Krylov Expressivity in Quantum Reservoir Computing and Quantum Extreme Learning

Abstract

Quantum machine learning utilizes the high-dimensional space of quantum systems, attracting significant research interest. This study employs Krylov complexity to analyze task performance in quantum machine learning. We calculate the spread complexity and effective dimension of the Krylov space, introducing the effective dimension as an easy-to-compute, measurable, and upper-bounded expressivity measure. Our analysis covers quantum reservoir computers and quantum extreme learning machines, showing that increasing effective dimension correlates with improved performance. We validate this with the Lorenz cross-prediction task, observing reduced error with higher effective dimensions. Lastly, we compare the spread complexity, the effective dimension, and the fidelity as expressivity measures and show that fidelity is not suitable, while spread complexity can qualitatively explain task performance. Only the effective dimension captures the phase space accurately and exhibits the same saturation as task performance for similar evolution times.