  Quantum simulation of different exotic topological phases of quantum matter
on a noisy intermediate-scale quantum (NISQ) processor is attracting growing
interest. Here, we develop a one-dimensional 43-qubit superconducting quantum
processor, named as Chuang-tzu, to simulate and characterize emergent
topological states. By engineering diagonal
Aubry-Andr$\acute{\mathrm{e}}$-Harper (AAH) models, we experimentally
demonstrate the Hofstadter butterfly energy spectrum. Using Floquet
engineering, we verify the existence of the topological zero modes in the
commensurate off-diagonal AAH models, which have never been experimentally
realized before. Remarkably, the qubit number over 40 in our quantum processor
is large enough to capture the substantial topological features of a quantum
system from its complex band structure, including Dirac points, the energy
gap's closing, the difference between even and odd number of sites, and the
distinction between edge and bulk states. Our results establish a versatile
hybrid quantum simulation approach to exploring quantum topological systems in
the NISQ era.
