Therefore, $\Gamma_Z$ leads to an excitation called charge anyon in the Toric code model which is geometrically represented by a string in the lattice. Now, consider a coherent superposition from the ground state and the above excited state in the following form:

![(Color online) Toric code can be defined on a lattice with an open boundary where face operators in the boundary are three-local. A string $\gamma$ which starts from the boundary leads to an excitation in the vertex in the end point of $\gamma$. There are two $X$-type loop operators $L_+$ and $L_-$ which cross the string $\gamma$ for even and odd number of times, respectively.[]{data-label="loop"}](loop){width="6cm" height="6cm"}

$$\label{str}
|\psi_+ \ra =\frac{1}{\sqrt{2}}(|G\ra +\Gamma_Z |G\ra ).$$

Next, we consider the above state as an initial state which has been affected by a bit-flip noise. As we explained, the bit-flip noise can be described by patterns including open strings and loops. Similar to the Toric code state $|G\ra$, the effect of open strings on the state $|\psi_+ \ra$ leads to flux anyons. However, the effect of loops is different here. We divide the noise patterns that correspond to loops to two subsets, see Fig.(\[loop\]). The first subset, denoted by $L_+$, contains loops 