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arXiv:2601.08724

Kernel Learning for Regression via Quantum Annealing Based Spectral Sampling

Yasushi Hasegawa, Masayuki Ohzeki

quant-phcs.LGSubmitted Jan 13, 2026• Announced Jan 14, 2026• 0 citations
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Abstract

While quantum annealing (QA) has been developed for combinatorial optimization, practical QA devices operate at finite temperature and under noise, and their outputs can be regarded as stochastic samples close to a Gibbs--Boltzmann distribution. In this study, we propose a QA-in-the-loop kernel learning framework that integrates QA not merely as a substitute for Markov-chain Monte Carlo sampling but as a component that directly determines the learned kernel for regression. Based on Bochner's theorem, a shift-invariant kernel is represented as an expectation over a spectral distribution, and random Fourier features (RFF) approximate the kernel by sampling frequencies. We model the spectral distribution with a (multi-layer) restricted Boltzmann machine (RBM), generate discrete RBM samples using QA, and map them to continuous frequencies via a Gaussian--Bernoulli transformation. Using the resulting RFF, we construct a data-adaptive kernel and perform Nadaraya--Watson (NW) regression. Because the RFF approximation based on $\cos(\bmω^{\top}Δ\bm{x})$ can yield small negative values and cancellation across neighbors, the Nadaraya--Watson denominator $\sum_j k_{ij}$ may become close to zero. We therefore employ nonnegative squared-kernel weights $w_{ij}=k(\bm{x}_i,\bm{x}_j)^2$, which also enhances the contrast of kernel weights. The kernel parameters are trained by minimizing the leave-one-out NW mean squared error, and we additionally evaluate local linear regression with the same squared-kernel weights at inference. Experiments on multiple benchmark regression datasets demonstrate a decrease in training loss, accompanied by structural changes in the kernel matrix, and show that the learned kernel tends to improve $R^2$ and RMSE over the baseline Gaussian-kernel NW. Increasing the number of random features at inference further enhances accuracy.

