Quantum-Feature Kernel Framework

• Quantum-Feature Kernel Framework is a family of quantum machine-learning constructions that embed inputs into quantum states to compute similarity via fidelity or trace.
• It employs various kernel families, including global fidelity, projected, and continuous-variable kernels, to adapt to diverse hardware platforms and data types.
• The framework leverages classical kernel learners, such as SVMs, integrating quantum feature maps with spectral and alignment diagnostics for enhanced performance.

Quantum-Feature Kernel Framework denotes a family of quantum machine-learning constructions in which an input is embedded into a quantum state or operator, similarity is defined by an overlap, a trace, or a measurement-derived bilinear form, and the resulting positive semidefinite Gram matrix is used by a classical kernel learner such as an SVM or kernel ridge regressor. Across the literature, the framework appears in qubit feature maps, trace-induced operator kernels, continuous-variable encodings, NMR implementations, neutral-atom graph kernels, generator-based kernels, and hardware-aware circuit-selection pipelines (Schuld, 2021, Gan et al., 2023, Henderson et al., 2024, Sabarad et al., 2024, Liu et al., 26 Jun 2025, Altmann et al., 30 Jan 2026).

1. Formal definition and mathematical basis

In its most common qubit form, a classical input $x$ is embedded by a data-dependent unitary into a pure state,

$$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$

This yields two canonical kernels: the amplitude-overlap kernel

$k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$

and the fidelity kernel

$$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$

The fidelity form is real, nonnegative, and positive semidefinite, and it is the default object in much of the modern literature because it admits direct estimation by prepare–invert or SWAP-test style circuits (Schuld, 2021).

A more general statement replaces state vectors by operator-valued features. In the unified trace-induced setting, a kernel can be written as

$$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$

or, equivalently, as a weighted measurement-feature expansion

$$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$

This form subsumes global fidelity kernels, projected local kernels, and other trace-induced constructions, while preserving positive semidefiniteness by Hilbert–Schmidt inner-product structure and by closure of kernels under nonnegative sums and products (Gan et al., 2023).

The same logic extends beyond state overlaps. In NMR implementations, the feature map is operator-valued,

$$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$

so the kernel is still a Hilbert–Schmidt inner product, but on encoded operators rather than encoded state vectors (Sabarad et al., 2024). In all of these cases, the representer theorem applies: training reduces to a classical kernel machine whose predictor is an expansion over training examples, rather than a fully unconstrained quantum variational model (Schuld, 2021).

2. Principal kernel families

A central branch of the framework is the trace-induced taxonomy. The global fidelity quantum kernel

$k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$

compares full states, whereas local projected quantum kernels restrict the comparison to reduced density matrices or local operator supports. The unified theory expresses these kernels as positive combinations of elementary “Lego” kernels

$$k_i(x,x')=\operatorname{tr}[\rho(x)A_i]\operatorname{tr}[\rho(x')A_i],$$

and organizes expressivity by the number of active Lego components, $p$, or by $$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$0-body operator support in Pauli-basis constructions (Gan et al., 2023).

Projected kernels form a second major branch. The review literature emphasizes projected quantum kernels built from local reduced density matrices, for example

$$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$1

which bias the model toward local information and can be estimated with local measurements rather than global overlap circuits (Tanner et al., 9 Apr 2026). Related measurement-based projected kernels also appear in software frameworks and application papers, where expectation values are first extracted and then fed to a classical Gaussian or RBF-style kernel (Marcantonio et al., 2022).

Continuous-variable variants replace qubit amplitudes by Segal–Bargmann functions. In that setting, a pure-state kernel is

$$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$2

and every finite-stellar-rank CV quantum kernel factorizes into a Gaussian term and an algebraic term. For displaced Fock encodings, this becomes the closed-form kernel

$$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$3

with $$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$4 the Laguerre polynomial. The same line of work introduces stellar rank as a hierarchy of non-Gaussian capacity and shows that infinite-stellar-rank kernels can be approximated arbitrarily well by finite-rank ones (Henderson et al., 2024).

Kerr-based CV kernels implement another non-Gaussian family. They use self-Kerr and cross-Kerr generators to map classical inputs to non-Gaussian bosonic states and define

$$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$5

or, in the mixed-state case,

$$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$6

These kernels are explicitly tied to Wigner negativity, cat-like interference, and oscillatory $$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$7 phase structure, which the paper treats as a resource for expressivity and potential hardness of classical simulation (Wood et al., 2024).

