Primitive quantum gates for dihedral gauge theories
M. Sohaib Alam, Stuart Hadfield, Henry Lamm, Andy C. Y. Li · Physical review. D/Physical review. D. · 2022
We describe the simulation of dihedral gauge theories on digital quantum computers. The non-Abelian discrete gauge group ${D}_{N}$---the dihedral group---serves as an approximation to $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ 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 ${D}_{N}$, 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 $n=\mathrm{log}N$. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of ${D}_{4}$. The estimated fidelity of all ${D}_{4}$ gates was found to exceed 80%.