Engineering Quantum Reservoirs through Krylov Complexity, Expressivity and Observability

TL;DR

This paper investigates how Krylov-based information measures can explain quantum reservoir computing performance, highlighting Krylov observability as a fast and effective predictor, especially under undersampling conditions.

Contribution

The study introduces and compares Krylov expressivity and observability, demonstrating that Krylov observability closely matches task performance and is computationally more efficient.

Findings

Krylov fidelity and spread complexity only explain short-term task performance.

Krylov observability correlates strongly with information processing capacity.

Krylov observability is three orders of magnitude faster to compute.

Abstract

This study employs Krylov-based information measures to understand task performance in quantum reservoir computing, a sub-field of quantum machine learning. In our study we show that fidelity and spread complexity can only explain the task performance for short time evolutions of the quantum systems. We then discuss two measures, Krylov expressivity and Krylov observability, and compare them to task performance and the information processing capacity. Our results show that Krylov observability exhibits almost identical behavior to information processing capacity, while being three orders of times faster to compute. In the case when the system is undersampled Krylov observability best captures the behavior of the task performance.