  We investigate many-body dynamics where the evolution is governed by unitary
circuits through the lens of `Krylov complexity', a recently proposed measure
of complexity and quantum chaos. We extend the formalism of Krylov complexity
to unitary circuit dynamics and focus on Floquet circuits arising as the
Trotter decomposition of Hamiltonian dynamics. For short Trotter steps the
results from Hamiltonian dynamics are recovered, whereas a large Trotter step
results in different universal behavior characterized by the existence of local
maximally ergodic operators: operators with vanishing autocorrelation
functions, as exemplified in dual-unitary circuits. These operators exhibit
maximal complexity growth, act as a memoryless bath for the dynamics, and can
be directly probed in current quantum computing setups. These two regimes are
separated by a crossover in chaotic systems. Conversely, we find that free
integrable systems exhibit a nonanalytic transition between these different
regimes, where maximally ergodic operators appear at a critical Trotter step.
