  The no-knowledge quantum feedback was proposed by Szigeti et al., Phys. Rev.
Lett. 113, 020407 (2014), as a measurement-based feedback protocol for
decoherence suppression for an open quantum system. By continuously measuring
environmental noises and feeding back controls on the system, the protocol can
completely reverse the measurement backaction and therefore suppress the
system's decoherence. However, the complete decoherence cancellation was shown
only for the instantaneous feedback, which is impractical in real experiments.
Therefore, in this work, we generalize the original work and investigate how
the decoherence suppression can be degraded with unavoidable delay times, by
analyzing non-Markovian average dynamics. We present analytical expressions for
the average dynamics and numerically analyze the effects of the delayed
feedback for a coherently driven two-level system, coupled to a bosonic bath
via a Hermitian coupling operator. We also find that, when the qubit's unitary
dynamics does not commute with the measurement and feedback controls, the
decoherence rate can be either suppressed or amplified, depending on the delay
time.
