Signal processing in continuous syndrome of the three qubit quantum error correction code*

We develop several protocols for quantum error correction of three qubit code with continuous syndrome measurements, including boxcar filter, double exponential smoothing and Bayesian filter. These protocols are realistic and efficient enough for rapid experimental realization of quantum error correction code. We also discuss the continuous feedback protocol based on Bayesian filtering, which requires real time correction. It is found numerically that for fast measurements delaying the correction until the qubit recall time is almost of the same fidelity as feeding back with pulses in real time. Thus with Bayesian filtering waiting until the end of the run will save considerable resources without losing too much fidelity.