Balancing Expressivity and Learnability in Quantum Kernel Bandit Optimization

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.

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 $S(K)/\log n$ 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 $d_σ(K)$, 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 $S/\log n$ 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 $3.2\%$ and mean $5.2\%$ in $S/\log n$ across $24$ configurations at $n_q = 4$, with matchgate and IQP within $5\%$ mean and a single HE configuration returning a $30\%$ outlier that drops to $0.5\%$ on rerun (attributed to calibration drift); the same diagnostic transfers to a second Heron backend (mean error $2.7\%$) and to a $n_q = 6$ scale-up on the original backend (mean error $1.7\%$). No error mitigation is applied throughout.

Kernel methods compare inputs through feature maps. Quantum kernels follow the same principle: input data are encoded into quantum states, which define quantum feature representations in Hilbert spaces. Kernel values are then obtained by estimating inner products between these states using suitable quantum circuit measurements. As a result, quantum kernels may be intractable to compute classically while remaining efficiently computable on quantum hardware, potentially leading to a quantum advantage. However, designing effective quantum kernels remains a major challenge. Many quantum kernels, such as the fidelity kernel, suffer from exponential concentration. This results in near-identity kernel matrices that fail to capture meaningful data correlations and lead to overfitting and poor generalization. In this paper, we propose a novel strategy for constructing quantum kernels that achieve good generalization performance, drawing inspiration from benign overfitting in classical machine learning. We introduce the concept of Local-Global quantum kernels, which combine two components: a local quantum kernel based on measurements of small subsystems, and a global quantum kernel derived from full-system measurements. To support the effectiveness of the proposed construction, we show theoretically and empirically that Local-Global quantum kernels exhibit benign overfitting.