  Simulating the topological phases of matter in synthetic quantum simulators
is a topic of considerable interest. Given the universality of digital quantum
simulators, the prospect of digitally simulating exotic topological phases is
greatly enhanced. However, it is still an open question how to realize digital
quantum simulation of topological phases of matter. Here, using common single-
and two-qubit elementary quantum gates, we propose and demonstrate an approach
to design topologically protected quantum circuits on the current generation of
noisy quantum processors where spin-orbital coupling and related topological
matter can be digitally simulated. In particular, a low-depth topological
quantum circuit is performed on both IBM and Rigetti quantum processors. In the
experiments, we not only observe but also distinguish the 0 and $\pi$ energy
topological edge states by measuring qubit excitation distribution at the
output of the circuits.
