Incoherent dynamics in the toric code subject to disorder

We numerically study the effects of two forms of quenched disorder on the anyons of the toric code. First, a class of codes based on random lattices of stabilizer operators is presented and shown to be superior to the standard square-lattice toric code for certain forms of biased noise. It is further argued that these codes are close to optimal, in that they tightly reach the upper bound of error thresholds beyond which no correctable Calderbank-Shore-Steane codes can exist. Additionally, we study the classical motion of anyons in toric codes with randomly distributed on-site potentials. In the presence of repulsive long-range interaction between the anyons, a surprising increase in the lifetime of encoded states with disorder strength is reported and explained by an entirely incoherent mechanism. Finally, the coherent transport of the anyons in the presence of both forms of disorder is investigated and a significant suppression of the anyon motion is found.