the endpoints of that string.[]{data-label="flux"}](flux){width="6cm" height="6cm"}

It is useful to give a geometrical description for the effect of the bit-noise in the Toric code state. As it has been shown in Fig.(\[flux\]), when an $X$ operator is applied to a qubit, it does not commute with two face operators which involve that qubit. Therefore, the system goes to an excited state which can be represented by two excitations, which are called flux anyons, in the corresponding faces. Consequently, the effect of the bit-flip operator on a qubit is geometrically described by a string in the dual lattice whose endpoints are flux anyons. In the same way, the effect of bit-flip noise on other qubits can be described by a pattern of strings whose endpoints show excitations. However, there can also be some closed strings (loops) where the pattern of effect of bit-flip operators is described by loops. Such patterns do not lead to an excitation and the Toric code state remains stable under such patterns. Consequently, the state $\rho$ is a mixture of excited states and the ground state of the Toric code model. In particular, one can compute the fidelity of $\rho$ with the ground state i. e. $F=\la 