We　investigate　a　square-lattice　architecture　of　superconducting　transmon　qubits　with　interqubit　interactions　mediated　by　inductive　couplers.　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.　First,　we　focus　on　the　three-leg　ladder　which　only　has　three　rows　and　investigate　the　chiral　dynamics　therein　for　the　single-particle　ground　state　when　the　effective　magnetic　flux　varies.　We　find　what　we　call　the　＂staggered　vortex-Meissner　phase　transition,＂　where　the　vortex　number　can　typically　stagger　a　few　times　between　one　(defined　as　＂vortex　phase＂)　and　larger　integers　(defined　as　＂Meissner　phase＂)　when　the　effective　magnetic　flux　changes　between　-pi　and　pi.　This　phenomenon,　actually　not　a　phase　transition　by　definition,　is　quite　different　from　the　vortex-Meissner　phase　transition　in　the　two-leg　ladder　that,　in　contrast,　possesses　only　two　rows　and　is　usually　treated　as　the　quasi-two-dimensional　model.　Also,　we　find　that　the　chiral　current　relies　on　the　effective　magnetic　flux　according　to　a　squeezed　sinusoidal　function.　Both　the　staggering　of　the　vortex　number　and　squeezing　of　the　chiral　current　can　be　controllable　by　the　coupling　ratio,　which　is　defined　by　the　coupling-strength　ratio　between　the　column　direction　and　row　direction.　Second,　we　continue　to　increase　the　number　of　rows　beyond　three,　and　the　topological　band　structure　to　be　anticipated　at　an　infinite　number　of　rows　begins　to　occur　even　for　a　relatively　small　number　(10　or　so　for　typical　parameters)　of　rows.　This　heralds　a　small　circuit　scale　enough　to　observe　the　topological　band.　The　behavior　of　the　edge　state　in　the　band　gap　can　be　interpreted　by　the　topological　Chern　number,　which　can　be　calculated　through　integrating　the　Berry　curvature　with　respect　to　the　first　Brillouin　zone.　Last,　we　present　a　systematic　method　on　how　to　measure　the　topological　band　structure　based　on　time-　and　space-domain　Fourier　transformation　of　the　wave　function　after　properly　exciting　the　qubits,　which　should　be　helpful　for　comprehensively　analyzing　the　topological　physics　since　all　the　topological　properties　are　mainly　contained　in　the　band　structure.　Our　results　offer　an　avenue　for　simulating　two-dimensional　topological　physics　on　the　state-of-the-art　superconducting　quantum　chips.