in the form of $H=-\sum_F B_F -\sum_v A_v$. Now, suppose that the initial state is the same as the ground state $|G\ra$ and then a Pauli operator $X$ is applied to each qubit with probability $p$. Here we use the same notation of random bond Ising model because we are going to map these problems to each other. Therefore, the initial state will change to the following state after the bit-flip noise: $$\label{bit}
\rho=\sum_{\mathcal{E}} W_{\mathcal{E}}(p) \hat{\mathcal{E}}(X)|G\ra \la G| \hat{\mathcal{E}}(X),$$ where $\mathcal{E}$ refers to a pattern of qubits which are affected by the noise and $\hat{\mathcal{E}}(X)$ refers to a product of $X$ operators which are applied to the qubits belonging to the error pattern $\mathcal{E}$. $W_{\mathcal{E}}(p)$ is the probability that an error patter $\mathcal{E}$ happens. If the number of qubits which have been affected by the noise is equal to $l$ and the total number of qubits is $M$, $W_{\mathcal{E}}(p)$ will be equal to $p^l (1-p)^{M-l}$.

![(Color online) An $X$ operator on a qubit in the Toric code does not commute with two face operators which are neighbors of that qubit. It leads to two excitations in the neighboring faces. A string of $X$ operators also leads to two excitations in 