Primitive Quantum Gates for an SU(3) Discrete Subgroup: $\mathrm{\Sigma }(36\times 3)$

We construct the primitive gate set for the digital quantum simulation of the 108-element $\mathrm{\Sigma }(36\times 3)$ group. This is the first time a nonabelian crystal-like subgroup of $SU(3)$ has been constructed for quantum simulation. The gauge link registers and necessary primitives – the inversion gate, the group multiplication gate, the trace gate, and the $\mathrm{\Sigma }(36\times 3)$ Fourier transform – are presented for both an eight-qubit encoding and a heterogeneous three-qutrit plus two-qubit register. For the latter, a specialized compiler was developed for decomposing arbitrary unitaries onto this architecture.