code can be mapped to a random-bond Ising model, and a threshold probability $f_c = 0.11$ is obtained. In the thermodynamic limit, retrieval of the encoded information is impossible if the relative number of errors is above this value. Below $f_c$, recovery is achieved with probability one. Numerically, we find a similar threshold $f_c\approx 0.1$ for the same error model, see Fig. \[Z\_vs\_f\]. This shows that our implementation of the minimum-weight error correction scheme works close to optimal.

![\[Z\_vs\_f\] (Color online.) Average of the corrected operator $Z_{\rm ec}$ for a model with independent $\sigma_x$-errors occurring with probability $f$ at each spin. The dashed-dotted, dashed, and solid curves refer to our numerical simulations with lattice sizes $L=40, 100, 200$, respectively. The error correction fails at a value $f_c\simeq 0.1$, which is slightly smaller than the value $0.11$ from Ref. [@Dennis2002]. In the inset, we plot the value of $f_c$ from simulations of the non-interacting toric code in contact with a bath at temperature $T$ and $\gamma(0)=\gamma(2J)$. The fraction $f_c$ is extracted at the time $\tau$ when $\langle Z_{\rm ec} \rangle$ decays to zero in the limit of large $L$ (see Fig. \[nonint\_fig\]). This value is always smaller than $f=0.11$ and depends 