Topological quantum codes on compact surfaces with genus g≥2

In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g≥2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2–5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs Ks, for s≡1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3–5.