Primitive Quantum Gates for Dihedral Gauge Theories

TL;DR

This paper presents methods for simulating dihedral gauge theories on quantum computers, including efficient circuit constructions and experimental benchmarks on Rigetti hardware.

Contribution

It introduces quantum circuits for nonabelian Fourier transforms and group operations for dihedral groups, with resource scaling analysis and experimental validation.

Findings

Quantum gates for D4 achieved over 80% fidelity.

Resource requirements scale linearly or polynomially with log N.

Experimental benchmarks demonstrate practical feasibility.

Abstract

We describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group $D_N$ -- the dihedral group -- serves as an approximation to $U(1)\times\mathbb{Z}_2$ 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 nonabelian 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 $n=\log N$. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of $D_4$. The fidelity of all $D_4$ gates was found to exceed $80\%$.