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arXiv:2606.28833

Active Quantum Kernel Acquisition for Gaussian Process Regression

Jian Xu, Artur Miroszewski, John William Paisley, Delu Zeng, Qibin Zhao

Published Jun 27, 2026• 0 citations
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Abstract

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 $|α_iα_j|$, 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 $10$--$21\%$ 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 ($-13$--$15\%$ at low budget, $p<0.05$ 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.

DOI
10.48550/arxiv.2606.28833

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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.
Estimating an N×NN \times NN×N quantum kernel from circuit fidelities requires Θ(N2S)Θ(N^2 S)Θ(N2S) measurement shots, the dominant bottleneck for deployment on near-term hardware. Existing budget-saving methods (Nyström-QKE, ShoFaR, kernel-target alignment) sub-sample \emph{which} entries to measure but allocate shots \emph{uniformly} within their chosen subset, ignoring how much each entry drives the downstream classifier. We close this gap with two contributions. \textbf{First, a complete regime decomposition} for shot-budgeted quantum kernel learning: a principled menu of when each allocator wins. Our method, \emph{AQKA}, dominates the budget-limited regime (B≲16npairsB \lesssim 16 n_{\mathrm{pairs}}B≲16npairs​) on sparse-sensitivity KRR, with the gap \emph{growing} from +8+8+8 to +25+25+25 pts over uniform as NNN scales 225→1000225{\to}1000225→1000 and reaching +26+26+26--323232 pts on an \texttt{ibm\_pittsburgh} (156-qubit Heron) hardware kernel; Nyström-QKE wins at saturating budgets on planted-sparse via low-rank reconstruction; ShoFaR is competitive only at extreme low budgets. \textbf{Second, a closed-form pair-level acquisition theory}: sij⋆∝∣gij∣Kij(1−Kij)s_{ij}^{\star} \propto |g_{ij}|\sqrt{K_{ij}(1-K_{ij})}sij⋆​∝∣gij​∣Kij​(1−Kij​)​ with explicit gradient gijg_{ij}gij​ for KRR (Lemma~1, ∣βiαj+βjαi∣Kij(1−Kij)|β_iα_j+β_jα_i|\sqrt{K_{ij}(1-K_{ij})}∣βi​αj​+βj​αi​∣Kij​(1−Kij​)​) and SVM via the envelope theorem (∣ηi∗ηj∗∣Kij(1−Kij)|η_i^*η_j^*|\sqrt{K_{ij}(1-K_{ij})}∣ηi∗​ηj∗​∣Kij​(1−Kij​)​); a \emph{corrected} sparsity-aware Cauchy--Schwarz rate ρ≤2m/Nρ\le 2m/Nρ≤2m/N matching empirics (vs.\ the naive m2/N2m^2/N^2m2/N2); an explicit-constant plug-in regret bound (Theorem~2); and a tighter SVM ceiling ρSVM≤msv2/N2ρ^{\mathrm{SVM}} \le m_{\mathrm{sv}}^2/N^2ρSVM≤msv2​/N2. We close with the first multi-seed live online adaptive shot allocation on quantum hardware: +17.0±4.8+17.0 \pm 4.8+17.0±4.8 pts at N=20N{=}20N=20 on \texttt{ibm\_aachen} (3.5σ3.5σ3.5σ, 5 seeds), with the advantage holding at N=30N{=}30N=30 at higher budget on \texttt{ibm\_berlin} (+14.0±8.5+14.0 \pm 8.5+14.0±8.5 pts, 5 seeds).
Given a set of input states, we consider the task of predicting the expectation value of a Pauli observable at the output of an unknown quantum evolution, using only a limited number of measurements. Recently, quantum Gaussian process (QGP) regression was introduced for this task across various classes of unitary evolution. Here, we extend the QGP framework beyond unitary dynamics. In particular, we prove convergence of the channel's outputs to a QGP and derive the associated closed-form kernel under a uniform (Lebesgue measure) prior over quantum channels. The kernel's dimensional factor, however, dictates the required observation precision. While manageable when the channel and observable are restricted to small subsystems, exponential suppression precludes learning when the subsystem grows extensively with the system size. Since the Lebesgue prior is overly broad for many applications, we propose an empirical Bayes heuristic that replaces the dimensional factor with a learnable scale parameter while retaining the kernel's state-overlap correlation structure. In numerical simulations of up to 64 qubits, channel QGP regression with the Lebesgue kernel exhibits a strong inductive bias for local channels, enabling faithful extrapolation. For global 64-qubit channels, the rescaled kernel restores learnability, with predictions improving systematically with the shot budget. Results from a noisy quantum computer further demonstrate the robustness of QGP regression under experimental conditions. Beyond regression, we validate QGPs as Bayesian-optimization surrogates for state preparation under noisy XXZ dynamics.
