  We numerically investigate the two-dimensional q-state quantum Potts model on
the infinite square lattice by using the infinite projected entangled-pair
state (iPEPS) algorithm. We show that the quantum fidelity, defined as an
overlap measurement between an arbitrary reference state and the iPEPS
groundstate of the system, can detect q-fold degenerate groundstates for the Zq
broken-symmetry phase. Accordingly, a multiple-bifurcation of the quantum
groundstate fidelity is shown to occur as the transverse magnetic field varies
from the symmetry phase to the broken-symmetry phase, which means that a
multiple-bifurcation point corresponds to a critical point. A (dis-)continuous
behavior of quantum fidelity at phase transition points characterizes a
(dis-)continuous phase transition. Similar to the characteristic behavior of
the quantum fidelity, the magnetizations, as order parameters, obtained from
the degenerate groundstates exhibit multiple bifurcation at critical points.
Each order parameter is also explicitly demonstrated to transform under the
subgroup of the Zq symmetry group. We find that the q-state quantum Potts model
on the square lattice undergoes a discontinuous (first-order) phase transition
for q = 3 and q = 4, and a continuous phase transition for q = 2 (the 2D
quantum transverse Ising model).
