Localization of Toric Code Defects

We explore the possibility of passive error correction in the toric code model. We first show that even coherent dynamics, stemming from spin interactions or the coupling to an external magnetic field, lead to logical errors. We then argue that Anderson localization of the defects, arising from unavoidable fluctuations of the coupling constants, provides a remedy. This protection is demonstrated by using general analytical arguments that are complemented with numerical results which show that self-correcting memory can in principle be achieved in the limit of a nonzero density of identical defects.

Figure 1

Left: Anderson localization of the two particles ($\Delta =50$, $L=10$, $t_{\max }=60$). Inset: Dynamics of the two particles in the absence of a random background potential. Right: Comparison between the one-particle dynamics with and without the exotic braiding phases ($\Delta =50$, $L=40$, $t_{\max }=100$). The figure suggests that localization is not greatly affected by exotic braiding phases.

Figure 2

Plot of the critical disorder ${\Delta }_c$ as a function of the particle density $n$. We assume nearest neighbor hopping [e.g., caused by (3)] together with an i.i.d. background potential whose values are uniformly distributed on $[0,2\Delta ]$. The boundary conditions are periodic.