Efficient decoding of topological color codes

Abstract

Color codes are a class of topological quantum codes with a high error threshold and a large set of transversal encoded gates and are thus suitable for fault-tolerant quantum computation in two-dimensional architectures. Recently, computationally efficient decoders for the color codes were proposed. We describe an alternate efficient iterative decoder for topological color codes and apply it to the color code on the hexagonal lattice embedded on a torus. In numerical simulations, we find an error threshold of 7.8% for independent dephasing and spin-flip errors.

Improved HDRG decoders for qudit and non-Abelian quantum error correction

“…They were prominently used for correcting codes with fractal logical operators [12], for which no alternative decoding procedure was available. These decoders have since been referred to as 'hard-decision renormalization group' or 'HDRG' decoders [13].…”

Fault-tolerant thresholds for quantum error correction with the surface code

“…Naturally, there are many avenues for further work. Given the recent proliferation of alternative decoding algorithms for topological codes, such as the surface code [36][37][38][39][40][41][42][43], it would be valuable to determine circuit-level thresholds for these algorithms, making it easier to understand their practical costs and benefits. It may also be possible to improve these thresholds by accounting for additional correlations present in some noise models (for example, the correlation between X and Z errors in depolarizing noise) [36].…”

“…Notwithstanding the recent development of several alternative decoding algorithms for topological codes [36][37][38][39][40][41][42][43], we restrict ourselves to decoding via Edmonds' minimum-weight perfect matching algorithm [44]. Also, we do not consider other topological codes, such as color codes [45], instead, referring the interested reader to the recent article of Landahl et al [46].…”

Dimensional jump in quantum error correction

“…It is interesting to mention that error correction has been substantially explored for 2D color codes [17][18][19][20][21][22][23][24], whereas for 3D little is known [12].…”