Variational Quantum Circuit Kernel

• Variational quantum circuit kernels are defined via state overlaps, Fourier-inspired feature maps, or tangent features from training dynamics.
• They enable diverse applications in quantum machine learning, including quantum SVMs, variational classifiers, and surrogate modeling for VQE.
• Empirical evaluations show that hybrid kernel constructions can improve accuracy and convergence, highlighting their practical impact in noisy and simulated settings.

Searching arXiv for papers on variational quantum circuit kernels and closely related kernel formulations. A variational quantum circuit kernel denotes a kernel associated with a parameterized quantum circuit, either through the overlap of data-encoding states, through an explicit feature map induced by the circuit’s Fourier or frequency structure, or through the inner product of tangent features obtained by linearizing training around an initialization point. In this sense, the topic sits at the intersection of variational quantum machine learning, quantum kernel methods, and classical surrogate modeling: the same circuit family can be viewed as a predictor, as a kernel-induced feature map, or as an object whose training dynamics generate an emergent kernel (Shirai et al., 2021, Sweke et al., 31 Mar 2025, Smith et al., 2022).

1. Conceptual scope and principal constructions

The literature does not use a single construction for the term. Instead, several mathematically distinct kernels are associated with variational or parameterized quantum circuits. Conventional quantum kernels arise from a data feature state and are evaluated as state overlaps. PQC-inspired kernels arise when the circuit output is written as a linear model over an explicit frequency-based feature map. Quantum tangent kernels and quantum neural tangent kernels arise from first-order or perturbative descriptions of training dynamics. More recent work makes the embedding itself trainable, producing hybrid kernel constructions that remain SVM- or kernel-matrix-based while introducing variational parameters or learned projections (Shirai et al., 2021, Sweke et al., 31 Mar 2025, Altmann et al., 30 Jan 2026).

Construction	Defining object	Source of the kernel
Conventional quantum kernel	$|\langle \phi(x_i)|\phi(x_j)\rangle|^2$	Data encoding
PQC-inspired kernel	$\langle \phi(x),A\phi(x')\rangle$	Explicit Fourier/frequency features
Quantum tangent kernel	$$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$	Training dynamics near initialization
Trainable hybrid kernel	Kernel circuit plus variational layers or learned projection	Joint embedding and optimization

A central distinction is that not all quantum machine learning models are kernel methods. The conventional feature-map kernel depends only on data encoding, whereas tangent-kernel formulations depend on how the full deep circuit responds to parameter perturbations during training. This distinction is important because it separates fixed-feature kernelization from lazy-training or near-linearization analyses of variational circuits (Shirai et al., 2021).

2. Feature-map kernels from parameterized circuits

A standard variational quantum predictor is written as

$$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$

In the conventional quantum kernel setting, one assumes

$$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$

so that the data feature state is

$$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$

and the kernel is

$$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$

This is the fidelity-style kernel that underlies quantum kernel SVM formulations and many quantum feature-map constructions (Shirai et al., 2021).

A broader kernel correspondence emerges when the circuit output admits a Fourier-like expansion. For a PQC model

$$f_{\theta}(x)=\langle 0|U^{\dagger}(x,\theta)OU(x,\theta)|0\rangle,$$

one may write

$$f_\theta(x)=\sum_{\omega\in\tilde{\Omega}} c_{\omega}(\theta)e^{i\langle\omega,x\rangle},$$

and, using the $\pm\omega$ symmetry, rewrite this as a real linear model

$\langle \phi(x),A\phi(x')\rangle$0

with

$\langle \phi(x),A\phi(x')\rangle$1

From any feature map $\langle \phi(x),A\phi(x')\rangle$2 and positive semidefinite matrix $\langle \phi(x),A\phi(x')\rangle$3, one then defines

$\langle \phi(x),A\phi(x')\rangle$4

For a PQC with frequency set $\langle \phi(x),A\phi(x')\rangle$5, this yields a PQC-inspired kernel $\langle \phi(x),A\phi(x')\rangle$6; for reweighted kernels with diagonal reweighting matrix $\langle \phi(x),A\phi(x')\rangle$7,

$\langle \phi(x),A\phi(x')\rangle$8

This kernel is normalized, $\langle \phi(x),A\phi(x')\rangle$9, and shift-invariant because it depends only on $$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$0 (Sweke et al., 31 Mar 2025).

A distinct kernel-theoretic interpretation appears in the ansatz-independent variational quantum classifier literature. There, a VQC with encoded state

$$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$1

and output

$$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$2

is shown to correspond to a structured subset of a quadratic kernel model. The paper’s direct correspondence identifies the encoded amplitudes with kernel features and the transformed observables with quadratic kernel weights, and on that basis introduces the unitary kernel method (UKM), which directly optimizes the unitary operator. Within that framework, the performance of quantum circuit learning is stated to be bounded from above by the UKM because the ansatz-dependent circuit explores only a restricted subset of the full unitary space (Miyahara et al., 2021).

