Quantum Annealing Kernel Learning

• Quantum annealing-inspired kernel learning is a technique that leverages quantum annealer dynamics and RBM-based spectral density estimation to create adaptive, data-driven kernel functions.
• Key methodologies include quantum Boltzmann sampling, joint quantum-classical optimization, and the use of Ising Hamiltonians to encode data characteristics.
• Experimental results demonstrate enhanced performance in classification and regression tasks compared to classical fixed-kernel approaches, despite current hardware limitations.

Quantum annealing-inspired kernel learning utilizes the physical processes and modeling capabilities of quantum annealers to construct and optimize data-driven kernel functions. By leveraging quantum Boltzmann sampling, quantum annealer dynamics, and joint quantum-classical optimization, these methods aim to synthesize kernels that adapt to the data, often outperforming fixed classical baselines. This paradigm encompasses frameworks that employ quantum-annealing-sampled restricted Boltzmann machines (RBMs) to engineer spectral densities (notably for random Fourier feature approaches), approaches that encode data into Ising Hamiltonians and extract kernels from annealer readouts, and hybrid quantum pipelines integrating gate-based quantum processing with annealing-based optimization.

1. Shift-Invariant Kernels, Bochner’s Theorem, and Random Fourier Features

A foundation of quantum annealing-inspired kernel methods is the spectral representation of shift-invariant kernels. For any real-valued, continuous, shift-invariant, positive-definite kernel on $\mathbb{R}^d$, Bochner’s theorem states

$$k(x, x') = k(x - x') = \int_{\mathbb{R}^d} p(\omega) e^{i\omega^\top(x-x')} d\omega,$$

where $p(\omega) \ge 0$ is a probability density over frequencies $\omega$. Equivalently,

$$k(x, x') = \mathbb{E}_{\omega \sim p}\left[\cos(\omega^\top x - \omega^\top x')\right].$$

Random Fourier features (RFF) approximate this expectation by sampling $m$ frequencies $\omega_1, \dots, \omega_m \sim p(\omega)$ and random phases $b_1, \dots, b_m \sim \mathrm{Unif}[0, 2\pi]$:

$$z(x) = \sqrt{\frac{2}{m}} \left[\cos(\omega_1^\top x + b_1), \dots, \cos(\omega_m^\top x + b_m)\right]^\top,$$

with $k(x, x') \approx z(x)^\top z(x')$. In classical settings, $p(\omega)$ is often chosen as a fixed Gaussian.

2. Boltzmann-Machine-Parametrized Spectral Distributions and Quantum Annealer Sampling

Quantum annealing-enabled kernel learning replaces the fixed spectral prior $p(\omega)$ with a data-adaptive distribution parameterized by an RBM. The key architecture consists of a binary RBM over visible and hidden units:

$$P(v, h | \mathbf{b}, \mathbf{c}, W) = \frac{1}{Z(\mathbf{b}, \mathbf{c}, W)} \exp[-E(v, h)],$$

with $$E(v, h) = -\mathbf{b}^\top v - \mathbf{c}^\top h - v^\top W h$$.

The visible unit configuration $v$ is mapped to a continuous frequency $\omega \in \mathbb{R}^{N_\omega}$ through a Gaussian–Bernoulli transformation:

$$P(\omega | v, a, U, \sigma) = \prod_{i=1}^{N_\omega} \mathcal{N}(\omega_i; \mu_i(v), \sigma_i^2), \quad \mu(v) = a + U v.$$

The combined model defines the joint probability $P(\omega, v, h|\theta)$, with $$\theta = \{a, \mathbf{b}, \mathbf{c}, W, U, \sigma\}$$. Training this model involves quantum annealer hardware (e.g., D-Wave Advantage), which samples from the Ising representation of the RBM at finite (effective) temperature, approximating draws from the Gibbs distribution. This quantum-annealer-based sampler is used for parameter-gradient estimation and ultimately sculpts $p_\theta(\omega)$ to optimize downstream performance (Hasegawa et al., 2023, Hasegawa et al., 13 Jan 2026).

3. Training Objectives and Optimization Algorithms

The primary objective for classification tasks is to maximize the alignment of the learned kernel with class labels. The training loss function typically takes the form

$$L(\theta) = -\frac{1}{N^2} \sum_{i, j} y_i y_j \mathbb{E}_{\omega \sim p_\theta}[\cos(\omega^\top (x_i - x_j))].$$

Gradients with respect to $\theta$ are estimated via the score function (log-derivative) trick, leveraging samples $(\omega, v, h)$ from the RBM+Gaussian provided by the quantum annealer. Parameter updates are performed via gradient descent. The quantum annealer is called once per gradient step to supply these samples, and the training loop iteratively optimizes kernel alignment (Hasegawa et al., 2023).

For regression, squared-kernel weights

$w_{ij} = [k_{ij}]^2$

are used in the Nadaraya–Watson estimator to enhance robustness to sign cancellation due to RFF cosine approximation. The training objective minimizes the leave-one-out mean squared prediction error, and the gradient is computed by backpropagation through the Monte Carlo kernel estimator. The quantum annealer, via RBM embedding, is an integral part of the model’s differentiable loop (Hasegawa et al., 13 Jan 2026).

