  Quantum walks constitute a versatile platform for simulating transport
phenomena on discrete graphs including topological material properties while
providing a high control over the relevant parameters at the same time. To
experimentally access and directly measure the topological invariants of
quantum walks we implement the scattering scheme proposed by Tarasinski et
al.[Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk
experiment. The tunable coin operation provides opportunity to reach distinct
topological phases, and accordingly to observe the corresponding topological
phase transitions. The ability to read-out the position and the coin state
distribution, complemented by explicit interferometric sign measurements,
allowed the reconstruction of the scattered reflection amplitudes and thus the
computation of the associated bulk topological invariants. As predicted we also
find localised states at the edges between two bulks belonging to different
topological phases. In order to analyse the impact of disorder we have measured
invariants of two different types of disordered samples in large ensemble
measurements, demonstrating their constancy in one disorder regime and a
continuous transition with increasing disorder strength for the second disorder
sample.