3. Tangent-kernel formulations and lazy-training regimes

For deep variational circuits that alternate data encoding and trainable layers, the circuit may not be reducible to a single $$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$3 factorization. One example is

$$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$4

In this setting, the kernel may emerge from training dynamics rather than from a fixed data feature state. If parameters remain close to their initialization, a first-order expansion gives

$$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$5

which identifies the quantum tangent feature map as

$$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$6

and the quantum tangent kernel as

$$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$7

The kernel can be computed on quantum hardware using the parameter-shift rule. In numerical simulations, deeper circuits showed smaller parameter movement from initialization on a 10-qubit MNIST binary classification task, and on an ansatz-generated dataset an SVM with deep QTK achieved accuracy $$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$8, compared with $$\nabla_\theta y(x_i,\theta_0)^T\nabla_\theta y(x_j,\theta_0)$$9 for the conventional quantum kernel and $$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$0 for shallow QTK (Shirai et al., 2021).

A related but broader formalism is the quantum neural tangent kernel. For a variational ansatz

$$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$1

and residual

$$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$2

gradient descent yields

$$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$3

In the frozen or lazy-training limit, where parameter updates remain small and the kernel is approximately constant, the residual obeys

$$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$4

with convergence rate

$$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$5

The same work introduces quadratic corrections and a quantum meta-kernel, or dQNTK, to capture the onset of representation learning when the kernel evolves during training. It also defines a large-width limit for hybrid quantum-classical kernels, in which a hybrid network can be approximately Gaussian under randomized initialization and suitable orthogonality conditions on quantum outputs (Liu et al., 2021).

4. Trainable and hybrid variational kernel models

A direct attempt to merge fixed quantum kernels with variational training is the quantum variational kernel support vector machine. In that framework, the quantum kernel SVM uses

$$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$6

evaluated with angle embedding and a SWAP test, while the variational SVM trains a parameterized circuit $$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$7 by minimizing

$$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$8

The proposed QVK-SVM combines these paradigms through a circuit composed of AngleEmbedding, Adjoint of AngleEmbedding, and StronglyEntanglingLayers, with gradients obtained by the parameter-shift method and a hinge-loss-driven objective. On the Iris dataset, the reported test-set metrics were as follows (Innan et al., 2023).

Model	Accuracy	F1 score
QK-SVM	96.34%	91.64%
QV-SVM	95.43%	89.99%
QVK-SVM	98.48%	94.24%

A different line of work makes the embedding itself learnable while preserving the kernel-method viewpoint. Quantum Generator Kernels introduce Variational Generator Groups built from a complete Hermitian generator basis $$y(\mathbf{x},\boldsymbol{\theta})=\langle 0^n|U^\dagger(\mathbf{x},\boldsymbol{\theta})\,O\,U(\mathbf{x},\boldsymbol{\theta})|0^n\rangle.$$9 spanning $$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$0. Each group $$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$1 is merged into a Hermitian operator $$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$2, parameterized as

$$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$3

The embedded feature state is

$$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$4

and the kernel is

$$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$5

In the learnable form, the parameters are produced by a classical affine projection

$$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$6

so that

$$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$7

Training is staged: first pre-train $$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$8 by Kernel Target Alignment, then train an SVM on the resulting kernel matrix. On five benchmarks—moons, circles, bank, MNIST, and CIFAR10—the method was reported as the strongest quantum method in both clean and noisy hardware-simulated settings; for example, clean test accuracies included $$U(\mathbf{x},\boldsymbol{\theta})=V(\boldsymbol{\theta})U_\phi(\mathbf{x}),$$9 on MNIST and $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$0 on CIFAR10 (Altmann et al., 30 Jan 2026).

5. Classical surrogate models, tensor networks, and dequantization

Variational quantum circuit kernels are also used as surrogate models for variational quantum algorithms. For VQE, the objective

$$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$1

can be modeled by a Gaussian process equipped with a classically evaluated quantum kernel. Two kernels were studied: $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$2 For a Pauli-rotation ansatz, the paper constructs an explicit angle feature vector $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$3 and shows that both kernels are quadratic forms in $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$4, while the VQE energy itself is linear in the same Fourier-like basis: $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$5 This feature-space alignment explains why the state kernel performed best in global GP regression and Bayesian optimization. In a 4-qubit, 16-parameter VQE example, the state kernel frequently reached energy errors below $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$6 relative to the optimum, whereas classical kernels often remained around $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$7 error and many runs failed badly with $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$8 errors. Against SPSA, the quantum-kernel Bayesian optimization reached comparable or better accuracy in about $$|\phi(\mathbf{x})\rangle=U_\phi(\mathbf{x})|0^n\rangle,$$9 energy evaluations, while in the best cases SPSA matched the best state-kernel results only after roughly $$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$0 evaluations (Smith et al., 2022).

