  Identifying topological properties is a major challenge because, by
definition, topological states do not have a local order parameter. While a
generic solution to this challenge is not available yet, a broad class of
topological states, namely symmetry-protected topological (SPT) states, can be
identified by distinctive degeneracies in their entanglement spectrum. Here, we
propose and realize two complementary protocols to probe these degeneracies
based on, respectively, symmetry-resolved entanglement entropies and
measurement-based computational algorithms. The two protocols link quantum
information processing to the classification of SPT phases of matter. They
invoke the creation of a cluster state, and are implemented on an IBM quantum
computer. The experimental findings are compared to noisy simulations, allowing
us to study the stability of topological states to perturbations and noise.