Comments
15pages, 5 figures, 4 tables

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The Boltzmann machine is one of the various applications using quantum annealer. We propose an application of the Boltzmann machine to the kernel matrix used in various machine-learning techniques. We focus on the fact that shift-invariant kernel functions can be expressed in terms of the expected value of a spectral distribution by the Fourier transformation. Using this transformation, random Fourier feature (RFF) samples the frequencies and approximates the kernel function. In this paper, furthermore, we propose a method to obtain a spectral distribution suitable for the data using a Boltzmann machine. As a result, we show that the prediction accuracy is comparable to that of the method using the Gaussian distribution. We also show that it is possible to create a spectral distribution that could not be feasible with the Gaussian distribution.
Probabilistic machine learning models are distinguished by their ability to integrate prior knowledge of noise statistics, smoothness parameters, and training data uncertainty. A common approach involves modeling data with Gaussian processes; however, their computational complexity quickly becomes intractable as the training dataset grows. To address this limitation, we introduce a quantum-assisted algorithm for sparse Gaussian process regression based on the random Fourier feature kernel approximation. We start by encoding the data matrix into a quantum state using a multi-controlled unitary operation, which encodes the classical representation of the random Fourier features matrix used for kernel approximation. We then employ a quantum principal component analysis along with a quantum phase estimation technique to extract the spectral decomposition of the kernel matrix. We apply a conditional rotation operator to the ancillary qubit based on the eigenvalue. We then use Hadamard and swap tests to compute the mean and variance of the posterior Gaussian distribution. We achieve a polynomialorder computational speedup relative to the classical method.
Quantum kernel estimation on near-term hardware is shot-budgeted: every entry of the kernel Gram matrix is a Bernoulli expectation that must be sampled with a finite number of circuit executions. Recent work on quantum kernel classification has shown that allocating shots non-uniformly across kernel entries, weighted by their downstream task sensitivity, can reduce the shot budget required to reach a target accuracy. We extend this idea to Gaussian process (GP) regression, a setting whose downstream quantities (full-spectrum posterior variance, log-determinant, marginal likelihood) couple to kernel error more tightly than the sign-only outputs of classification. We derive three closed-form pair-level sensitivities predictive coupling ∣αiαj∣|α_iα_j|∣αi​αj​∣, leave-one-out residual, and marginal-likelihood gradient and plug them into a Neyman-style minimum-variance allocation rule. To prevent catastrophic over-concentration when the warm-up sensitivity estimate is itself noisy, we add a high uniform coverage floor justified by a Frobenius lower bound on the missing-entry perturbation. On four UCI benchmarks and two synthetic RBF + Bernoulli controlled studies, the resulting allocator delivers 101010--21%21\%21% test-RMSE improvement over uniform allocation across the moderate-budget regime. The gain transfers (i) to genuine ZZ and Pauli-Z quantum kernels on quantum-natural data (−13-13−13--15%15\%15% at low budget, p<0.05p<0.05p<0.05 paired) and (ii) to four downstream tasks (Bayesian quadrature, heteroscedastic regression, hyperparameter learning, multi-output Cokriging). On UCI features embedded into a ZZ kernel the gain disappears, consistent with the exponential-concentration regime where shot allocation has nothing to exploit.
We propose a scheme to calibrate the internal parameters of a quantum annealer to obtain well-approximated samples for training a restricted Boltzmann machine (RBM). Empirically, samples from quantum annealers obey the Boltzmann distribution, making them suitable for RBM training. Quantum annealers utilize physical phenomena to generate a large number of samples in a short time. However, hardware imperfections make it challenging to obtain accurate samples. Existing research often estimates the inverse temperature for the compensation. Our scheme efficiently utilizes samples for RBM training also to estimate internal parameters. Furthermore, we consider additional parameters and demonstrate that they improve sample quality. We evaluate our approach by comparing the obtained samples with classical Gibbs sampling, which theoretically generates accurate samples. Our results indicate that our scheme demonstrates performance on par with Gibbs sampling. In addition, the training results with our estimation scheme outperform those of the contrastive divergence algorithm, a standard training algorithm for RBM.
Abstract Boltzmann machines, a class of machine learning models, are the basis of several deep learning methods that have been successfully applied to both supervised and unsupervised machine learning tasks. These models assume that some given dataset is generated according to a Boltzmann distribution, and the goal of the training procedure is to learn the set of parameters that most closely match the input data distribution. Training such models is difficult due to the intractability of traditional sampling techniques, and proposals using quantum annealers for sampling hope to mitigate the cost associated with sampling. However, real physical devices will inevitably be coupled to the environment, and the strength of this coupling affects the effective temperature of the distributions from which a quantum annealer samples. To counteract this problem, error correction schemes that can effectively reduce the temperature are needed if there is to be some benefit in using quantum annealing for problems at a larger scale, where we might expect the effective temperature of the device to be too high. To this end, we have applied nested quantum annealing correction (NQAC) to do unsupervised learning with a small bars and stripes dataset, and to do supervised learning with a coarse-grained MNIST dataset, which consists of black-and-white images of hand-written integers. For both datasets we demonstrate improved training and a concomitant effective temperature reduction at higher noise levels relative to the unencoded case. We also find better performance overall with longer anneal times and offer an interpretation of the results based on a comparison to simulated quantum annealing and spin vector Monte Carlo. A counterintuitive aspect of our results is that the output distribution generally becomes less Gibbs-like with increasing nesting level and increasing anneal times, which shows that improved training performance can be achieved without equilibration to the target Gibbs distribution.
There is currently a huge effort to understand the potential and limitations of variational quantum machine learning (QML) based on the optimization of parameterized quantum circuits. Recent proposals toward dequantizing variational QML models for regression problems include approaches based on kernel methods with carefully chosen kernel functions, approximated via Random Fourier Features (RFF). In this short note we show that for a wide range of instances, this approach can be simplified. More specifically, we show that the kernels whose evaluation is approximated by RFF in this dequantization algorithm can in fact often be evaluated exactly and efficiently classically. As such, our results enhance the toolkit for kernel-based dequantization of variational QML.