Family of Quantum Codes with Exotic Transversal Gates.

Abstract

Recently, an algorithm has been constructed that shows that the binary icosahedral group 2I together with a T-like gate forms the most efficient single-qubit universal gate set. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. However, no such code has ever been demonstrated in the literature. We fill this void by constructing a family of distance d=3 codes that all implement 2I transversally. A surprising feature of this family is that the codes can be deduced entirely from symmetry considerations that only 2I affords.