  We propose the $(3+1)$-dimensional $\mathbb{Z}_3$ lattice gauge theory
coupled with the 2-flavor Wilson-Dirac fermion as a toy model for studying
quantum chromodynamics (QCD) at nonzero density. We study its phase diagram in
the space of the lattice gauge couplings $g^2$ and the quark chemical
potentials $\mu$ and discuss the similarity and difference compared with
anticipated behaviors of actual QCD. This model also provides a testing ground
for various algorithms of the numerical Hamiltonian formalism as its Hilbert
space is finite-dimensional in a finite box.
