Projective plane and planar quantum codes
Michael H. Freedman, David A. Meyer
TL;DR
It is observed that Shor's code can be constructed in a planar domain and generalize to planar constructions of higher-genus codes for multiple qubits.
Abstract
Cellulations of the projective plane RP^2 define single qubit topological quantum error correcting codes since there is a unique essential cycle in H_1(RP^2;Z_2). We construct three of the smallest such codes, show they are inequivalent, and identify one of them as Shor's original 9 qubit repetition code. We observe that Shor's code can be constructed in a planar domain and generalize to planar constructions of higher genus codes for multiple qubits.