A Family of Quantum Codes with Exotic Transversal Gates

TL;DR

This paper constructs a family of quantum error-correcting codes that implement the binary icosahedral group transversally, enabling fault-tolerant quantum gates based on a recently identified universal gate set.

Contribution

It introduces a new family of distance 3 quantum codes that implement the binary icosahedral group transversally, derived solely from symmetry considerations.

Findings

Codes have distance d=3.

Codes implement the binary icosahedral group transversally.

Construction relies on symmetry properties of the group.

Abstract

Recently an algorithm has been constructed that shows 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 $\ico$ 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.