  Using an efficient one and two qubit gate simulator, operating on graphical
processing units, we investigate ergodic properties of a quantum Ising spin 1/2
model on a two dimensional lattice, which is periodically driven by a
$\delta$-pulsed transverse magnetic field. We consider three different
dynamical properties: (i) level density and (ii) level spacing distribution of
the Floquet quasienergy spectrum, as well as (iii) time-averaged
autocorrelation function of components of the magnetization. Varying the
parameters of the model, we found transitions between ordered (non ergodic) and
quantum chaotic (ergodic) phases, but the transitions between flat and non-flat
spectral density {\em do not} correspond to transitions between ergodic and
non-ergodic local observables. Even more surprisingly, we found nice agreement
of level spacing distribution with the Wigner surmise of random matrix theory
for almost all values of parameters except where the model is essentially
noninteracting, even in the regions where local observables are not ergodic or
where spectral density is non-flat. These findings put in question the
versatility of the interpretation of level spacing distribution in many-body
systems and stress the importance of the concept of locality.
