A Family of Quantum Codes with Exotic Transversal Gates

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.

Significant Revision. Modernized tables. Updated proof techniques

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• Semidefinite programming bounds on the size of entanglement-assisted codeword stabilized quantum codes
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