  We investigate how the Kitaev spin liquid state described by Majorana
fermions coupled with $Z_2$ gauge fields is affected by non-Kitaev interactions
which exist in real candidate materials of the Kitaev magnet. It is found that
the off-diagonal exchange interaction referred to as the $\Gamma'$ term
dramatically changes the Majorana band structure in the case of the
antiferromagnetic Kitaev interaction, and gives rise to a topological phase
transition from a non-Abelian topological phase with the Chern number equal to
$\pm 1$ to an Abelian phase with the Chern number equal to $\pm 2$, in which
the $Z_{2}$ vortices behave as Abelian anyons. On the other hand, other
non-Kitaev interactions such as the Heisenberg interaction and the $\Gamma$
term, only affect the bandwidth of the Majorana band as long as the spin liquid
state is not destabilized.
