  Time-periodic (Floquet) drive is a powerful method to engineer quantum phases
of matter, including fundamentally non-equilibrium states that are impossible
in static Hamiltonian systems. One characteristic example is the anomalous
Floquet insulator, which exhibits topologically quantized chiral edge states
similar to a Chern insulator, yet is amenable to bulk localization. We study
the response of this topological system to time-dependent noise, which breaks
the topologically protecting Floquet symmetry. Surprisingly, we find that the
quantized response, given by partially filling the fermionic system and
measuring charge pumped per cycle, remains quantized up to finite noise
amplitude. We trace this robust topology to an interplay between diffusion and
Pauli blocking of edge state decay, which we expect should be robust against
interactions. We determine the boundaries of the topological phase for a system
with spatial disorder numerically through level statistics, and corroborate our
results in the limit of vanishing disorder through an analytical Floquet
superoperator approach. This approach suggests an interpretation of the state
of the system as a non-Hermitian Floquet topological phase. We comment on
quantization of other topological responses in the absence of Floquet symmetry
and potential experimental realizations.
