diffusion and an expression of the memory lifetime as a function of system parameters is provided in Sec. \[sec:lifetime\]. The main results of our paper are in Secs. \[sec:mf\] and \[sec:lifetime\_interacting\] where, first by an analytical mean-field treatment and then by direct numerical simulation, we demonstrate the self-correcting properties of our model. Section \[sec:long\_range\] contains a discussion of the possible implementations of the long-range anyon interaction and Sec. \[sec:conclusion\] concludes the paper with our final remarks.

![(Color online.) Quantum memory based on the toric code. Illustrated is an $8 \times 8$ lattice (periodic boundary conditions) with a total of 128 spins-$\frac{1}{2}$ \[gray (smaller) circles\] on its edges. The four-body plaquette and star operators are indicated in the background. A particular choice for all logical operators $X_1$, $Z_1$, $X_2$, and $Z_2$ is shown, although we will focus only on the decay of $Z_1 \equiv Z$ (see main text). A number of spins is affected by $\sigma_x$-errors (solid dots), leading to excited plaquettes, or ‘plaquette anyons’ (striped plaquettes). Measuring the plaquette operators yields the positions of the excited plaquettes, but reveals no information about how they were originally paired or which path (indicated by the framed plaquettes) they took. A minimum-weight error 