Codes and Protocols for Distilling T , controlled- S , and Toffoli Gates
Jeongwan Haah, Matthew B. Hastings · Quantum · 2018
We present several different codes and protocols to distill T , controlled- S , and Toffoli (or C C Z ) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal T . We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency γ → 1 . We also present a Reed-Muller based construction of these codes which obtains a worse γ but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of C C Z magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of. Several examples, including a Reed-Muller code for T -to-Toffoli distillation, punctured Reed-Muller codes for T -gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a 512 T-gate to 10 Toffoli gate code with distance 8 as well as triorthogonal codes with parameters [ [ 887 , 137 , 5 ] ] , [ [ 912 , 112 , 6 ] ] , [ [ 937 , 87 , 7 ] ] with very low prefactors in front of the leading order error terms in those codes.