Primitive Quantum Gates for an $SU(2)$ Discrete Subgroup: Binary Octahedral

We construct a primitive gate set for the digital quantum simulation of the 48-element binary octahedral ($\mathbb{B}\mathbb{O}$) group. This nonabelian discrete group better approximates $SU(2)$ lattice gauge theory than previous work on the binary tetrahedral group at the cost of one additional qubit – for a total of six – per gauge link. The necessary primitives are the inversion gate, the group multiplication gate, the trace gate, and the $\mathbb{B}\mathbb{O}$ Fourier transform.