  Quantum theory sets a bound on the minimal time evolution between initial and
target states. This bound is called as quantum speed limit time. It is used to
quantify maximal speed of quantum evolution. The quantum evolution will be
faster, if quantum speed limit time decreases. In this work, we study the
quantum speed limit time of a quantum state in the presence of disturbance
effects in an environment. We use the model which is provided by Masashi Ban in
\href{https://doi.org/10.1103/PhysRevA.99.012116}{Phys. Rev. A 99, 012116
(2019)}. In this model two quantum systems $\mathcal{A}$ and $\mathcal{S}$
interact with environment sequentially. At first, quantum system $\mathcal{A}$
interacts with the environment $\mathcal{E}$ as an auxiliary system then
quantum system $\mathcal{S}$ interacts with disturbed environment immediately.
In this work, we consider dephasing coupling with two types of environment with
different spectral density: Ohmic and Lorentzian. We observe that,
non-Markovian effects will be appear in the dynamics of quantum system
$\mathcal{S}$ by the interaction of quantum system $\mathcal{A}$ with the
environment. Given the fact that quantum speed limit time reduces due to
non-Markovian effects, we show that disturbance effects will reduce the quantum
speed limit time.
