## Ensemble average of fidelity and langle vec{J} angle over 2000 trajectories of the single-qutrit dephasing model under different UDD control sequences

**Figure 1.** Ensemble average of fidelity and \langle \vec{J}\rangle over 2000 trajectories of the single-qutrit dephasing model under different UDD control sequences. We choose the initial state |\psi _0\rangle =\frac{1}{\sqrt{3}}(|0\rangle +|1\rangle +|2\rangle ) and the environment memory parameter γ = 1. The black solid line represents the fidelity, the red dotted line denotes 〈*J _{x}*〉, the green dashed line denotes 〈

*J*〉 and the blue dot–dashed line denotes 〈

_{y}*J*〉.

_{z}**Abstract**

In this paper, we use the quantum state diffusion (QSD) equation to implement the Uhrig dynamical decoupling to a three-level quantum system coupled to a non-Markovian reservoir comprising of infinite numbers of degrees of freedom. For this purpose, we first reformulate the non-Markovian QSD to incorporate the effect of the external control fields. With this stochastic QSD approach, we demonstrate that an unknown state of the three-level quantum system can be universally protected against both coloured phase and amplitude noises when the control-pulse sequences and control operators are properly designed. The advantage of using non-Markovian QSD equations is that the control dynamics of open quantum systems can be treated exactly without using Trotter product formula and be efficiently simulated even when the environment is comprised of infinite numbers of degrees of freedom. We also show how the control efficacy depends on the environment memory time and the designed time points of applied control pulses.