Robustness of concatenated five-qubit QEC protocol for noise with fluctuation

In quantum error correction, the description of noise channel cannot be completely accurate, and fluctuation always appears in noise channel. In some typical cases, it is shown that the concatenated five-qubit QEC protocol is an efficient and robust protocol against noise channels with fluctuation, and our numerical results indicate that both standard deviation of effective channel fidelity and standard deviations of diagonal elements of quantum process matrix (Pauli Form) decay exponentially as the concatenated level increases, and attenuation ratios of the standard deviations are almost only dependent on the value of average effective channel fidelity. Meanwhile, standard deviations of off-diagonal elements of quantum process matrix (Pauli Form) decay more quickly (at least $70$ times faster) than standard deviations of diagonal elements, and the effective channels are approximate to depolarizing channel as the concatenated level increases.