  We construct a primitive gate set for the digital quantum simulation of the
binary tetrahedral ($\mathbb{BT}$) group on two quantum architectures. This
nonabelian discrete group serves as a crude approximation to $SU(2)$ lattice
gauge theory while requiring five qubits or one quicosotetrit per gauge link.
The necessary basic primitives are the inversion gate, the group multiplication
gate, the trace gate, and the $\mathbb{BT}$ Fourier transform over
$\mathbb{BT}$. We experimentally benchmark the inversion and trace gates on ibm
nairobi, with estimated fidelities between $14-55\%$, depending on the input
state.