Gaussian processes (GPs) are flexible nonparametric Bayesian priors over functions and are widely used in regression, time-series modelling, and Bayesian optimisation. Yet their performance is often bottlenecked by the choice of covariance kernel, especially when the relevant correlations are unknown or inherently non-stationary. In this work, we put forward a NISQ-compatible framework that uses a quantum device as a generator of local quantum kernels for GP learning. At a high level, shallow, geometry-aware randomly parametrized circuits with area-law entanglement encode inputs into many-qubit states, and few-body Pauli measurements-estimated efficiently via classical shadows-produce quantum feature vectors that are subsequently wrapped into classical scalar kernels. A multiplekernel learning layer then fits a convex mixture of these quantum kernels by maximizing the GP marginal likelihood, so that all heavy optimisation remains on the classical side, while the quantum hardware is used solely for kernel estimation. On synthetic regression tasks, we find that the learned mixture can approximate both stationary and non-stationary covariances, achieving high kernel alignment and accurate predictions. Taken together, these results suggest that local quantum kernels can in principle form a family of flexible GP covariance learners within realistic NISQ resource budgets.
Two recent results have reshaped quantum Gaussian processes (QGPs). On the one hand, \citet{lowe2025assessing} rule out the exponential speedups claimed by HHL-based QGP regression in the typical, well-conditioned regime; on the other, an independent line of work shows that highly expressive quantum kernels suffer posterior pathologies that break Bayesian optimization. We show that these seemingly unrelated phenomena are governed by the same quantity: the normalized spectral entropy S(K)/log⁡nS(K)/\log nS(K)/logn of the kernel Gram matrix. We prove a Cauchy--Schwarz tail bound on Nyström approximation error, a finite-sample variance-contraction identity in terms of Bach's degrees of freedom dσ(K)d_σ(K)dσ​(K), and a characterization of the \emph{target-dependent} optimal entropy via the intrinsic dimension of the target in the kernel eigenbasis. Empirically, the diagnostic is kernel-agnostic: hardware-efficient, matchgate, IQP \emph{and} RBF/Matérn/RFF/deep-kernel families all collapse onto identical S/log⁡nS/\log nS/logn curves on dequantization, ECE, and variance-contraction panels. The NLL sweet spot lives at high entropy for smooth targets and at low entropy for band-limited quantum-data targets. The diagnostic transfers from simulator to IBM Heron hardware with median absolute error 3.2%3.2\%3.2% and mean 5.2%5.2\%5.2% in S/log⁡nS/\log nS/logn across 242424 configurations at nq=4n_q = 4nq​=4, with matchgate and IQP within 5%5\%5% mean and a single HE configuration returning a 30%30\%30% outlier that drops to 0.5%0.5\%0.5% on rerun (attributed to calibration drift); the same diagnostic transfers to a second Heron backend (mean error 2.7%2.7\%2.7%) and to a nq=6n_q = 6nq​=6 scale-up on the original backend (mean error 1.7%1.7\%1.7%). No error mitigation is applied throughout.
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⁡(ω⊤Δx)\cos(\bmω^{\top}Δ\bm{x})cos(ω⊤Δx) can yield small negative values and cancellation across neighbors, the Nadaraya--Watson denominator ∑jkij\sum_j k_{ij}∑j​kij​ may become close to zero. We therefore employ nonnegative squared-kernel weights wij=k(xi,xj)2w_{ij}=k(\bm{x}_i,\bm{x}_j)^2wij​=k(xi​,xj​)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 R2R^2R2 and RMSE over the baseline Gaussian-kernel NW. Increasing the number of random features at inference further enhances accuracy.
Abstract Gaussian process regression is a well-established Bayesian machine learning method. We propose a new approach to Gaussian process regression using quantum kernels based on parameterized quantum circuits. By employing a hardware-efficient feature map and careful regularization of the Gram matrix, we demonstrate that the variance information of the resulting quantum Gaussian process can be preserved. We also show that quantum Gaussian processes can be used as a surrogate model for Bayesian optimization, a task that critically relies on the variance of the surrogate model. To demonstrate the performance of this quantum Bayesian optimization algorithm, we apply it to the hyperparameter optimization of a machine learning model which performs regression on a real-world dataset. We benchmark the quantum Bayesian optimization against its classical counterpart and show that quantum version can match its performance.