Observing topological zero modes on a 41-qubit superconducting processor

Programming different exotic topological phases of quantum matter on a noisy intermediate-scale quantum (NISQ) processor represents a practical advantage of quantum simulation. The capabilities to perform accurate quantum operations and addressable readouts on a hybrid analogue-digital quantum simulator enable to program and characterise emergent topological states that are very challenging in real materials. Here, we develop a one-dimensional (1D) 43-qubit superconducting quantum processor, named as Chuang-tzu, to engineer different types of Aubry-Andr$\acute{\text{e}}$-Harper (AAH) models. By engineering instances of diagonal AAH models using up to 41 qubits, we experimentally demonstrate the Hofstadter butterfly energy spectrum, predicted for a two-dimensional (2D) electron gas under a perpendicular magnetic field. In addition, we apply Floquet engineering to simulate gapless commensurate off-diagonal AAH models, directly measure the band structures and witness the localisation of a boundary excitation during its quantum walks (QWs). With the bulk-edge correspondence, we verify the existence of topological zero-energy edge modes in the gapless commensurate AAH models, which have never been experimentally observed before. Remarkably, the qubit number over 40 in our quantum processor is large enough to capture the substantial topological features of a 1D quantum many-body system from its complex band structure, including Dirac points, the energy gap's closing, the difference between the even and odd number of sites, and the distinction between edge and bulk states. Using a superconducting quantum processor assisted by highly controllable Floquet engineering, our results establish a versatile hybrid simulation approach to exploring quantum topological many-body systems in the NISQ era.