G| \rho |G\ra$ which is also a measure of stability of the Toric code state against the bit-flip noise. This quantity will be in fact equal to total probability that bit-flip noise generates a pattern of loops in the dual lattice.

By the above geometrical description for noise patterns, it seems that some concepts like stability against noise might be related to some geometrical problems. Furthermore, other geometrical problems can emerge when we consider other initial states in Eq.(\[bit\]). For example, we define a specific quantum state as a coherent superposition of the ground state and an excited state of the Toric code model, an excitation which is generated by applying $Z$ operators in the ground state. To this end, as it has been shown in Fig.(\[loop\]), consider a Toric code model with an open boundary condition. Then consider a string $\gamma$ which starts from the boundary and ends to a vertex of the lattice, see Fig.(\[loop\]). Then, we construct the string operator $\Gamma_Z =\prod_{e \in \gamma}Z_e$ corresponding to such a string. This operator commutes with all face operators. It also commutes with all vertex operators except the vertex operator corresponding to the vertex in the endpoint of the string. 