correction procedure (see Sec. \[app:simulation\_details\]) applies $\sigma_x$-operators to the spins marked by the larger orange circles. While the vertically striped green anyons are annihilated ‘properly’ (with a trivial loop of errors remaining from the top pair and no error from the bottom pair), the horizontally striped red pair is connected around a topologically non-trivial loop on the torus. Although this last pair is annihilated as well, an uncorrected $\sigma_x$-error remains on the logical $Z$ string, having thereby introduced a logical error in the state stored in the memory.[]{data-label="fig:toric code"}](toric_code.pdf){width="0.95\columnwidth"}

Repulsive long range interactions in the Kitaev model {#sec:model}
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The model under study is defined on a $L\times L$ square lattice with periodic boundary conditions (a ‘torus’), and a spin-$\frac{1}{2}$ is placed on each of its $2L^2$ edges. Starting from the toric code model [@Kitaev2003], we consider the more general stabilizer Hamiltonian $$\label{H0}
H_0= \frac12 \sum_{p p'} U_{pp'} n_p n_{p'} + \frac12 \sum_{s s'} V_{ss'} n_s n_{s'},$$ where $n_p=(1-\prod_{i\in p} \sigma_{z,i})/2$, $n_s=(1-\prod_{i\in s} \sigma_{x,i})/2$, and $\sigma_{x, i}, \sigma_{z, i}$ denote the usual single-spin $x$ and $z$ Pauli operators applied to spin $i$. The indices $p$ and $p'$ run over all ‘plaquettes’ (involving the four spins on the edges of a 