Simulation of Topological X-Gates via Braiding of Majorana Zero Modes in an Interacting Quantum Dot System

Motivated by recent advances in quantum-dot platforms, we propose an experimentally feasible setup for implementing topological $\sqrt{X}$ and $X$ quantum gates in an interacting $Y$-shaped quantum-dot array. The proposed architecture enables initialization, braiding, and readout of a Majorana-zero-mode (MZM)-based qubit controlled by gate-tunable potentials. Using many-body time-dependent exact diagonalization, we analyze the braiding and fusion dynamics of MZMs in the presence of nearest-neighbor Coulomb interactions and pairing disorder. We compute diabatic errors, braiding fidelity, and the time- and space-resolved electron and hole components of the local density of states to monitor the braiding process. Our results show that even weak interactions or pairing disorder cause oscillations in the braiding fidelity, thereby setting an upper limit on the braiding speed. Finally, we show that comparing fusion outcomes before and after braiding provides an experimentally accessible signature of the non-Abelian nature of MZMs in quantum-dot systems.