper, we propose a novel strategy for constructing quantum kernels that achieve good generalization performance, drawing inspiration from benign overfitting in classical machine learning. We introduce the concept of Local-Global quantum kernels, which combine two components: a local quantum kernel based on measurements of small subsystems, and a global quantum kernel derived from full-system measurements. To support the effectiveness of the proposed construction, we show theoretically and empirically that Local-Global quantum kernels exhibit benign overfitting.
Joachim Tomasi, Sandrine Anthoine, Hachem Kadri
Aix-Marseille Univ, CNRS, LIS, Marseille, France. · Aix Marseille Univ, CNRS, I2M, Marseille, France.