  Quantum kernel methods are a proposal for achieving quantum computational
advantage in machine learning. They are based on a hybrid classical-quantum
computation where a function called the quantum kernel is estimated by a
quantum device while the rest of the computation is performed classically.
Quantum advantages may be achieved through this method only if the quantum
kernel function cannot be estimated efficiently on a classical computer. In
this paper, we provide sufficient conditions for the efficient classical
estimation of quantum kernel functions for bosonic systems. Specifically, we
show that if the negativity in the phase-space quasi-probability distributions
of data-encoding quantum states associated with the quantum kernel scales at
most polynomially with the size of the quantum circuit, then the kernel
function can be estimated efficiently classically. We consider quantum optical
examples involving linear-optical networks with and without adaptive
non-Gaussian measurements and investigate the effects of loss on the efficiency
of the classical simulation. Our results underpin the role of the negativity in
phase-space quasi-probability distributions as an essential resource in quantum
machine learning based on kernel methods.
