Parameterized Quantum Kernel (pQK)

• Parameterized quantum kernel (pQK) methods are hybrid models that use quantum circuits to map data into feature spaces, creating similarity measures for classical learning.
• The framework spans fixed fidelity, trainable, and projected variants, each tailored to specific applications like clustering, classification, and pattern recognition.
• Recent developments focus on optimizing kernel geometry and scalability through techniques such as kernel target alignment and Nyström approximations.

Parameterized quantum kernel (pQK) denotes a family of hybrid quantum-classical kernel constructions in which a quantum feature map—typically realized by a parameterized quantum circuit—induces the similarity function used by a classical learner. In the literature, the term spans at least three distinct regimes: fixed fidelity kernels whose circuit is parameterized only by the input data; trainable kernels in which learnable parameters alter the feature map itself; and projected quantum kernel pipelines in which quantum states are first measured into a classical representation and only then kernelized. Reviews on parameterized quantum circuits present this progression as the move from a fixed quantum feature map to a learnable kernel, while later work instantiates it in clustering, materials science, cybersecurity, biology, and image or sensor-data classification (Benedetti et al., 2019, Slabbert et al., 9 Jul 2025, Rahman et al., 15 Jun 2026).

1. Terminological scope and conceptual variants

Reviews of hybrid quantum-classical learning describe a quantum kernel PQC as a circuit that prepares $|0\rangle$, applies a data-encoding map $U_d(x)$, applies the adjoint map $U_d^\dagger(x')$, and estimates the probability of returning to $|0^n\rangle$. In this formulation, the quantum computer supplies kernel entries, while preprocessing, optimization, and downstream prediction remain classical (Chang, 2022).

Application papers use “parameterized” in more than one sense. In autonomous materials science, the “parameterized quantum kernel” is explicitly not a trained or variational kernel with adjustable weights optimized during learning: $U(x)$ is parameterized by the input x-ray diffraction intensities, the circuit structure is fixed, and the kernel is the return probability of $U^\dagger(x_2)U(x_1)$ to $|0\rangle$ (Adams et al., 16 Jan 2026). By contrast, spectral clustering work uses pQK for a trainable fidelity kernel in which angle encodings are multiplied by learned scaling parameters, and QUACK treats pQK as a trainable quantum kernel classifier whose kernel depends on variational parameters $\boldsymbol{w},\boldsymbol{b}$, not just on the input data (Slabbert et al., 9 Jul 2025, Tscharke et al., 2024).

A further branch replaces direct state-overlap kernels by projected constructions. In CAR T-cell cytotoxicity prediction and IoT classification, Projected Quantum Kernel methods first embed classical data into a quantum state, then measure local observables or single-qubit reduced density matrices, and finally apply a classical kernel or SVM in the projected feature space rather than on direct quantum fidelity (Utro et al., 30 Jul 2025, d'Amore et al., 20 May 2025). Accordingly, pQK is not a single canonical formalism but a cluster of related kernelized quantum feature-map methods.

2. Mathematical forms of pQK

The canonical overlap-based construction maps data to a feature state $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$ and defines the kernel by state fidelity:

$$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$

This is the basic quantum-kernel form emphasized in the PQC review literature and in expositions of quantum kernel PQCs (Benedetti et al., 2019, Chang, 2022).

Trainable pQKs modify the feature map itself. In spectral clustering, each feature $U_d(x)$0 is first scaled and normalized to $U_d(x)$1, then encoded by three single-qubit rotations, with trainable scale factors:

$U_d(x)$2

For $U_d(x)$3 qubits, this yields $U_d(x)$4 parameters, and the kernel remains a fidelity:

$U_d(x)$5

The same parameter set is reused for both $U_d(x)$6 and $U_d(x)$7, which preserves symmetry and keeps the model compact (Slabbert et al., 9 Jul 2025).

Other pQK constructions tailor the similarity function more directly. The distance-based quantum classifier with tailored kernel defines

$U_d(x)$8

allowing both an arbitrary power $U_d(x)$9 and arbitrary nonnegative sample weights $U_d^\dagger(x')$0, so that the decision statistic becomes a weighted power sum of fidelities (Blank et al., 2019). In DNA comparison, the kernel is explicitly parameterized by a permutation-invariant trainable circuit,

$U_d^\dagger(x')$1

with the parameterized layer designed to preserve the permutation symmetry associated with Levenshtein distance (Shi et al., 7 Mar 2025).

Projected kernels depart from direct overlap. In the IoT formulation, one prepares $U_d^\dagger(x')$2, measures observables $U_d^\dagger(x')$3 on selected subsystems $U_d^\dagger(x')$4, forms a classical vector of expectation values $U_d^\dagger(x')$5, and defines a Gaussian kernel on those projected quantities. In the CAR T setting, the same idea is expressed as $U_d^\dagger(x')$6, where $U_d^\dagger(x')$7 is reconstructed from local $U_d^\dagger(x')$8, $U_d^\dagger(x')$9, and $|0^n\rangle$0 measurements (d'Amore et al., 20 May 2025, Utro et al., 30 Jul 2025).

