  Phases of matter with non-trivial topological order are predicted to exhibit
a variety of exotic phenomena, such as the existence robust localized bound
states in 1D systems, and edge states in 2D systems, which are expected to
display spin-helicity, immunity to back-scattering, and weak anti-localization.
In this Letter, we present an experimental observation of topological
structures generated via the controlled implementation of two consecutive
non-commuting rotations in photonic discrete-time quantum walks. The second
rotation introduces valley-like Dirac points in the system, allowing to create
the non-trivial topological pattern. By choosing specific values for the
rotations, it is possible to coherently drive the system between topological
sectors characterized by different topological invariants. We probe the full
topological landscape, demonstrating the emergence of localized bound states
hosted at the topological boundaries, and the existence of extremely localized
or delocalized non-Gaussian quantum states. Our results pave the way for the
study of valley-based electronics and applications of topological mechanisms in
robust optical-device engineering.
