  Recently, quantum simulation of low-dimensional lattice gauge theories (LGTs)
has attracted many interests, which may improve our understanding of strongly
correlated quantum many-body systems. Here, we propose an implementation to
approximate $\mathbb{Z}_2$ LGT on superconducting quantum circuits, where the
effective theory is a mixture of a LGT and a gauge-broken term. Using matrix
product state based methods, both the ground state properties and quench
dynamics are systematically investigated. With an increase of the transverse
(electric) field, the system displays a quantum phase transition from a
disordered phase to a translational symmetry breaking phase. In the ordered
phase, an approximate Gauss law of the $\mathbb{Z}_2$ LGT emerges in the ground
state. Moreover, to shed light on the experiments, we also study the quench
dynamics, where there is a dynamical signature of the spontaneous translational
symmetry breaking. The spreading of the single particle of matter degree is
diffusive under the weak transverse field, while it is ballistic with small
velocity for the strong field. Furthermore, due to the emergent Gauss law under
the strong transverse field, the matter degree can also exhibit confinement
dynamics which leads to a strong suppression of the nearest-neighbor hopping.
Our results pave the way for simulating the LGT on superconducting circuits,
including the quantum phase transition and quench dynamics.
