that cross the string $\gamma$ an even number of times and the second subset, denoted by $L_-$, contains loops that cross the string $\gamma$ for an odd number of times. Since the $X$-type loop operators corresponding to the first subset commute with the operator $\Gamma_Z$, they preserve the state $|\psi_+ \ra$. On the other hand, the $X$-type loop operators corresponding to the second subset anti-commute with the $\Gamma_Z$ and therefore, they convert the state $|\psi _+\ra$ to a new state in the form of $|\psi_- \ra =\frac{1}{\sqrt{2}}(|G\ra -\Gamma_Z |G\ra$. It will be interesting to consider the effect of bit-flip noise only in a subspace generated by $|\psi_+ \ra$ and $| \psi_- \ra$. The initial state is a coherent state in this subspace and after bit-flip channel the quantum state in this subspace is converted to a completely mixed state. It has been shown that there is a quantum phase transition from coherent to the mixed phase in a critical value of the probability $p_{cr}$ [@zareimon]. Such a result has been derived by a mapping from the order parameter of the Ising model to an order parameter in Toric code state which characterizes the coherence in the above subspace. In the 