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.

Figure 1

Gadgets that move a qubit into an ancilla, which are useful for code conversion.

Figure 2

Example of a circuit to perform a logical $T$ gate on a qubit encoded in the self-dual code via code conversion to a doubled code and back. Every operation is transversal in this procedure, with the cnot gates involving the doubled code using only the first self-dual block of the code.

Figure 3

An example circuit for preparing high-quality encoded magic states with transversal $T$ gates using the doubled code which are then converted to the self-dual code and consumed as in normal magic-state procedures.