From Krylov Complexity to Observability: Capturing Phase Space Dimension with Applications in Quantum Reservoir Computing

TL;DR

This paper introduces Krylov observability as a new measure of phase space dimension in quantum systems, demonstrating its effectiveness in quantum reservoir computing and its relation to information processing capacity.

Contribution

It proposes Krylov observability as a novel metric for phase space dimension and applies it to quantum reservoir computing, linking operator complexity with data expressivity.

Findings

Krylov observability closely mirrors information processing capacity

It enables faster computation times in quantum reservoir computing

Validates the use of operator complexity as a measure of system dynamics

Abstract

We demonstrate that time-evolved operators can construct a Krylov space to compute Operator complexity and introduce Krylov observability as a measure of effective phase space dimension in quantum systems. We test Krylov observability in the framework of quantum reservoir computing and show that it closely mirrors information processing capacity, a data-driven expressivity metric, while achieving computation times that are orders of magnitude faster. Our results validate Operator complexity and give the interpretation that data in a quantum reservoir is mapped onto the Krylov space.