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∕d² for surface codes Cₛ and color codes Cc in several instances. On the torus, typical values are Cₛ=2 and Cc=3∕2, but we find that the optimal values are Cₛ=1 and Cc=9∕8. For planar codes, a typical value is Cₛ=2, while we find that the optimal values are Cₛ=1 and Cc=3∕4. In general, a color code encodes twice as many logical qubits as does a surface code.