  The Kitaev model realizes a quantum spin liquid where the spin excitations
are fractionalized into itinerant Majorana fermions and localized
$\mathbb{Z}_2$ vortices. Quantum entanglement between the fractional
excitations can be utilized for decoherence-free topological quantum
computation. Of particular interest is the anyonic statistics realized by
braiding the vortex excitations under a magnetic field. Despite the promising
potential, the practical methodology for creation and control of the vortex
excitations remains elusive thus far. Here we theoretically propose how one can
create and move the vortices in the Kitaev spin liquid. We find that the
vortices are induced by a local modulation of the exchange interaction;
especially, the local Dzyaloshinskii-Moriya (symmetric off-diagonal)
interaction can create vortices most efficiently in the (anti)ferromagnetic
Kitaev model, as it effectively flips the sign of the Kitaev interaction. We
test this idea by performing the {\it ab initio} calculation for a candidate
material $\alpha$-RuCl$_3$ through the manipulation of the ligand positions
that breaks the inversion symmetry and induces the local Dzyaloshinskii-Moriya
interaction. We also demonstrate a braiding of vortices by adiabatically and
successively changing the local bond modulations.
