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

Integrability to chaos transition through Krylov approach for state evolution