Bringing Order through Disorder: Localization of Errors in Topological Quantum Memories

Anderson localization emerges in quantum systems when randomized parameters cause the exponential suppression of motion. Here we consider this phenomenon in topological models and establish its usefulness for protecting topologically encoded quantum information. For concreteness we employ the toric code. It is known that in the absence of a magnetic field this can tolerate a finite initial density of anyonic errors, but in the presence of a field anyonic quantum walks are induced and the tolerable density becomes zero. However, if the disorder inherent in the code is taken into account, we demonstrate that the induced localization allows the topological quantum memory to regain a finite critical anyon density and the memory to remain stable for arbitrarily long times. We anticipate that disorder inherent in any physical realization of topological systems will help to strengthen the fault tolerance of quantum memories.

Figure 1

The average value of the localization length $l$ as the disorder $\gamma /h$ is increased, with each point calculated over 100 samples. The different curves show the results for different system sizes $L$, with the units of $l$ corresponding to the lattice spacing.

Figure 2

The behavior and approximate values of the critical anyon density ${\rho }_c$ as a function of the localization length $l$. This is calculated by numerically solving for the side length ${\lambda }_c$ at which the probability $p$ of anyons moving between squares reaches the critical value $p_c$ where error correction fails.