Neural quantum kernels: training quantum kernels with quantum neural networks

Quantum and classical machine learning have been naturally connected through kernel methods, which have also served as proof-of-concept for quantum advantage. Quantum embeddings encode classical data into quantum feature states, enabling the construction of embedding quantum kernels (EQKs) by measuring vector similarities and projected quantum kernels (PQKs) through projections of these states. However, in both approaches, the model is influenced by the choice of the embedding. In this work, we propose using the training of a quantum neural network (QNN) to construct neural quantum kernels: both neural EQKs and neural PQKs, which are problem-inspired kernels. Unlike previous methods in the literature, our approach requires the kernel matrix to be constructed only once. We present several strategies for constructing neural quantum kernels and propose a scalable method to train an $n$-qubit data re-uploading quantum neural network (QNN). We provide numerical evidence of the performance of these models under noisy conditions. Additionally, we demonstrate how neural quantum kernels can alleviate exponential concentration and enhance generalization capabilities compared to problem-agnostic kernels, positioning them as a scalable and robust solution for quantum machine learning applications.