Quantum Fisher kernel for mitigating the vanishing similarity issue

Abstract

Abstract Quantum kernel methods exploit quantum computers to calculate quantum kernels (QKs) for the use of kernel-based learning models.
Despite a potential quantum advantage of the method, the commonly used fidelity-based QK suffers from a detrimental issue, which we call the vanishing similarity issue; the exponential decay of the expectation value and the variance of the QK deteriorates implementation feasibility and trainability of the model with the increase of the number of qubits.
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 can avoid the issue when shallow alternating layered ansatzes are used.
In addition, the Fourier analysis numerically elucidates that the QFK can have the expressivity comparable to the fidelity-based QK. 
Moreover, we demonstrate synthetic classification tasks where QFK outperforms the fidelity-based QK in performance due to the absence of vanishing similarity.
These results indicate that QFK paves the way for practical applications of quantum machine learning toward possible quantum advantages.