We compare the performance of continuous coherent feedback, implemented using an ideal single-qubit controller, to that of continuous measurement-based feedback for the task of controlling the state of a single qubit. Here the basic dynamical resource is the ability to couple the system to a traveling-wave field (for example, a transmission line) via a system observable, and the fundamental limitation is the maximum rate that is available for this coupling. We focus on the question of the best achievable control given ideal controllers. To obtain a fair comparison we acknowledge that the amplification involved in measurement-based control allows the controller to use macroscopic fields to apply feedback forces to the system, so it is natural to allow these feedback forces to be much larger than the mesoscopic coupling to the transmission line that mediates both the measurement for measurement-based control and the coupling to the mesoscopic controller for coherent control. Interestingly our numerical results indicate that under the above platform for comparison, coherent feedback is able to exactly match the performance of measurement-based feedback given ideal controllers. We also discuss various properties of, and control mechanisms for, coherent feedback networks.
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