4. Construction of Feature Maps and Learned Kernels

After optimization, spectral frequency vectors $\omega_1, \dots, \omega_S$ are drawn from the annealer-sampled RBM+Gaussian. The data-driven feature map is constructed as

$$\phi(x) = \frac{1}{\sqrt{S}}[\cos(\omega_1^\top x), \dots, \cos(\omega_S^\top x), \sin(\omega_1^\top x), \dots, \sin(\omega_S^\top x)]^\top.$$

Optionally, random phases may be added. The resulting empirical kernel is $k(x, x') = \phi(x)^\top \phi(x')$.

Empirical histograms of learned frequency components frequently display multimodal or heavy-tailed distributions, in contrast to the unimodal Gaussian characteristic of fixed-kernel RFF. These learned spectral densities manifest as marked changes in the kernel matrix structure and improved downstream performance (classification accuracy, regression $R^2$, RMSE) over fixed-Gaussian baselines on small- and medium-scale datasets (Hasegawa et al., 2023, Hasegawa et al., 13 Jan 2026).

5. Quantum Annealing of Data-Empowered Hamiltonians and Annealing-Dynamics-Induced Kernels

An alternative quantum annealing-inspired kernelization approach encodes classical data into the parameters ($h_i$, $J_{ij}$) of an Ising Hamiltonian:

$$H(x) = \sum_{i=1}^{N} h_i(x)\, \sigma_i^z + \sum_{i < j} J_{ij}(x)\, \sigma_i^z \sigma_j^z,$$

embedding data geometry into the energy landscape. The quantum annealer then executes a time-dependent evolution

$H(t; x) = A(t/T)\,\hat H_1 + B(t/T)\,H_2(x),$

with $\hat H_1$ the transverse-field term and $A(s), B(s)$ defining the annealing schedule. Measuring in the computational basis yields a classical distribution $P(s|x)$, forming the feature map $\phi(x) = P(x)$, from which

$K(x, x') = \sum_{s} P(s|x)\,P(s|x')$

serves as a kernel function. The participation ratio

$R(x) = \frac{1}{\sum_{s}[P(s|x)]^2}$

quantifies the effective model complexity and is tunable via annealing time and energy scale (Sakurai et al., 14 Jan 2026).

6. Experimental Protocols, Empirical Results, and Performance Metrics

Empirical validation spans classification (Fashion-MNIST, Digits/MNIST, Breast Cancer) and regression (bodyfat, Mackey–Glass series, energy efficiency) benchmarks.

• RBM+QA-learned RFF kernels outperform or match tuned Gaussian RFF approaches, with improved alignment, block-structured kernel matrices, and higher classification accuracies and regression $R^2$ (Hasegawa et al., 2023, Hasegawa et al., 13 Jan 2026).
• The learned spectral densities frequently show multimodal or correlated structure, unattainable by fixed Gaussian RFF.
• Increasing the number of random features ($S$) at inference enhances accuracy, consistently reducing kernel Monte Carlo error.
• For data-encoded Hamiltonian kernels, short (non-adiabatic) annealing times give high participation ratio $R$ and correspondingly rich feature spaces, maximizing classification accuracy at modest shot budgets; too-small or too-large $R$ results in underfitting or noisy models, respectively (Sakurai et al., 14 Jan 2026).
• Hybrid pipelines that combine gate-based quantum feature maps with annealing-based SVM dual solvers achieve high kernel-target alignment and competitive F1-scores (e.g., F1 = 90%), closely mirroring classical baselines (Bifulco et al., 5 Sep 2025).

Summary statistics and empirical findings are summarized in the following table:

Method/paper	Core Kernel Mechanism	Key Empirical Observation
(Hasegawa et al., 2023)	RBM+QA spectral density learning	Kernel adapts to data; multi-peaked spectra; improved classification accuracy
(Hasegawa et al., 13 Jan 2026)	QA-RBM RFF for regression	Data-adaptive kernel; increased $S$ enhances $R^2$, reduces RMSE
(Sakurai et al., 14 Jan 2026)	Annealer probability feature map	Short annealing maximizes accuracy; participation ratio $R$ is a complexity knob
(Bifulco et al., 5 Sep 2025)	Gate-based + annealing SVM pipeline	Top KTA maps correlate with F1; minimal loss vs. classical SVM

7. Limitations, Scalability, and Future Research Directions

The primary technical limitation of current annealing-inspired kernel learners is hardware-constrained model size, set by qubit count and connectivity—e.g., RBMs with $N_v = N_h = 4$—and sampling rates. Each gradient evaluation requires a quantum annealing call, and evaluating kernel matrices or gradients involves $\mathcal{O}(N^2 S)$ computational steps.

Further, RBM parameterization is restricted by present quantum annealer graph structures. Scalability to larger datasets and deeper models depends on advances in hardware embedding (e.g., chain-strength strategies), exploitation of richer quantum effects, and hybridization with classical contrastive divergence or continuous-time annealing (Hasegawa et al., 2023, Hasegawa et al., 13 Jan 2026, Bifulco et al., 5 Sep 2025).

Future extensions include:

• Learning richer, potentially non-shift-invariant kernels by augmenting the RBM prior or stacking multiple Boltzmann layers.
• Generalizing the annealing-based pipelines to other kernel machines (e.g., kernel PCA, multi-class SVM).
• Optimizing annealing schedules or incorporating non-stoquastic drivers to finely control kernel complexity.
• Developing an “annealing-kernel toolbox” exposing physical annealer parameters as trainable kernel hyperparameters (Sakurai et al., 14 Jan 2026).

A plausible implication is that increasing annealer resources and exploiting more quantum-coherent sampling will enable exploration of kernel regimes inaccessible to classical heuristics.