  We apply a notion of quantum complexity, called "Krylov complexity", to study
the evolution of systems from integrability to chaos. For this purpose we
investigate the integrable XXZ spin chain, enriched with an integrability
breaking deformation that allows one to interpolate between integrable and
chaotic behavior. K-complexity can act as a probe of the integrable or chaotic
nature of the underlying system via its late-time saturation value that is
suppressed in the integrable phase and increases as the system is driven to the
chaotic phase. We furthermore ascribe the (under-)saturation of the late-time
bound to the amount of disorder present in the Lanczos sequence, by mapping the
complexity evolution to an auxiliary off-diagonal Anderson hopping model. We
compare the late-time saturation of K-complexity in the chaotic phase with that
of random matrix ensembles and find that the chaotic system indeed approaches
the RMT behavior in the appropriate symmetry class. We investigate the
dependence of the results on the two key ingredients of K-complexity: the
dynamics of the Hamiltonian and the character of the operator whose time
dependence is followed.

High Energy Physics - Theory;Strongly Correlated Electrons;Quantum Physics