  Owing to the presence of exceptional points (EPs), non-Hermitian (NH) systems
can display intriguing topological phenomena without Hermitian analogs.
However, experimental characteristics of exceptional topological invariants
have been restricted to second-order EPs (EP2s) in classical or semiclassical
systems. We here propose an NH multi-qubit model with higher-order EPs, each of
which is underlain by a multifold-degenerate multipartite entangled eigenstate.
We implement the three-qubit model by controllably coupling a superconducting
qubit to two microwave resonators, one serving as a Hermitian photonic qubit
while the other as an NH qubit. We experimentally quantify the topological
invariant for an EP3, by mapping out the complex eigenspectra along a loop
surrounding this EP3 in the parameter space. The nonclassicality of the
realized topology is manifested by the observed quantum correlations in the
corresponding eigenstates. Our results extend research of exceptional topology
to fully quantum-mechanical models with multi-partite entangled eigenstates. We
further demonstrate the non-reciprocal transmission of a single photon, during
which the photon is nonlocally shared by three individual elements.
