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

Abstract

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 $C\ensuremath{\mathrel{:=}}n∕{d}^{2}$ 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.