Phase-space negativity as a computational resource for quantum kernel methods

TL;DR

This paper investigates how phase-space negativity in quantum states influences the classical simulability of quantum kernel methods, highlighting negativity as a key resource for quantum advantage in machine learning.

Contribution

It provides sufficient conditions based on phase-space properties for classical estimation of quantum kernels, emphasizing negativity as a computational resource.

Findings

Negative volume, non-classical depth, and excess range are signatures of phase-space negativity.

Loss affects the efficiency of classical simulation of quantum kernels.

Negativity in phase-space distributions is essential for quantum advantage.

Abstract

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 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. These conditions are based on phase-space properties of data-encoding quantum states associated with the quantum kernels: negative volume, non-classical depth, and excess range, which are shown to be three signatures of phase-space negativity. 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.