From Krylov complexity to observability: Capturing phase space dimension with applications in quantum reservoir computing

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 are mapped onto the Krylov space.