In this work, we investigate the dynamical process of an open quantum system under the action of quantum cyclic $n$-switch in generalized amplitude-damping channels and depolarizing channels. The quantum cyclic $n$-switch operation is introduced by an external control system. Our results show that the quantum switch can regulate the dynamical behavior of the system dynamics and achieve non-Markovian speedup dynamics. Increasing the dimension of the control system can significantly enhance the evolution speed of the system's non-Markovian dynamics. For the generalized amplitude-damping channels, when the influence of thermal noise is intense, the greater the quantum cyclic $n$-switch is, the lower the initial coherence is, and the more conducive it is to the acceleration of the quantum state evolution speed. In the depolarizing channels, under the action of quantum cyclic $n$-switch, the evolution speed of the system dynamics is directly related to the non-Markovianity and is independent of the initial coherence of the system.

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• Open quantum systems & decoherence
• Quantum information processing