Adaptive Low-Rank Quantum Kernel Acquisition under Finite-Shot Measurements: Perturbation Guarantees for QSVMs

Abstract

Quantum kernel methods require estimating entries of a dense Gram matrix through repeated circuit measurements, making kernel construction expensive and statistically noisy. For a dataset of size $n$, dense construction requires $O(n^2)$ kernel queries, while existing approximation methods typically assume inexpensive access to kernel entries or use non-adaptive sampling. We study adaptive low-rank quantum kernel acquisition when the kernel is accessible only through a finite-shot quantum overlap oracle. Our framework, ICAQKL, uses residual-driven incomplete Cholesky construction to acquire informative kernel columns sequentially and form a positive-semidefinite rank-$r$ surrogate, reducing the training-kernel acquisition cost to $O(nrN_s)$ circuit executions for $r\ll n$, where $N_s$ is the number of shots per queried entry. The central difficulty is that finite-shot errors perturb both the queried kernel values and the adaptive pivot sequence that determines the low-rank factor. We develop a non-asymptotic perturbation framework for this stochastic adaptive process, establishing concentration over the realized queries, stability of noisy pivot selection and factor construction, and propagation of the coupled sampling and approximation errors to the QSVM objective, optimizer, prediction function, and classification margin. The resulting rank--shot conditions characterize when ICAQKL approaches the full QSVM induced by the same quantum feature map. Experiments on benchmark and biomedical datasets show favorable accuracy--query-cost tradeoffs, competitive performance with the full QSVM, and competitive or improved performance relative to the evaluated Nyström baselines under comparable settings. ICAQKL is intended to enable measurement-efficient and theoretically controlled use of a selected quantum kernel, rather than to establish quantum advantage over classical kernels.