Operator Growth in Krylov Space

Investigating the physics of many-body quantum systems requires facing the irreducible complexity of their dynamics. This feature can be quantified through the operator growth undergone by the typical observables of the system under unitary dynamics. As a result of the time evolution, the initial information is delocalized and cannot be recovered through local measurements, a process named information scrambling. In this chapter, we introduce a measure of operator growth known as Krylov complexity. The evolution in Krylov space is mapped to a hopping problem on a one-dimensional lattice, and the complexity of the operator is quantified by its mean position. We further discuss how the presence of symmetries in the generator affects the dynamics and determines the complexity growth. By doing so, we investigate how complexity is related to another prominent feature of quantum dynamics, namely, to the notion of chaos.