  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. 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.
