Active Quantum Kernel Acquisition for Gaussian Process Regression

Jian Xu; Artur Miroszewski; John Paisley; Delu Zeng; Qibin Zhao

cs.LGJun 27, 2026

Active Quantum Kernel Acquisition for Gaussian Process Regression

Authors: Jian Xu, Artur Miroszewski, John Paisley, Delu Zeng, Qibin Zhao

Organizations: RIKEN iTHEMS · RIKEN AIP · European Space Agency · Columbia University · South China University of Technology

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.

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Estimating an

N \times N

quantum kernel from circuit fidelities requires

Θ(N^2 S)

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 \lesssim 16 n_{\mathrm{pairs}}

) on sparse-sensitivity KRR, with the gap \emph{growing} from

+8

+25

pts over uniform as

N

scales

225{\to}1000

and reaching

+26

32

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s_{ij}^{\star} \propto |g_{ij}|\sqrt{K_{ij}(1-K_{ij})}

with explicit gradient

g_{ij}

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ρ^{\mathrm{SVM}} \le m_{\mathrm{sv}}^2/N^2

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+17.0 \pm 4.8

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N{=}20

on \texttt{ibm_aachen} (

3.5σ

, 5 seeds), with the advantage holding at

N{=}30

at higher budget on \texttt{ibm_berlin} (

+14.0 \pm 8.5

pts, 5 seeds).

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S(K)/\log n

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d_σ(K)

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3.2\%

and mean

5.2\%

S/\log n

across

24

configurations at

n_q = 4

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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\%

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n_q = 6

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1.7\%

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