  The detection of topological phases of matter becomes a central issue in
recent years. Conventionally, the realization of a specific topological phase
in condensed matter physics relies on probing the underlying surface band
dispersion or quantum transport signature of a real material, which may be
imperfect or even absent. On the other hand, quantum simulation offers an
alternative approach to directly measure the topological invariant on a
universal quantum computer. However, experimentally demonstrating
high-dimensional topological phases remains a challenge due to the technical
limitations of current experimental platforms. Here, we investigate the
three-dimensional topological insulators in the AIII (chiral unitary) symmetry
class which yet lack experimental realization. Using the nuclear magnetic
resonance system, we experimentally demonstrate their topological properties,
where a dynamical quenching approach is adopted and the dynamical bulk-boundary
correspondence in the momentum space is observed. As a result, the topological
invariants are measured with high precision on the band-inversion surface,
exhibiting robustness to the decoherence effect. Our work paves the way towards
the quantum simulation of topological phases of matter in higher dimensions and
more complex systems through controllable quantum phases transitions.
