Quantum-Classical Dual Kernel

• Quantum-classical dual kernels are convex combinations of quantum and classical kernels that integrate data-encoded quantum state overlaps with classical geometric features.
• They leverage quantum expressivity to enhance feature mapping while using classical kernels, like RBF, to mitigate noise and exponential concentration challenges.
• Empirical results show that adaptively tuning the mixing parameter stabilizes generalization and improves classification accuracy across various feature dimensions.

A quantum-classical dual kernel, often termed a hybrid or mixed kernel, refers to a convex linear combination of a quantum kernel—arising from overlaps of data-encoded quantum states—and a classical kernel, such as the radial basis function (RBF) or polynomial kernels. This dual construction enables support vector machines (SVMs) and other kernel-based learning algorithms to combine potentially non-classically simulable quantum feature spaces with robust, well-understood classical geometric structures. Dual kernels have emerged as a practical means to mitigate the exponential concentration and hardware noise associated with deep quantum feature maps, while leveraging quantum expressivity where beneficial. The approach is now central to quantum machine learning methodologies for both theoretical studies of quantum advantage and scalable implementations on near-term devices.

1. Mathematical Foundation of Quantum-Classical Dual Kernels

Let $\mathcal{X} \subset \mathbb{R}^d$ denote the input data space. In the dual kernel paradigm, two base kernels are defined:

• A classical kernel $K_c(x, x')$, commonly RBF:

$$K_c(x, x') = \exp\left(-\gamma \| x - x' \|^2\right)$$

• A quantum kernel $K_q(x, x')$ via overlaps of quantum feature states:

$$|\psi(x)\rangle = U(x) |0\rangle^{\otimes n}; \quad K_q(x, x') = |\langle \psi(x) | \psi(x') \rangle|^2$$

The dual kernel is then formed as a convex linear combination:

$$K_{q\!-\!c}(x, x') = \alpha K_q(x, x') + (1-\alpha) K_c(x, x'),\quad \alpha \in [0,1]$$

$\alpha$ is treated as a mixing hyperparameter (or, in MKL, as an optimized combination vector over several kernels), and can be learned adaptively or set via cross-validation to maximize validation accuracy (Ghukasyan et al., 2023, Sam et al., 1 Feb 2026).

2. Dual Kernel Training and Multiple Kernel Learning Formulations

The SVM dual optimization with the hybrid kernel follows the classical form:

$$\min_{\alpha_i} \frac{1}{2} \sum_{i,j} \alpha_i \alpha_j y_i y_j K_{q\!-\!c}(x_i, x_j) - \sum_i \alpha_i\ \text{subject to} \; \sum_i \alpha_i y_i = 0, \; 0 \leq \alpha_i \leq C$$

with decision function $$f(x) = \sum_{i} \alpha_i y_i K_{q\!-\!c}(x_i, x) + b$$.

In a more general MKL context, the hybrid kernel can blend multiple classical and quantum kernels:

$K(x, x') = \sum_{r=1}^R \gamma_r K_r(x, x')$

with $\gamma_r \ge 0, \sum_r \gamma_r = 1$. The corresponding min–max problem (e.g., EasyMKL) jointly optimizes the kernel weights $\gamma$ and dual SVM variables via convex programming (Ghukasyan et al., 2023). For parameterized quantum kernels, joint optimization of both $\gamma$ and variational kernel parameters $\theta$ is achieved through differentiable layer embedding (QCC-net), enabling backpropagation through the cone program for end-to-end learning.

3. Quantum and Classical Feature Map Realizations

Classical kernels include:

• Linear: $K_{lin}(x,x') = \langle x, x' \rangle$
• Polynomial: $$K_{poly}(x,x') = (\theta_0 \langle x, x' \rangle + \theta_1)^3$$
• RBF: $K_{rbf}(x,x') = \exp(-\theta_2 \| x-x' \|^2 )$ (Ghukasyan et al., 2023)

Quantum kernels are defined by a data-dependent unitary embedding $U(x)$. Principal schemes:

• RX embedding: $U(x)=\bigotimes_{i=1}^{d} R_x(x_i)$ (no trainable params)
• IQP embedding: $$V(x) = \exp[-\frac{i}{2}\sum_{p<q} x_p x_q Z_p Z_q] \bigotimes_{i=1}^d R_z(x_i)$$ (no trainable params)
• QAOA-type embedding: $$|\Phi(x; \theta)\rangle = W(\theta) \bigotimes_{i=1}^d R_x(x_i)|0\rangle$$, with $W(\theta)$ entangling and local rotations, and $2d$ variational parameters (Ghukasyan et al., 2023, Sam et al., 1 Feb 2026, Xu et al., 7 May 2025)

Experimental implementations include NMR quantum registers encoding classical vectors as multiple-quantum coherences, and optical hardware mapping input features to Fock state amplitudes (Sabarad et al., 2024, Bartkiewicz et al., 2019).

4. Large-Scale Simulation, Scaling, and Stability

Crucially, as the number of features and corresponding qubits increases, pure quantum kernels suffer from exponential concentration, where the kernel values $K_q(x_i, x_j)$ become nearly uniform and the kernel matrix approaches rank-deficiency (Sam et al., 1 Feb 2026, Egginger et al., 2023). Hardware noise and shot noise further degrade kernel discrimination at large $n$.