Because exact classical evaluation becomes hard for larger systems, the same work proposes an MPS approximation. With

$$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$1

the approximated state kernel is

$$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$2

and the contraction cost scales as

$$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$3

The paper explicitly notes that other tensor networks, such as PEPS or tree tensor networks, could be used for more complicated connectivity (Smith et al., 2022).

A stronger dequantization result is obtained for PQC-inspired regression kernels. For data encodings of the form

$$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$4

the frequency set factorizes as a Cartesian product,

$$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$5

When the reweighting $$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$6 is induced from a symmetric MPS weighting of the full frequency set, the kernel can be evaluated exactly and efficiently classically. The central observation is stated as: given $$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$7, let $$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$8 be induced from a symmetric MPS weighting $$K_q(\mathbf{x}_i,\mathbf{x}_j)=\mathrm{Tr}\big(\rho(\mathbf{x}_i)\rho(\mathbf{x}_j)\big)=|\langle \phi(\mathbf{x}_i)|\phi(\mathbf{x}_j)\rangle|^2, \qquad \rho(\mathbf{x})=|\phi(\mathbf{x})\rangle\langle \phi(\mathbf{x})|.$$9. Then $$f_{\theta}(x)=\langle 0|U^{\dagger}(x,\theta)OU(x,\theta)|0\rangle,$$0 can be evaluated exactly and efficiently classically. The resulting contraction-based complexity is

$$f_{\theta}(x)=\langle 0|U^{\dagger}(x,\theta)OU(x,\theta)|0\rangle,$$1

with polynomial bond dimension $$f_{\theta}(x)=\langle 0|U^{\dagger}(x,\theta)OU(x,\theta)|0\rangle,$$2. The immediate implication is not that Random Fourier Features were incorrect, but that for this class of kernels they are not necessary for efficiency: the kernel method can be derandomized and used directly via exact tensor-network contraction (Sweke et al., 31 Mar 2025).

6. Empirical scope, misconceptions, and continuing issues

A recurrent misconception is that a variational quantum circuit kernel is necessarily a fixed overlap kernel. The literature instead supports at least three distinct regimes: fixed data-encoding kernels, explicit PQC-inspired kernels built from a Fourier feature map, and tangent-kernel descriptions in which the kernel is generated by training dynamics. Another misconception is that variational quantum classifiers sit outside classical kernel methods. One line of work argues that a VQC is mathematically a structured subset of a quadratic kernel model, and that ansatz dependence imposes an additional restriction relative to direct optimization over the unitary operator (Miyahara et al., 2021).

Several limitations are explicit. For QTK, there is not yet an established infinite-width analytic limit in general. For QGK, all kernel methods remain $$f_{\theta}(x)=\langle 0|U^{\dagger}(x,\theta)OU(x,\theta)|0\rangle,$$3 in sample count, large-depth circuits remain hard on current hardware, and the classical simulation cost is still substantial. The reported classical cost bound is

$$f_{\theta}(x)=\langle 0|U^{\dagger}(x,\theta)OU(x,\theta)|0\rangle,$$4

and the authors note that kernel approximations such as Nyström or random features may be needed for very large $$f_{\theta}(x)=\langle 0|U^{\dagger}(x,\theta)OU(x,\theta)|0\rangle,$$5 (Shirai et al., 2021, Altmann et al., 30 Jan 2026).

Automated kernel-circuit design is an additional development rather than a settled endpoint. An agentic LLM system has been used to search for pure quantum feature maps for image classification under explicit constraints. The learned feature maps were evaluated as kernels

$$f_{\theta}(x)=\langle 0|U^{\dagger}(x,\theta)OU(x,\theta)|0\rangle,$$6

with SVM downstream classification. The best generated feature map, after trial-and-error optimization over 15 trials, outperformed representative quantum feature maps at 10 qubits and, when scaled to 14 qubits, surpassed the classical RBF kernel on MNIST, Fashion-MNIST, and CIFAR-10. At the same time, practical deployment was explicitly left open because the study targeted controlled, noiseless simulation settings and noted that device-specific constraints and robustness to noise would be required for real hardware use (Sakka et al., 11 Jun 2026).

Taken together, these results define the variational quantum circuit kernel not as a single object but as a family of kernel constructions tied to different aspects of parameterized quantum circuits: state overlaps, explicit circuit-induced feature maps, lazy-training linearizations, learned generator embeddings, and classically tractable tensor-network surrogates. This suggests that the kernel viewpoint is both a modeling tool and a diagnostic one: it characterizes expressivity, clarifies when training behaves linearly, and, in some regimes, removes the need for quantum optimization or even for quantum evaluation altogether.