3. Training objectives and optimization strategies

Once the kernel ceases to be fixed, the learning problem becomes the optimization of the kernel geometry itself. In labeled spectral clustering, the objective is kernel target alignment (KTA),

$|0^n\rangle$1

Rather than using gradient descent, the method performs grid search over the parameterized angles to maximize alignment and thereby tune the affinity matrix passed to a classical spectral clustering pipeline in scikit-learn. The same work notes that if labels were unavailable, one could instead optimize label-free criteria such as spectral gap, cluster coherence, or inter-cluster separation (Slabbert et al., 9 Jul 2025).

QUACK replaces full Gram-matrix alignment by centroid alignment. Its encoding angles take the affine form

$|0^n\rangle$2

and the model alternates two optimizations: Kernel Alignment Optimization over $|0^n\rangle$3 at fixed class centroid, and Centroid Optimization over the centroid at fixed kernel parameters. The alignment statistic is computed on the vector of sample-to-centroid kernel values rather than on a full $|0^n\rangle$4 matrix. This yields a trainable quantum kernel classifier in which the centroids and the kernel co-adapt during training (Tscharke et al., 2024).

In scalable malware classification, pQK is the nonlinear similarity engine inside a longer supervised pipeline rather than the sole trainable object. Static Portable Executable features are one-hot encoded or standardized, projected by Linear Discriminant Analysis into a compact class-aware representation, encoded by an 8-qubit parameterized feature map with ring-based entanglement and $|0^n\rangle$5 repetitions, transformed into Nyström features, and then classified by multiclass ridge regression. The kernel itself is the squared fidelity $|0^n\rangle$6, while the classifier is trained in the resulting approximate feature space (Rahman et al., 15 Jun 2026).

For trained implicit kernel models, a separate line of work rewrites the predictor as

$|0^n\rangle$7

then diagonalizes the observable and constructs an explicit quantum surrogate circuit by an extended automatic quantum circuit encoding procedure. This does not introduce a new pQK, but it makes a trained quantum-kernel predictor amenable to circuit-level compilation (Nakayama et al., 2024).

4. Scalability, compilation, and hardware realization

The principal computational obstacle for quantum kernel methods is the quadratic cost of constructing full kernel matrices. QUACK addresses this by evaluating kernels only between training samples and learned class centroids. The paper states training complexity $|0^n\rangle$8 and test complexity $|0^n\rangle$9, instead of $U(x)$0 and $U(x)$1, and benchmarks this strategy on eight binary datasets, including EMNIST, FMNIST, and MNIST with up to 784 features and no dimensionality reduction (Tscharke et al., 2024).

The malware framework addresses the same scaling problem by Nyström approximation. The full quantum kernel matrix $U(x)$2 is replaced by

$U(x)$3

with landmark regularization $U(x)$4, explicit Nyström features $U(x)$5, and complexity reduced from full $U(x)$6 evaluations and memory to $U(x)$7 evaluations and $U(x)$8 memory. The reported experimental setting uses $U(x)$9 samples, $U^\dagger(x_2)U(x_1)$0 classes, $U^\dagger(x_2)U(x_1)$1 qubits, $U^\dagger(x_2)U(x_1)$2 repetitions, and $U^\dagger(x_2)U(x_1)$3 landmarks (Rahman et al., 15 Jun 2026).

Explicit surrogate construction offers a different notion of scalability. After training a kernel model, the EQS framework approximates the observable $U^\dagger(x_2)U(x_1)$4 by a low-rank eigen-decomposition and builds a circuit $U^\dagger(x_2)U(x_1)$5 such that $U^\dagger(x_2)U(x_1)$6. Prediction is then reduced from $U^\dagger(x_2)U(x_1)$7 kernel evaluations to one quantum circuit execution plus measurement of a diagonal observable (Nakayama et al., 2024).

Hardware-specific compilation has also been developed. In the gate-based neutral-atom QKE implementation, a ZZFeatureMap on $U^\dagger(x_2)U(x_1)$8 qubits with $U^\dagger(x_2)U(x_1)$9 repetitions is compiled into pulse sequences on a Pasqal Chadoq2 architecture by deriving single-qubit and two-qubit gates from Raman and Rydberg channels. Each kernel entry is estimated with $|0\rangle$0 shots, and the average QKE sequence runtime is reported as about $|0\rangle$1 per pair (Russo et al., 2023).

5. Application domains and empirical behavior

Empirical studies show that pQK performance is strongly task-dependent and closely tied to inductive bias, data regime, and kernel construction.

