Fast Decoders for Topological Quantum Codes

We present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects. Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size , our algorithm runs in time log compared to ⁶ needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold. ©2010 The American Physical Society