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.
Department of Physics, University of Wisconsin–Madison, Madison, WI 53706, USA · Department of Chemistry, University of Wisconsin–Madison, Madison, WI 53706, USA