The study of non-Hermitian systems with parity-time (PT) symmetry is a rapidly developing frontier. Realized in recent experiments, PT-symmetric classical optical systems with balanced gain and loss hold great promise for future applications. Here we report the experimental realization of passive PT-symmetric quantum dynamics for single photons by temporally alternating photon losses in the quantum walk interferometers. The ability to impose PT symmetry allows us to realize and investigate Floquet topological phases driven by PT-symmetric quantum walks. 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 contribute towards the realization of quantum mechanical PT-synthetic devices and suggest exciting possibilities for the exploration of the topological properties of non-Hermitian systems using discrete-time quantum walks.