problem in the Toric code model.

Mapping to the Toric code model under bit-flip channels {#sec3}
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By now, we have only rewritten the partition function as well as the order parameter of the random bond Ising model in a quantum formalism. However, the above quantum formalism seems only a mathematical mapping and we need to introduce a physical meaning for the order parameter in the quantum side. In this section, we show that the order parameter can characterize coherence of a quantum state in the Toric code model which is under two sequences of bit-flip channels. To this end, first we need to introduce the bit-flip channel and consider its effect on the Toric code model. This section involves two subsections where in the first one we introduce the noisy Toric code and in the second one we come back to the quantum formalism of the order parameter and derive the main result of the paper.

Toric code model and bit-flip noise
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As we briefly mentioned in the first section, Toric code model is defined by two kinds of stabilizer operators $B_F$ and $A_v$. Furthermore, the Toric code state $|G\ra$ is also a ground state of a Hamiltonian 