Quantum walks (QWs) provide a powerful tool as a quantum simulator to study topological phases. However, some interesting topological phenomena in the two dimensional (2D) QW that do not exist in the one dimensional case, e.g., the edge-state-enhanced transport, have not been demonstrated experimentally. Here we have observed 2D topological bound states with vanishing Chern numbers and confirmed the robustness of these bound states with respect to perturbations and disorder, which go beyond what has been known in static systems and are unique to periodically driven systems. Our studies open up an avenue to explore topological properties in multidimensional QWs.
We study the topological properties of one-dimensional discrete-time quantum walks with Fibonacci quasiperiodic modulation. Spectral analysis under open boundary conditions reveals isolated edge modes that coexist at both zero and
π energies within bulk gaps. Using the mean chiral displacement (MCD) as a dynamical bulk probe, we obtain a fractal, butterfly-like phase diagram, indicating a nontrivial bulk topology. However, since the MCD involves wave-packet averaging, its direct correspondence with boundary-localized states remains ambiguous. To resolve this issue, we determine the integer topological invariant by employing the Schur function formalism, a scattering-based approach defined on the unit disk. This analysis identifies a topological phase with winding number
W=2 inside the ``wings'' of the butterfly diagram, demonstrating that the coexistence of zero- and
π-energy edge modes is an intrinsic bulk property. Moreover, we find that the topological index crucially depends on the phason degree of freedom: the winding number can take values of
W=0,
2, and even
4 depending on the local surface termination. This behavior originates from the surface impedance, which yields an effective winding number ranges from complete masking by reflective boundaries (
W=0) to enhancement through surface resonances (
W=4). Finally, we show that although individual surface terminations lead to anisotropic phase diagrams, their ensemble average restores the overall geometric structure observed in the MCD phase diagram. Our results establish a complete bulk-edge correspondence in quasiperiodic quantum walks and provide a guiding principle for surface topological engineering, where edge transport can be controlled purely through boundary manipulation.