  Quantum speed limit time (QSLT) can be used to characterize the intrinsic
minimal time interval for a quantum system evolving from an initial state to a
target state. We investigate the QSLT of the open system in Schwarzschild
space-time. We show that, in some typical noisy channels,the Hawking effect can
be beneficial to the evolution of the system. For an initial entangled state,
the evolution speed of the system can be enhanced in the depolarizing, bit
flip, and bit-phase flip channels as the Hawking temperature increases, which
are in sharp contrast to the phase flip channel. Moreover, the optimal initial
entanglement exists in other noise channels except the phase flip channel,
which minimizes the QSLT of the system and thus leads to the maximum evolution
speed of the system.
