Active Quantum Kernel Improves Gaussian Process Regression Accuracy.

Quantum kernel estimation, particularly on current quantum hardware, is constrained by a limited "shot budget," meaning each element of the kernel matrix requires a finite number of circuit executions. Previous work showed that intelligently distributing these shots based on their importance to a classification task can improve accuracy while reducing budget. This paper applies and extends this concept to Gaussian Process (GP) regression. For GP regression, downstream metrics like posterior variance and marginal likelihood are more tightly coupled to kernel error than in classification. The researchers derived three specific sensitivities—predictive coupling, leave-one-out residual, and marginal-likelihood gradient—to guide shot allocation. These sensitivities are integrated into a minimum-variance allocation rule, with a uniform coverage floor added to prevent over-concentration due to noisy initial estimates. Evaluations on various benchmarks, including UCI datasets and quantum-natural data with ZZ and Pauli-Z kernels, demonstrated substantial improvements. The active allocation method yielded a 10-21% reduction in test-RMSE compared to uniform allocation in moderate-budget scenarios. This gain also translated to other tasks like Bayesian quadrature and hyperparameter learning, highlighting the method's broad applicability in quantum machine learning.