On fundamental aspects of quantum extreme learning machines

TL;DR

This paper analyzes the expressivity and scalability of Quantum Extreme Learning Machines (QELMs), revealing fundamental limitations related to Fourier frequency representation and the impact of quantum reservoir properties on their effectiveness.

Contribution

It provides a Fourier series decomposition of QELMs, identifies key factors limiting their expressivity and scalability, and highlights the effects of randomness and noise on their performance.

Findings

Fourier frequencies depend on data encoding scheme.

Expressivity limited by number of Fourier frequencies and observables.

Random quantum reservoirs can cause exponential concentration, hindering scalability.

Abstract

Quantum Extreme Learning Machines (QELMs) have emerged as a promising framework for quantum machine learning. Their appeal lies in the rich feature map induced by the dynamics of a quantum substrate - the quantum reservoir - and the efficient post-measurement training via linear regression. Here we study the expressivity of QELMs by decomposing the prediction of QELMs into a Fourier series. We show that the achievable Fourier frequencies are determined by the data encoding scheme, while Fourier coefficients depend on both the reservoir and the measurement. Notably, the expressivity of QELMs is fundamentally limited by the number of Fourier frequencies and the number of observables, while the complexity of the prediction hinges on the reservoir. As a cautionary note on scalability, we identify four sources that can lead to the exponential concentration of the observables as the system size grows (randomness, hardware noise, entanglement, and global measurements) and show how this can turn QELMs into useless input-agnostic oracles. In particular, our result on the reservoir-induced concentration strongly indicates that quantum reservoirs drawn from a highly random ensemble make QELM models unscalable. Our analysis elucidates the potential and fundamental limitations of QELMs, and lays the groundwork for systematically exploring quantum reservoir systems for other machine learning tasks.