Abstract
The Steane code and quantum Golay code have been identified as good candidates for fault-tolerant quantum computing via code concatenation. These two codes have transversal implementations of all Clifford gates, but require some other scheme for fault-tolerant gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault-tolerantly without the use of magic state distillation. Analogously to how the Steane code can be fault-tolerantly converted to and from the Reed-Muller code which has a transversal gate, the Golay code can be converted to a triorthogonal code with a transversal gate. A crucial ingredient to this procedure is the triorthogonal code, which can itself be seen as related to the self-dual 2D color code. Additionally, a method for code conversion based on a transversal CNOT between the codes, rather than stabilizer measurements, is described.