  We investigate a square-lattice architecture of superconducting transmon
qubits with inter-qubit interactions mediated by inductive couplers. Therein,
the inductive couling between the qubit and couplers is suggested to be
designed into the gradiometer form to intigimate the flux noise orginating from
the environment. Via periodically modulating the couplers,the Abelian gauge
potential, termed effective magnetic flux, can be synthesized artificially,
making the system an excellent platform for simulating two-dimensional
topological physics. In the simplest two-dimensional model, the double (or
three-leg) ladder, the staggered vortex-Meissner phase transition different
from that in the two-leg ladder can be found in the single-particle ground
state as the effective magnetic flux varies. Besides, the large coupling ratio
between the interleg and intraleg coupling strengths also makes the chiral
current resemble squeezed sinusoidal functions. If the row number is further
increased, the topological band structure anticipated at massive rows begins to
occur even for a relatively small number of rows (ten or so for the considered
parameters). This heralds a small circuit scale to observe the topological
band. The edge state in the band gap is determined by the topological Chern
number and can be calculated through integrating the Berry curvature with
respect to the first Brillouin zone. Besides, we present a systematic method on
how to measure the topological band structure based on time- and space-domain
Frourier transformation of the wave function after properly excited. The result
offers an avenue for simulating two-dimensional topological physics on the
state-of-the-art superconducting quantum chips.
