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 systems 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 qubits 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.
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