Training embedding quantum kernels with data re-uploading quantum neural networks

Kernel methods play a crucial role in machine learning and the EmbeddingQuantum Kernels (EQKs), an extension to quantum systems, have shown verypromising performance. However, choosing the right embedding for EQKs ischallenging. We address this by proposing a $p$-qubit Quantum Neural Network(QNN) based on data re-uploading to identify the optimal $q$-qubit EQK for atask ($p$-to-$q$). This method requires constructing the kernel matrix onlyonce, offering improved efficiency. In particular, we focus on two cases:$n$-to-$n$, where we propose a scalable approach to train an $n$-qubit QNN, and$1$-to-$n$, demonstrating that the training of a single-qubit QNN can beleveraged to construct powerful EQKs.