## 1 Introduction  
Quantum dots (QDs), also known as \"artificial atoms\", are primary candidates for quantum information technologies [1]. An important structure is based on the electron spin states of a QD in the so-called Voigt geometry, which form the basic qubit, where one applies a magnetic field at a direction perpendicular to the growth axis of the QD [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. The basic energy level scheme for a negatively charged QD in the Voigt configuration (see Figure 1a) consists of four energy levels. The lower levels \(|1\rangle\) and \(|2\rangle\) are for the electron spin states, and the two upper levels are for the trion states. Naturally, the QD is in an incoherent mixture state, with equal population distribution among the two spin states \(|1\rangle\) and \(|2\rangle\). An important problem in this system is initialization, i.e., the creation of a particular electron spin state [10; 14; 15; 16; 17; 19; 20; 21; 22], which can be succeeded by the optical pumping method. Specifically, a linearly polarized optical field excites resonantly the QD system and couples one of the electron spin states to a specific trion state (let us assume it couples state \(|1\rangle\) to state \(|4\rangle\)). Then, by spontaneous emission pumping of the population, the other spin state (in our example state \(|2\rangle\)) is obtained after a few decay times. During the pumping process, the population also returns back to state \(|1\rangle\) due to spontaneous decay. This population needs to be pumped again, and this slows down the initialization process. [PERSON] et al. proposed the idea to accelerate the optical initialization process by utilizing a preferential Purcell-accelerated deexcitation towards the desired target state [14] (in our example state \(|2\rangle\)). The key to their proposal is to create an anisotropic Purcell-enhanced decay rate from the trion state to the target electron spin state by coupling the QD with a photonic structure. This idea has been explored by coupling the QD with various microphotonic and nanophotonic structures, like a semiconductor micropillar cavity, a photonic crystal nanocavity [15], a metallic prolate spheroidal nanoparticle [16], a metallic bow tie nanoantenna [17], and a planar metallo-dielectric nanostructure [19]. Recently, our group proposed the coupling of a QD in the Voigt geometry with a graphene monolayer, the basic two-dimensional material, for obtaining high fidelity electron spin initialization in short time intervals [22]. Here, we analyze a different two-dimensional material for accelerating the optical pumping process in a QD in the Voigt configuration. Specifically, the QD is placed next to a two-dimensional semiconductor, a molybdenum disulfide (MoS\({}_{2}\)) monolayer, as shown in Figure 1b. MoS\({}_{2}\) is an atomically thin, two-dimensional, direct bandgap semiconductor [23; 24; 25]. It features a bandgap in the visible, strong excitonic resonances, and high oscillator strengths. MoS\({}_{2}\) also supports exciton-polaritons, and it has been predicted theoretically that it can modify the spontaneous decay rates for nearby quantum emitters [26; 27; 28; 29]. Experiments with coupled quantum emitters (quantum dots and molecules) and MoS\({}_{2}\) layers have also appeared in the literature [30; 31; 32; 33]. The spontaneous decay rates for a quantum emitter near a MoS\({}_{2}\) layer can be anisotropic for electric dipole moments parallel or perpendicular to the monolayer [28], which makes the MoS\({}_{2}\) layer a potential candidate for accelerating the spin initialization process in a nearby QD in the Voigt geometry. Here, we first compute the Purcell factors for the QD near a MoS\({}_{2}\) monolayer. Then, we solve the density matrix equations with the modified Purcell factors and investigate the fidelity of spin initialization under optical pumping conditions. We examine two different cases of interaction of the QD with the applied optical field.