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

Kernel methods are essential for understanding quantum machine learning models. In the realm of quantum kernels, Quantum Embedding Kernels (QEK) are created by calculating inner products between quantum feature states, making them well-suited for actual quantum devices. Nevertheless, the primary challenge with kernel methods lies in selecting the appropriate embedding, as the effectiveness of a QEK depends on the specific learning task. To address this challenge, various methods have been developed to train a quantum kernel constructed from a parametrized embedding ansatz. However, these techniques involve computing the kernel matrix, which results in a quadratic increase in complexity with respect to the number of training samples, at every training step. In response, we propose the utilization of a Quantum Neural Network (QNN), specifically a data re-uploading-based QNN, to identify the optimal kernel for a given task. Notably, even with the potential for iterative training strategies for the QNN, training a single-qubit QNN suffices to construct powerful multi-qubit QEKs with entanglement, all while requiring the construction of the kernel matrix only once. This method has been validated on various artificial datasets, and it has also demonstrated remarkably good performance on real image classification datasets.