Tensor network simulation frameworks now enable kernel-matrix construction for $n$ up to 784 qubits by efficient tensor contraction, slicing, and blockwise parallelization—demonstrated on Fashion-MNIST with quantum-classical dual kernels (Sam et al., 1 Feb 2026). The hybrid kernel preserves expressivity at small $n$ (quantum-dominated regime, $\alpha > 0.5$ for $n<128$) and shifts weight to the classical RBF term ($\alpha\sim0.3-0.4$ for $n>128$), thereby regularizing the kernel matrix and maintaining high classification accuracy up to $n=784$. This dual approach stabilizes generalization, avoids overfitting, and mitigates implosion under concentration.

5. Empirical Performance and Weight Dynamics

Empirical analysis across synthetic and real datasets reveals:

• On low-dimensional data, trained quantum kernels (especially parameterized QAOA-type) can dominate the mixture, with optimal quantum weights rising with $d$ if parameters are co-trained (Ghukasyan et al., 2023, Xu et al., 7 May 2025).
• For larger feature spaces, the hybrid kernel adaptively shifts towards the classical kernel, maintaining accuracy as pure quantum methods degrade (Sam et al., 1 Feb 2026).
• MKL solvers without parameter training assign near-equal weights; parameter optimization is essential for the quantum component to contribute utility in high-dimensions (Ghukasyan et al., 2023).
• In practice, the dual kernel consistently outperforms single-kernel baselines and mitigates quantum kernel collapse (Sam et al., 1 Feb 2026).

A representative summary of observed trends:

Regime / Parameterization	Quantum Kernel Weight ($\alpha$)	Generalization/Accuracy
Small $n$ ($n \lesssim 128$)	$\alpha > 0.5$ (quantum-heavy)	Quantum-classical > classical/quantum
Large $n$ ($n \gtrsim 128$)	$\alpha \sim 0.3-0.4$	Dual stable, quantum degrades
Random/non-parametric quantum	$\alpha \approx 0.5$	No clear optimization gain
Trained/parametric quantum	$\alpha$ grows with $d$	Trained quantum dominates at large $d$

6. Geometric Difference and Generalization Error Bounds

The geometric-difference metric $\Delta(K_Q, K_C)$, based on $$\sqrt{\| \sqrt{K_Q} K_C^{-1} \sqrt{K_Q} \|_\infty }$$, upper-bounds the generalization error gap between classical and quantum kernel-based learners (Egginger et al., 2023). Large $\Delta$ is necessary for quantum advantage, but not sufficient: empirical results show that real-world label structure often resides in low-$\Delta$ eigenspaces, precluding quantum outperformance. The metric also serves as a rapid prescreening tool to evaluate datasets for quantum suitability.

Hyperparameter studies demonstrate that maximizing kernel expressivity (large $\Delta$) is anti-correlated with accuracy in naturally labeled data; only artificially aligned labels with high $\Delta$ manifest a quantum over classical gap. Increasing feature subsystems' dimensionality and bandwidth hyperparameters enhances $\Delta$, but without guaranteed accuracy benefit in standard benchmarks.

7. Practical Implementations and Experimental Platforms

Quantum-classical dual kernels have been realized experimentally on NMR quantum registers, photonic circuits, and superconducting devices (Sabarad et al., 2024, Bartkiewicz et al., 2019, Ghukasyan et al., 2023). The standard hybrid workflow is:

1. Choose a feature map (quantum, classical, or both).
2. Construct kernel entries, evaluating quantum overlaps via subroutines (Hadamard/swap tests).
3. Assemble the Gram matrix and optimize dual SVM variables on a classical processor.
4. Tune kernel-mixing weights (and parameters if available) via cross-validation or end-to-end optimization (QCC-net).
5. Deploy decision functions for regression or classification.

The approach allows exponential Hilbert-space expressivity in the number of qubits, with practical cost and noise stability managed via tensor-network simulation, hyperparameter optimization, and blend adaptation. Experimental NMR results further validate quantum kernels on classical and quantum input data, demonstrating strong generalization (e.g., 94% accuracy in entanglement detection extrapolating beyond the training region) (Sabarad et al., 2024).

• "Quantum-Classical Multiple Kernel Learning" (Ghukasyan et al., 2023)
• "Scalable Tensor Network Simulation for Quantum-Classical Dual Kernel" (Sam et al., 1 Feb 2026)
• "A Hyperparameter Study for Quantum Kernel Methods" (Egginger et al., 2023)
• "Experimental Machine Learning with Classical and Quantum Data via NMR Quantum Kernels" (Sabarad et al., 2024)
• "Hybrid model of the kernel method for quantum computers" (Borba et al., 2024)
• "Quantum Classifiers with Trainable Kernel" (Xu et al., 7 May 2025)
• "Experimental kernel-based quantum machine learning in finite feature space" (Bartkiewicz et al., 2019)
• "Supervised quantum machine learning models are kernel methods" (Schuld, 2021)