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

Abstract: 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 to this procedure is the $[[49,1,5]]$ triorthogonal code, which can itself be seen as related to the self-dual $[[17,1,5]]$ 2D color code. Additionally, a method for code conversion based on a transversal CNOT between the codes, rather than stabilizer measurements, is described.