Benign Overfitting with Quantum Kernels

Abstract: Quantum kernels quantify similarity between data points by measuring the inner product between quantum states, computed through quantum circuit measurements. By embedding data into quantum systems, quantum kernel feature maps, that may be classically intractable to compute, could efficiently exploit high-dimensional Hilbert spaces to capture complex patterns. However, designing effective quantum feature maps remains a major challenge. Many quantum kernels, such as the fidelity kernel, suffer from exponential concentration, leading to 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. Our approach introduces the concept of local-global quantum kernels, which combine two complementary components: a local quantum kernel based on measurements of small subsystems and a global quantum kernel derived from full-system measurements. Through numerical experiments, we demonstrate that local-global quantum kernels exhibit benign overfitting, supporting the effectiveness of our approach in enhancing quantum kernel methods.