A more recent trainable construction is the Quantum Generator Kernel, in which the feature unitary is generated by grouped Lie-algebraic Hamiltonians,

$$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$8

with grouped generators $$|\psi(x)\rangle = U_\phi(x)|0\rangle^{\otimes n}, \qquad \rho(x)=|\psi(x)\rangle\langle\psi(x)|.$$9 drawn from a universal basis of $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$0. This retains a fidelity kernel,

$k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$1

but learns a linear compression $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$2 before quantum embedding, thereby decoupling large input dimension from qubit count (Altmann et al., 30 Jan 2026).

3. Architectural and hardware realizations

The framework is not tied to a single substrate. It has been realized or proposed on superconducting qubits, superconducting Kerr resonators, NMR spin registers, neutral-atom arrays, and large-scale tensor-network simulators.

Platform	Encoding/kernel mechanism	Representative detail
IBM superconducting devices	topology-respecting circuit kernels with GNN screening	IBM Perth, Lagos, Nairobi, Jakarta, Torino
Kerr resonators	Kerr and cross-Kerr CV overlap kernels	parity and photon-number sampling
NMR star topology	operator-valued kernel $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$3	10-qubit star register
Neutral atoms	Rydberg evolution kernels and GDQC kernels	local detuning for node attributes

On superconducting-qubit hardware, early work implemented both a quantum variational classifier and a direct quantum kernel estimator using shallow diagonal feature maps interleaved with Hadamards on a superconducting processor, with 100% success on two benchmark sets and 94.75% on a third for the kernel-SVM route (Havlicek et al., 2018). Later hardware-aware design work generalized this into HaQGNN, which constructs candidate circuits directly from device-native gates and connectivity, chooses low-noise subgraphs, and screens candidates with graph neural networks trained on device calibration data. Its native gate sets were $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$4 on IBM Torino and $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$5 on 7-qubit IBM devices (Liu et al., 26 Jun 2025).

The NMR realization uses a 10-qubit star-topology register and an operator kernel based on the central-spin observable $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$6. It reports one-dimensional regression with RMS errors of $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$7 and $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$8, and two-dimensional classification with hinge losses $k_A(x,x')=\langle \psi(x)|\psi(x')\rangle,$9 and $$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$0 (Sabarad et al., 2024).

Neutral-atom implementations embed graph structure into a Rydberg Hamiltonian. Edge structure is encoded through atomic positions and van der Waals couplings, whereas node attributes are encoded through local detunings. Two kernel families are defined there: the global-observable Quantum Evolution Kernel (QEK) and the local-observable Generalized-Distance Quantum-Correlation (GDQC) kernel. Pooling across time slices improves performance, and the best pooled results surpass attributed classical baselines on both MUTAG* and PTC_FM* (Djellabi et al., 11 Sep 2025).

For CV hardware, a superconducting Kerr platform replaces qubit circuits entirely with bosonic evolution under self-Kerr and cross-Kerr Hamiltonians. Kernel elements are sampled directly by displaced-parity or photon-number measurements after stochastic control, rather than by explicit qubit gate compilation (Wood et al., 2024).

Large-scale classical realization has also been demonstrated through matrix product state simulation. Using a linear-chain Hamiltonian-inspired ansatz, one study built quantum kernel models for 165 features and 6,400 training examples on the Elliptic Bitcoin dataset, completing training in three hours on 32 GPUs (Metcalf et al., 2024).

4. Learning objectives, optimization, and model selection

The default downstream learner is a classical kernel machine. The standard SVM dual,

$$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$1

appears across the literature, while kernel ridge regression uses

$$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$2

What differentiates frameworks is therefore not the classical solver but the way kernels are selected, trained, or estimated (Schuld, 2021).

