  The maximal evolution speed of any quantum system can be expressed by the
quantum speed limit time. In this paper, we consider a model in which the
system has a correlation with the environment. The influence of the initial
correlation between the system and environment on the quantum speed limit is
investigated. It is shown that the appearance of non-Markovianity effects
causes the speedup of quantum evolution. Moreover, we demonstrate the
dependence of quantum dynamical speedup on the quantum coherence of the
correlated initial state.
