  The stability of the topological order phase induced by the $Z_3$ Kitaev
model, which is a candidate for fault-tolerant quantum computation, against the
local order phase induced by the 3-State Potts model is studied. We show that
the low energy sector of the Kitaev-Potts model is mapped to the Potts model in
the presence of transverse magnetic field. Our study relies on two high-order
series expansion based on continuous unitary transformations in the limits of
small- and large-Potts couplings as well as mean-field approximation. Our
analysis reveals that the topological phase of the $Z_3$ Kitaev model breaks
down to the Potts model through a first order phase transition. We capture the
phase transition by analysis of the ground state energy, one-quasiparticle gap
and geometric measure of entanglement.