Kernel-target alignment (KTA) has become a central proxy for task relevance. In HaQGNN, for binary labels,

$$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$3

and for multiclass classification the target kernel uses $$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$4 on same-class pairs and $$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$5 otherwise. HaQGNN couples this performance proxy to a fidelity proxy, the Probability of Successful Trials,

$$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$6

obtained by circuit inversion on noisy simulators. Using GNN surrogates, it reports test $$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$7 and $$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$8, together with speedups of $$F(x,x') = |\langle \psi(x)|\psi(x')\rangle|^2 = \operatorname{tr}[\rho(x)\rho(x')].$$9 for PST prediction and $$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$0 for KTA prediction on 100 circuits (Liu et al., 26 Jun 2025).

QuKerNet replaces exhaustive circuit search with a neural predictor over image-like encodings of circuit layouts. It samples $$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$1 layouts for KTA supervision, evaluates about $$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$2 candidates, keeps the top-$$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$3 layouts, and fine-tunes trainable angles on the selected circuits. The reported predictor–accuracy correlation is very strong, with PCC $$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$4 for layout-only search and $$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$5 after parameter fine-tuning (Lei et al., 2024).

Trainable-kernel variants go further by optimizing the embedding itself. “Quantum Classifiers with Trainable Kernel” introduces a universally trainable quantum feature mapping with data re-uploading, a variational support-vector QSVM objective, and partially evenly weighted trial states to improve distinguishability and reduce the “reading out burden.” Its SV-QSVM formulation replaces uniform training-state superpositions by support-vector-weighted states, thereby avoiding the $$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$6 decision-value shrinkage identified for LS-QSVM at large $$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$7 (Xu et al., 7 May 2025).

On the software side, QuASK packages projected kernels, trainable kernels, and structure-optimized kernels into a command-line and Python-library workflow. It exposes alignment, geometric difference, approximate dimension, and model complexity as analysis tools, and provides overlap/SWAP-test kernels, projected kernels, gradient-based optimization, and structure search via simulated annealing or genetic algorithms (Marcantonio et al., 2022).

5. Expressivity, generalization, and spectral structure

A recurring question is how kernel expressivity relates to generalization. In the Lego-kernel framework, the number of active components $$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$8 controls both. The corresponding RKHSs satisfy a strict nesting relation,

$$k(x,x')=\langle \Phi(x),\Phi(x')\rangle_{HS} = \operatorname{tr}[\Phi(x)^\dagger \Phi(x')],$$9

and the generalization bound for binary classification contains an explicit $$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$0 complexity term,

$$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$1

identifying $$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$2 as a structural risk parameter rather than merely a descriptive one (Gan et al., 2023).

Spectral structure offers a complementary view. “Quantum Kernels are Spectral Tensor Networks” shows that layered quantum kernels admit finite Fourier expansions whose coefficient tensors can be factorized as matrix product operators. After grouping gate-level frequencies into feature-wise frequencies, the grouped kernel takes the form

$$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$3

and on a frequency-resolving grid the kernel-target alignment becomes the Frobenius cosine similarity between grouped Fourier coefficient tensors,

$$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$4

The numerical experiments reported there show that layered quantum kernels are often accurately representable with small bond dimension, so compressibility itself becomes a diagnostic of classical representability and tractability (Åsgrim et al., 18 Jun 2026).

In CV systems, stellar rank plays a related role. Finite-stellar-rank kernels factor into Gaussian and algebraic terms, higher stellar rank improves performance on annular data, and infinite-stellar-rank feature maps can still be approximated arbitrarily well by finite-rank ones. This suggests that non-Gaussian capacity is hierarchical rather than all-or-nothing (Henderson et al., 2024). Kerr-kernel work makes a parallel point in phase space by tying expressive decision structure to Wigner negativity rather than to Gaussian encodings, which remain classically simulable (Wood et al., 2024).

Modern theory also highlights failure modes. The review of non-variational supervised quantum kernel methods emphasizes exponential concentration, hardware noise, dequantization by tensor-network methods, and adverse kernel spectra as the main obstacles to practical quantum advantage. It also argues that any plausible separation requires both favorable generalization and hardness of kernel evaluation, not merely the use of a quantum feature space (Tanner et al., 9 Apr 2026).

Finally, high-dimensional asymptotics for quantum kernel ridge regression show that quantum kernels exhibit double descent. The deterministic-equivalent test risk contains a variance term proportional to $$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$5, which diverges near the interpolation threshold and is suppressed by explicit regularization $$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$6. The analysis makes spectral decay of the population covariance the key object governing the size of the interpolation peak (Kamisoyama et al., 19 Apr 2026).

