Classically Approximating Variational Quantum Machine Learning with Random Fourier Features

TL;DR

This paper introduces a classical sampling method using Random Fourier Features to approximate variational quantum circuits, challenging the assumption that VQCs inherently provide exponential quantum advantage in machine learning.

Contribution

It presents a theoretical framework and experimental validation for classically approximating VQCs with RFF, narrowing the conditions for quantum advantage.

Findings

Classical approximation efficiency increases with quantum spectrum size

The number of samples needed grows favorably with quantum feature space complexity

The approach questions the potential for quantum advantage in many VQC applications

Abstract

Many applications of quantum computing in the near term rely on variational quantum circuits (VQCs). They have been showcased as a promising model for reaching a quantum advantage in machine learning with current noisy intermediate scale quantum computers (NISQ). It is often believed that the power of VQCs relies on their exponentially large feature space, and extensive works have explored the expressiveness and trainability of VQCs in that regard. In our work, we propose a classical sampling method that may closely approximate a VQC with Hamiltonian encoding, given only the description of its architecture. It uses the seminal proposal of Random Fourier Features (RFF) and the fact that VQCs can be seen as large Fourier series. We provide general theoretical bounds for classically approximating models built from exponentially large quantum feature space by sampling a few frequencies to build an equivalent low dimensional kernel, and we show experimentally that this approximation is efficient for several encoding strategies. Precisely, we show that the number of required samples grows favorably with the size of the quantum spectrum. This tool therefore questions the hope for quantum advantage from VQCs in many cases, but conversely helps to narrow the conditions for their potential success. We expect VQCs with various and complex encoding Hamiltonians, or with large input dimension, to become more robust to classical approximations.