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.