Primitive Quantum Gates for an SU(2) Discrete Subgroup: BT

We construct a primitive gate set for the digital quantum simulation of the binary tetrahedral ($\mathbb{B}\mathbb{T}$) 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 $\mathbb{B}\mathbb{T}$ Fourier transform over $\mathbb{B}\mathbb{T}$. We experimentally benchmark the inversion and trace gates on ibm nairobi, with estimated fidelities between $14-55\mathrm{\%}$, depending on the input state.