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Equivalence of topological color codes (without translational symmetry) to surface codes

In a recent work, Bombin, Duclos-Cianci, and Poulin showed that every local translationally invariant 2D topological stabilizer code is locally equivalent to a finite number of copies of Kitaev’s toric code. In this paper, we focus on color codes and relax the constraint on translation invariance. We show that any color code can be mapped to exactly two copies of a related surface code. The surface code in our map is induced by the color code and easily derived from the color code. Furthermore, our map does not require any ancilla qubits for the surface codes. We also indicate the various degrees of freedom in constructing the map and the consequent variations.