Primitive quantum gates for dihedral gauge theories
M. Sohaib Alam, Stuart Hadfield, Henry Lamm, Andy C. Y. Li
Abstract
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%.