Hall 123

Equivalence of 2D color codes (without translational symmetry) to surface codes

Pradeep Sarvepalli

Indian Institute of Technology, Madras

Surface codes and color codes are among the most important quantum codes for achieving fault-tolerance in quantum computers. Both are topological quantum codes, but there are significant differences in their construction and properties which might lead one to believe that they are inequivalent. However, 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. For 2D color codes, Delfosse relaxed the constraint on translation invariance and mapped a 2D color code onto three surface codes. We propose an alternate map based on linear algebra. We show that any 2D color code can be mapped onto 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. Our results find application in the decoding of color codes.