  We numerically investigate non-Abelian braiding dynamics of vortices in
two-dimensional topological superconductors, such as $s$-wave superconductors
with Rashba spin-orbit coupling. Majorana zero modes (MZMs) hosted by the
vortices constitute a topological qubit, which offers a fundamental building
block of topological quantum computation. As the MZMs are protected by
$\mathbb{Z}_2$ invariant, however, the Majorana qubit and quantum gate
operations may be sensitive to intrinsic decoherence caused by quasiparticle
interference. Numerically simulating the time-dependent Bogoliubov-de Gennes
equation without assuming ${\it a \ priori}$ existence of MZMs, we examine
quantum noises on the unitary operators of non-abelian braiding dynamics due to
interactions with neighboring MZMs and other quasiparticle states. We
demonstrate that after the interchange of two vortices, the lowest vortex-bound
states accumulate the geometric phase $\pi/2$, and errors stemming from
dynamical phases are negligibly small, irrespective of interactions of MZMs.
Furthermore, we numerically simulate the braiding dynamics of four vortices in
two-dimensional topological superconductors, and discuss an optimal braiding
condition for realizing the high performance of non-Abelian statistics and
quantum gates operations of Majorana-based qubits.
