  In recent years, there have been many studies on local stabilizer codes.
Under the assumption of translation and scale invariance Yoshida classified
such codes. His result implies that translation invariant 2D color codes are
equivalent to copies of toric codes. Independently, Bombin, Duclos-Cianci, and
Poulin showed that a local translation 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 2D topological color codes and relax the assumption of
translation invariance. Using a linear algebraic framework we show that any 2D
color code (without boundaries) is locally equivalent to two copies of a
related surface code. This surface code is induced by color code. We study the
application of this equivalence to the decoding of 2D color codes over the bit
flip channel as well as the quantum erasure channel. We report the performance
of the color code on the square octagonal lattice over the quantum erasure
channel. Further, we provide explicit circuits that perform the transformation
between 2D color codes and surface codes. Our circuits do not require any
additional ancilla qubits.
