  A general framework for analyzing topology of quantum channels of
single-particle systems is developed to find a class of genuinely dynamical
topological phases that can be realized by means of discrete quantum feedback
control. We provide a symmetry classification of quantum channels by
identifying ten symmetry classes of discrete quantum feedback control with
projective measurements. We construct various types of topological feedback
control by using topological Maxwell's demons that achieve robust
feedback-controlled chiral or helical transport against noise and decoherence.
Topological feedback control thus offers a versatile tool for creating and
controlling nonequilibrium topological phases in open quantum systems that are
distinct from non-Hermitian and Lindbladian systems and should provide a
guiding principle for topology-based design of quantum feedback control.
