Code conversion with the quantum Golay code for a universal transversal gate set

The [7, 1, 3] Steane code and [23, 1, 7] 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 T 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 [7, 1, 3] Steane code can be fault tolerantly converted to and from the [15, 1, 3] Reed-Muller code which has a transversal T gate, the [23, 1, 7] Golay code can be converted to a [95, 1, 7] triorthogonal code with a transversal T gate. A crucial ingredient of this procedure is the [49, 1, 5] triorthogonal code, which can itself be seen as being related to the self-dual [17, 1, 5] two-dimensional color code. Additionally, a method for code conversion based on a transversal cnot between the codes, rather than stabilizer measurements, is described.