  Quantum kernel method is a machine learning model exploiting quantum
computers to calculate the quantum kernels (QKs) that measure the similarity
between data. Despite the potential quantum advantage of the method, the
commonly used fidelity-based QK suffers from a detrimental issue, which we call
the vanishing similarity issue; detecting the difference between data becomes
hard with the increase of the number of qubits, due to the exponential decrease
of the expectation and the variance of the QK. This implies the need to design
QKs alternative to the fidelity-based one. In this work, we propose a new class
of QKs called the quantum Fisher kernels (QFKs) that take into account the
geometric structure of the data source. We analytically and numerically
demonstrate that the QFK based on the anti-symmetric logarithmic derivatives
(ALDQFK) can avoid the issue when the alternating layered ansatzs (ALAs) are
used, while the fidelity-based QK cannot even with the ALAs. Moreover, the
Fourier analysis numerically elucidates that the ALDQFK can have expressivity
comparable to that of the fidelity-based QK. These results indicate that the
QFK paves the way for practical applications of quantum machine learning with
possible quantum advantages.
