the random-bond Ising model to a coherence parameter for the above noisy Toric code model. Finally, in Sec.(\[sec4\]) we explain the connection of our results with the error threshold of the Toric code and we conclude that the duality correspondence between classical spin models and quantum CSS states is helpful for investigation of the error threshold of quantum CSS states.

Quantum formalism for partition function of 2D Ising model {#sec1}
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In this section, we give a brief review of the quantum formalism for partition function of the two-dimensional (2D) Ising model. To this end, consider a 2D square lattice where classical spins of $s=\pm 1$ live on vertices (vertex spins). The classical Hamiltonian is defined as follows: $$\label{ran}
H=-\sum_{\la i, j\ra} J_{ij}s_i s_j,$$ where $\la i, j\ra$ refers to the nearest neighbors and $J_{ij}$ refers to the coupling constant corresponding to the interaction of $\la i, j\ra$. All thermodynamic properties of the above model can be derived from the partition function which is obtained by a summation of the Boltzmann weights corresponding to all spin configurations $\mathbf{s}$: $$\label{ranz}
\mathcal{Z}[\mathbf{J}]=\sum_{\mathbf{s}}e^{\beta \sum_{\la i, j\ra} J_{ij}s_i s_j},$$ where $\beta$ is equal to $\frac{1}{k_B T}$ and $k_B$ refers to the Boltzmann constant.

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