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We describe the simulation of dihedral gauge theories on digital quantum computers. The non-Abelian discrete gauge group D₍---the dihedral group---serves as an approximation to U (1) Z₂ lattice gauge theory. In order to carry out such a lattice simulation, we detail the construction of efficient quantum circuits to realize basic primitives including the non-Abelian Fourier transform over D₍, the trace operation, and the group multiplication and inversion operations. For each case the required quantum resources scale linearly or as low-degree polynomials in n=logN. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of D₄. The estimated fidelity of all D₄ gates was found to exceed 80%.
Alam et al. (Thu,) studied this question.