Domain	pQK form	Reported behavior
Autonomous materials science	25-qubit fidelity kernel on Fe–Ga–Pd x-ray diffraction data	Model complexity: simulated quantum $|0\rangle$2, measured quantum $|0\rangle$3, cosine similarity $|0\rangle$4, RBF $|0\rangle$5; the quantum kernel outperforms RBF for about $|0\rangle$6 to $|0\rangle$7 training points, but cosine similarity outperforms all others on this dataset (Adams et al., 16 Jan 2026)
Spectral clustering	Trainable angle-encoded fidelity kernel with KTA tuning	RBF strongly outperforms pQK on Blobs, Circles, Moons, and Iris; pQK is closer to RBF on SDSS and performs better on some recall and cluster-count metrics in higher-dimensional settings (Slabbert et al., 9 Jul 2025)
Malware family classification	LDA + fidelity kernel + Nyström + ridge regression	Accuracy $|0\rangle$8; macro F1 $|0\rangle$9, macro recall $\boldsymbol{w},\boldsymbol{b}$0, macro precision $\boldsymbol{w},\boldsymbol{b}$1; classical baselines on the same split include Softmax Regression $\boldsymbol{w},\boldsymbol{b}$2, Linear SVM $\boldsymbol{w},\boldsymbol{b}$3, KNN $\boldsymbol{w},\boldsymbol{b}$4, GaussianNB $\boldsymbol{w},\boldsymbol{b}$5 (Rahman et al., 15 Jun 2026)
CAR T-cell cytotoxicity	60/61-qubit projected kernel with SVM	PQK median F1 $\boldsymbol{w},\boldsymbol{b}$6 versus $\boldsymbol{w},\boldsymbol{b}$7 for original-data SVM; best-case F1 $\boldsymbol{w},\boldsymbol{b}$8 versus $\boldsymbol{w},\boldsymbol{b}$9; largest gains occur at position $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$0 and for motifs linked to CD40, IRAK1, LAT, and GAB1 in certain positions (Utro et al., 30 Jul 2025)
DNA sequence similarity	Permutation-invariant variational kernel for length-8 sequences	Over $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$1 order accuracy after $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$2 epochs; reported values rise from $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$3 at re-uploading $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$4 to $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$5 at re-uploading $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$6 (Shi et al., 7 Mar 2025)
IoT occupancy classification	Projected quantum kernel on measured expectation features	Best result $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$7 for the 3D FeatureMap with CNOTs and $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$8; classical SVM with RBF kernel gives $|\phi(x)\rangle = U_\phi(x)|0^n\rangle$9, and the improvement is reported as not statistically significant (d'Amore et al., 20 May 2025)

Earlier image-classification work cited in a later QNN study reported that Projected Quantum Kernel features on a filtered Fashion-MNIST task yielded quantum models with more than $$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$0 accuracy and better performance than the classical counterpart on that setup, whereas a QNN with NEQR preprocessing reached about $$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$1 and a classical NN about $$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$2 on the reduced classical dataset (Ganguly, 2022).

A plausible implication is that pQK methods are most competitive when the task structure aligns with the imposed quantum feature geometry: magnitude-sensitive XRD similarity, higher-dimensional clustering, class-aware malware projections, or low-information biological subregions are recurrent examples. The same record also shows that simple classical kernels remain strong baselines.

6. Relation to deep variational kernels, quantum advantage, and limitations

A distinct but closely related development is the quantum tangent kernel (QTK). For a deep alternating circuit with output

$$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$3

the model can enter a regime in which parameters remain close to the random initialization $$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$4, so that first-order linearization applies,

$$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$5

and the emergent kernel becomes

$$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$6

On an ansatz-generated dataset, deep QTK reportedly achieves test accuracy around $$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$7, compared with $$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$8 for the conventional quantum kernel and $$k(x,x')=\left|\langle 0^n|U_\phi^\dagger(x')U_\phi(x)|0^n\rangle\right|^2 =|\langle \phi(x')|\phi(x)\rangle|^2.$$9 for shallow QTK (Shirai et al., 2021). This suggests a conceptual boundary: conventional pQK usually parameterizes the feature map directly, whereas QTK arises as an emergent lazy-training kernel for deep variational circuits.

Theoretical and practical limitations recur across the literature. Reviews emphasize that quantum advantage is not automatic; it is most plausible when the quantum feature encoding is classically intractable and when the induced kernel is useful for the task. The same review identifies three concentration mechanisms—expressibility-induced concentration, global-measurement-induced concentration, and noise-induced concentration—which can make kernel values nearly indistinguishable and damage trainability (Chang, 2022). Empirical work reinforces the point. In Fe–Ga–Pd phase classification, the cosine similarity kernel outperforms both measured and simulated quantum kernels on the reported dataset; in spectral clustering, classical RBF dominates pQK on the simpler low-dimensional benchmarks; in CAR T prediction and IoT classification, the reported gains are modest and are not presented as definitive quantum advantage (Adams et al., 16 Jan 2026, Slabbert et al., 9 Jul 2025, Utro et al., 30 Jul 2025, d'Amore et al., 20 May 2025).

Across these results, pQK appears less as a single algorithm than as a design space linking quantum feature maps, kernel learning, and classical downstream solvers. Fixed fidelity kernels, trainable overlap kernels, centroid-aligned kernels, projected kernels, and tangent-kernel limits all occupy that space. The field’s central technical question remains the same in each variant: how to engineer a quantum-induced similarity measure that is expressive, trainable, hardware-compatible, and better matched to the data than carefully optimized classical alternatives.