  The complexity of quantum states under dynamical evolution can be
investigated by studying the spread with time of the state over a pre-defined
basis. It is known that this complexity is minimised by choosing the Krylov
basis, thus defining the spread complexity. We study the dynamics of spread
complexity for quantum maps using the Arnoldi iterative procedure. The main
illustrative quantum many-body model we use is the periodically kicked Ising
spin-chain with non-integrable deformations, a chaotic system where we look at
both local and non-local interactions. In the various cases we find distinctive
behaviour of the Arnoldi coefficients and spread complexity for regular vs.
chaotic dynamics: suppressed fluctuations in the Arnoldi coefficients as well
as larger saturation value in spread complexity in the chaotic case. We compare
the behaviour of the Krylov measures with that of standard spectral diagnostics
of chaos. We also study the effect of changing the driving frequency on the
complexity saturation.
