channels where a quenched disorder is induced by measurement of the errors after the first channel. On the other hand, the above transition is directly related to error threshold of the Toric code model. Accordingly, and since the noisy process can be applied to other topological CSS states, we conclude that the dual correspondence can also provide a useful tool for the study of error thresholds in different topological CSS codes.'
author:
- Mohammad Hossein Zarei
- Abolfazl Ramezanpour
title: |
    Noisy Toric code and random bond Ising model:\
    The error threshold in a dual picture
---

Introduction
============

It has been shown that quantum information theory has many fascinating connections to statistical physics which have led to several cross-fertilizations between these two important fields of physics. During past decades different problems including quantum algorithms for evaluation of partition functions or simulation of classical spin models [@Geraci2008; @Lidar1997; @Geraci2010; @algor; @durmar], mathematical mappings from quantum systems to classical statistical models [@Somma2007; @eis17; @castel2005; @gemma] and applications of percolation theory in quantum error correcting codes [@vodola] have been developed. Of the most interesting connections is a mapping from partition function of classical spin models to quantum stabilizer states [@Nest2007]. Such mappings 