phase transition. Such phase transition can be characterized by a local order parameter which is the averaged magnetization $m$, see Fig.(\[schem\]) as a schematic of phase diagram of random bond Ising model.

In the previous section we gave a quantum formalism for the partition function of such a model. In this section we are going to find a quantum formalism for the magnetization $m$ in order to consider consequences of phase transition in 2D Ising model in the quantum side. Such a study has been done for the Ising model without quenched disorder where the above phase transition is mapped to a coherent/non-coherent phase transition in the noisy Toric code state [@zareimon]. However, here we are going to consider random-bond Ising model which has a richer phase diagram and the magnetization is a function of the temperature and the probability of disorder $p$.

We start with a quenched configuration of couplings to calculate the magnetization and then we calculate the mean value of this quantity for all realizations of the quenched disorder. For an arbitrary spin, for example $s_n$: $$\label{deg}
m_n[\mathbf{J}]=\la s_n \ra=\frac{1}{\mathcal{Z}[\mathbf{J}]}\sum_{\mathbf{s}}s_n e^{\beta \sum_{\la i,j\ra }J_{ij}s_i s_j },$$

![(Color online) a) String $\gamma_n$ starts from a spin in the 