Optimal resources for topological two-dimensional stabilizer codes: Comparative study

We study the resources needed to construct topological two-dimensional stabilizer codes as a way to estimate in part their efficiency, and this leads us to perform a comparative study of surface codes and color codes. This study clarifies the similarities and differences between these two types of stabilizer code. We compute the topological error-correcting rate C:=n/d(2) for surface codes C-s and color codes C-c in several instances. On the torus, typical values are C-s=2 and C-c=3/2, but we find that the optimal values are C-s=1 and C-c=9/8. For planar codes, a typical value is C-s=2, while we find that the optimal values are C-s=1 and C-c=3/4. In general, a color code encodes twice as many logical qubits as does a surface code.