Power Characterization of Noisy Quantum Kernels

Abstract: Quantum kernel methods have been widely recognized as one of promising quantum machine learning algorithms that have potential to achieve quantum advantages. In this paper, we theoretically characterize the power of noisy quantum kernels and demonstrate that under global depolarization noise, for different input data the predictions of the optimal hypothesis inferred by the noisy quantum kernel approximately concentrate towards some fixed value. In particular, we depict the convergence rate in terms of the strength of quantum noise, the size of training samples, the number of qubits, the number of layers affected by quantum noises, as well as the number of measurement shots. Our results show that noises may make quantum kernel methods to only have poor prediction capability, even when the generalization error is small. Thus, we provide a crucial warning to employ noisy quantum kernel methods for quantum computation and the theoretical results can also serve as guidelines when developing practical quantum kernel algorithms for achieving quantum advantages.