We investigate the roles of the relativistic effect on the speed of evolution of a quantum system coupled with amplitude damping channels. We find that the relativistic effect speed-up the quantum evolution to a uniform evolution speed of open quantum systems for the damping parameter $p_{tau}lesssim p_{tau_{c0}}.$ Moreover, we point out a non-monotonic behavior of the quantum speed limit time (QSLT) with acceleration in the damping limit $p_{tau_{c0}}lesssim p_{tau}lesssim p_{tau_{c1}},$ where the relativistic effect first speed-up and then slow down the quantum evolution process of the damped system. For the damping strength $p_{tau_{c1}}lesssim p_{tau}$, we observe a monotonic increasing behavior of QSLT, leads to slow down the quantum evolution of the damped system. In addition, we examine the roles of the relativistic effect on the speed limit time for a system coupled with the phase damping channels.
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