  The quantum Hall effect, fundamental in modern condensed matter physics,
continuously inspires new theories and predicts emergent phases of matter. Here
we experimentally demonstrate three types of Chern insulators with synthetic
dimensions on a programable 30-qubit-ladder superconducting processor. We
directly measure the band structures of the 2D Chern insulator along synthetic
dimensions with various configurations of Aubry-Andr\'e-Harper chains and
observe dynamical localisation of edge excitations. With these two signatures
of topology, our experiments implement the bulk-edge correspondence in the
synthetic 2D Chern insulator. Moreover, we simulate two different bilayer Chern
insulators on the ladder-type superconducting processor. With the same and
opposite periodically modulated on-site potentials for two coupled chains, we
simulate topologically nontrivial edge states with zero Hall conductivity and a
Chern insulator with higher Chern numbers, respectively. Our work shows the
potential of using superconducting qubits for investigating different
intriguing topological phases of quantum matter.
