Fig.(\[k4\]-b). Here $X_e$ refers to the Pauli operator $\sigma_x$ which is applied to the edge qubit on edge $e$ emanating to vertex $v$, see Fig.(\[k4\]-b). Since each vertex operator $A_v$ has zero or two common qubits with $B_F$ operators, it is concluded that $[A_v ,B_F]=0$. Finally, if we apply $A_v$ to state $|G\ra =\prod_{F} \frac{1+B_F}{2}|+\ra ^{\otimes M}$ it will pass from $\prod_{F} \frac{1+B_F}{2}$ and since $A_v |+\ra ^M =|+\ra^M$, it is concluded that the $A_v$’s are also stabilizers of $|G\ra$. Therefore, state $|G\ra$ can also be written in terms of vertex operators in the following form: $$\label{innx}
|G\ra =\prod_{v} \frac{1+A_v}{2}|0\ra ^{\otimes M}.$$ In this way, the state $|G\ra$ is stabilized by vertex and face operators $A_v$ and $B_F$ which are exactly stabilizers of Toric code state as defined in [@Kitaev2003]. Consequently, the partition function of 2D Ising model has been mapped to inner product of a product state and the Toric code state where couplings of the Ising model are encoded in the product state and interaction pattern of the Ising model is encoded in the Toric code state.

![(Color online) Loop operators $T_z ^1$, $T_z ^2$, $T_x ^1$ and $T_x ^2$ correspond to non-trivial loops around the torus.[]{data-label="k1"}](k1){width="8cm" height="4cm"}

