Robustness of the concatenated quantum error-correction protocol against noise for channels affected by fluctuation

In quantum error correction, the description of noise channel cannot be completely accurate, and fluctuation always appears in noise channel. It is found that when fluctuation of physical noise channel is considered, the average effective channel is dependent only on the average of physical noise channel, and the average of physical noise channel here plays the role of the independent error model in the previous works. Now, one may conclude that in the independent error model, the results in previous works are also valid for average channel where fluctuation exists. In some typical cases, our numerical simulations in the concatenated QEC protocol with 5-qubit code, 7-qubit Steane code and 9-qubit Shor confirm this conjecture. For 5-qubit code, the effective channels are approximate to depolarizing channel as the concatenated level increases. For Steane code, the effective channels are approximate to one Pauli channel as the concatenated level increases. For Shor code, the effective channels are approximate to one of Pauli-$X$ and Pauli-$Z$ channels in each level, and in next concatenated level, the effective channels are approximate to the other. Meanwhile, for these codes, the numerical results indicate that the degree of approximation increases with the concatenated level increases, and the fluctuation of noise channel decays exponentially as concatenated QEC performed. On the error-correction threshold, attenuation ratio of standard deviation of channel fidelity roughly has a stable value. On the other hand, standard deviations of off-diagonal elements of quantum process matrix (Pauli Form) decay more quickly than standard deviations of diagonal elements.