  The quantum kernel method has attracted considerable attention in the field
of quantum machine learning. However, exploring the applicability of quantum
kernels in more realistic settings has been hindered by the number of physical
qubits current noisy quantum computers have, thereby limiting the number of
features encoded for quantum kernels. Hence, there is a need for an efficient,
application-specific simulator for quantum computing by using classical
technology. Here we focus on quantum kernels empirically designed for image
classification and demonstrate a field programmable gate arrays (FPGA)
implementation. We show that the quantum kernel estimation by our heterogeneous
CPU-FPGA computing is 470 times faster than that by a conventional CPU
implementation. The co-design of our application-specific quantum kernel and
its efficient FPGA implementation enabled us to perform one of the largest
numerical simulations of a gate-based quantum kernel in terms of features, up
to 780-dimensional features. We apply our quantum kernel to classification
tasks using Fashion-MNIST dataset and show that our quantum kernel is
comparable to Gaussian kernels with the optimized hyperparameter.
