  We investigate the robustness of a dynamical phase transition against quantum
fluctuations by studying the impact of a ferromagnetic nearest-neighbour spin
interaction in one spatial dimension on the non-equilibrium dynamical phase
diagram of the fully-connected quantum Ising model. In particular, we focus on
the transient dynamics after a quantum quench and study the pre-thermal state
via a combination of analytic time-dependent spin-wave theory and numerical
methods based on matrix product states. We find that, upon increasing the
strength of the quantum fluctuations, the dynamical critical point fans out
into a chaotic dynamical phase within which the asymptotic ordering is
characterised by strong sensitivity to the parameters and initial conditions.
We argue that such a phenomenon is general, as it arises from the impact of
quantum fluctuations on the mean-field out of equilibrium dynamics of any
system which exhibits a broken discrete symmetry.
