\gamma(\epsilon_s -
\epsilon_{x_i(s)})$, where $s$ is the current state of the system. (iii) We draw the time $\Delta t$ it takes for the next spin to flip from an exponential distribution, $\Delta t \sim {\rm Exp}(1/R)$, and then add this to the current total time. (iv) We calculate all individual spin flip probabilities $p_i = \gamma(\epsilon_s -
\epsilon_{x_i(s)})/R$ and flip a spin at random accordingly. After some initially specified time has been reached, we stop and have obtained a single ‘run’. The final data presented in this work is then generated by averaging over many (typically several thousand) runs.

Although continuous monitoring and error-correction are not required in a passive memory during the storage time, it is still beneficial to apply an error correction scheme once the memory is being read out. By $\langle Z_{\rm ec}\rangle(t)$, we denote in this work the average value of $Z$ we would have obtained if we had performed error correction at time $t$. The goal here is to properly annihilate corresponding anyons (by applying $\sigma_x$-operations), thereby reverting the undesired operations performed by anyon paths crossing the logical operator strings. However, since only the positions of the anyons are known, this correspondence has to be 