We prove the existence of topological quantum error correcting codes with encoding rates $k∕n$ asymptotically approaching the maximum possible value. Explicit constructions of these topological codes are presented using surfaces of arbitrary genus. We find a class of regular toric codes that are optimal. For physical implementations, we present planar topological codes.

• Received 10 February 2006

DOI:https://doi.org/10.1103/PhysRevA.73.062303