  The complexity of quantum evolutions can be understood by examining their
dispersion in a chosen basis. Recent research has stressed the fact that the
Krylov basis is particularly adept at minimizing this dispersion [V.
Balasubramanian et al, Physical Review D 106, 046007 (2022)]. This property
assigns a central role to the Krylov basis in the investigation of quantum
chaos. Here, we delve into the transition from integrability to chaos using the
Krylov approach, employing an Ising spin chain and a banded random matrix model
as our testing models. Our findings indicate that both the saturation of Krylov
complexity and the dispersion of the Lanczos coefficients can exhibit a
significant dependence on the initial condition. However, both quantities can
gauge dynamical quantum chaos with a proper choice of the initial state.