6. Applications, empirical record, and recurring limitations

The framework has been tested on speech, tabular data, images, time series, graphs, molecular data, finance, and synthetic benchmarks. In low-resource spoken command recognition, the Gaussian-QKL system achieved average accuracies of $$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$7 on Georgian, $$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$8 on Chuvash, $$k(x,x')=\sum_{\ell} w_\ell\, m_\ell(x)\,m_\ell(x'), \qquad m_\ell(x)=\operatorname{tr}[M_\ell \rho(x)].$$9 on Lithuanian, and $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$0 on Arabic, outperforming classical kernel metric learning and QCNN-DNN baselines in those settings (Yang et al., 2022).

Automated kernel-design systems report strong gains on standard vision and fraud benchmarks. QuKerNet improved top test accuracy across its search pipeline from $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$1 to $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$2 to $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$3 on tailored MNIST, from $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$4 to $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$5 to $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$6 on tailored Credit Card data, and from $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$7 to $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$8 to $$A(x)=U(x)A_0U^\dagger(x), \qquad k(x,x')=\operatorname{tr}[A(x)A(x')],$$9 on a synthetic dataset (Lei et al., 2024). HaQGNN reports the highest classification accuracy on Credit Card using 4 qubits on IBM Perth and IBM Torino, competitive-or-best performance on MNIST-5 across Perth, Lagos, Nairobi, Jakarta, and Torino, and significant gains on FMNIST-4 with 8 qubits on Torino (Liu et al., 26 Jun 2025).

Broader benchmark studies also report consistent improvements over classical kernels. On eight high-dimensional datasets, QAmp or QRBF outperformed tuned RBF kernels by $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$0 on Higgs Boson, $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$1 on QSAR, $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$2 on TCGA-LGG, $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$3 on Fashion T-shirt/Shirt, $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$4 on PhysioNet2017-NA, $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$5 on SEED-P12S1, and $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$6 on PROTEINS, while tying on MUTAG (Jiang et al., 13 Nov 2025).

Graph and molecular settings have been used to test kernels tied directly to physical dynamics. In neutral-atom graph learning, pooled GDQC and QEK variants reached weighted F1 scores of $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$7 on MUTAG* and $k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$8 on PTC_FM*, surpassing WL OA$k_{GF}(x,x')=\operatorname{tr}[\rho(x)\rho(x')]$9 at $$k_i(x,x')=\operatorname{tr}[\rho(x)A_i]\operatorname{tr}[\rho(x')A_i],$$0 and $$k_i(x,x')=\operatorname{tr}[\rho(x)A_i]\operatorname{tr}[\rho(x')A_i],$$1 respectively (Djellabi et al., 11 Sep 2025). Large-scale simulation work on the Elliptic Bitcoin dataset demonstrated that quantum-kernel performance improved as feature dimension and training size increased, with a $$k_i(x,x')=\operatorname{tr}[\rho(x)A_i]\operatorname{tr}[\rho(x')A_i],$$2 AUC gain from 100 to 165 features at 6,400 training samples, while keeping the entire workflow tractable through MPS simulation (Metcalf et al., 2024).

Despite this breadth, several limitations recur. Gram-matrix construction typically scales quadratically in the number of samples; hardware-aware pipelines require retraining across qubit counts and calibration epochs; neutral-atom graph methods depend on 2D unit-disk embeddability; Kerr-based kernels are sensitive to photon loss; and several application papers leave gate sets, shot counts, or hardware noise models unspecified, which affects reproducibility and deployment planning (Liu et al., 26 Jun 2025, Wood et al., 2024, Yang et al., 2022, Tanner et al., 9 Apr 2026).

Taken together, these works define Quantum-Feature Kernel Framework not as a single algorithm but as a research program: construct a quantum feature map matched to task and hardware, define a PSD similarity through fidelity, trace, or measurement statistics, use spectral or alignment diagnostics to control capacity, and exploit classical kernel solvers for training. The unifying claim is not that every such kernel yields an advantage, but that quantum hardware, quantum-inspired surrogates, and careful inductive-bias design can produce kernel geometries that are difficult to obtain by straightforward classical means and, in several benchmark regimes, empirically superior.