The study of non-Hermitian systems with parity-time (PT) symmetry is a rapidly developing frontier in recent years. Experimentally, PT-symmetric systems have been realized in classical optics by balancing gain and loss, which holds great promise for novel optical devices and networks. Here we report the first experimental realization of passive PT-symmetric quantum dynamics for single photons by temporally alternating photon losses in the quantum walk (QW) interferometers. The ability to impose PT symmetry allows us to realize and investigate Floquet topological phases driven by PT-symmetric QWs. We observe topological edge states between regions with different bulk topological properties and confirm the robustness of these edge states with respect to PT-symmetry-preserving perturbations and PT-symmetry-breaking static disorder. Our results pave the way for realizing quantum mechanical PT-synthetic devices and augur exciting possibilities for exploring topological properties of non-Hermitian systems using discrete-time QWs.