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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 Cₒ and color codes C₂ in several instances. On the torus, typical values are Cₒ=2 and C₂=3∕2, but we find that the optimal values are Cₒ=1 and C₂=9∕8. For planar codes, a typical value is Cₒ=2, while we find that the optimal values are Cₒ=1 and C₂=3∕4. In general, a color code encodes twice as many logical qubits as does a surface code.
Bombín et al. (Fri,) studied this question.