  Recently, a hypothesis on the complexity growth of unitarily evolving
operators was presented. This hypothesis states that in generic, non-integrable
many-body systems the so-called Lanczos coefficients associated with an
autocorrelation function grow asymptotically linear, with a logarithmic
correction in one-dimensional systems. In contrast, the growth is expected to
be slower in integrable or free models. In the paper at hand, we numerically
test this hypothesis for a variety of exemplary systems, including 1d and 2d
Ising models as well as 1d Heisenberg models. While we find the hypothesis to
be practically fulfilled for all considered Ising models, the onset of the
hypothesized universal behavior could not be observed in the attainable
numerical data for the Heisenberg model. The proposed linear bound on operator
growth eventually stems from geometric arguments involving the locality of the
Hamiltonian as well as the lattice configuration. We investigate such a
geometric bound and find that it is not sharply achieved for any considered
model.
