Discrete non-Abelian gauge theories in two-dimensional lattices and their realizations in Josephson-junction arrays

TL;DR

This paper explores lattice models with discrete non-Abelian gauge groups, their Hamiltonian formalism suitable for solid-state systems, and proposes Josephson-junction arrays as potential physical realizations for topologically protected quantum computing.

Contribution

It introduces a Hamiltonian formalism for non-Abelian gauge theories on lattices and suggests experimental implementations using Josephson-junction arrays for quantum computing.

Findings

Topological properties of non-Abelian excitations are highly nontrivial.

Proposed Josephson-junction array designs for realizing these models.

Potential for robust quantum computation using these systems.

Abstract

We discuss real-space lattice models equivalent to gauge theories with a discrete non-Abelian gauge group. We construct the Hamiltonian formalism which is appropriate for their solid-state physics implementation and outline their basic properties. The unusual physics of these systems is due to local constraints on the degrees of freedom which are variables localized on the links of the lattice. We discuss two types of constraints that become equivalent after a duality transformation for Abelian theories but are qualitatively different for non-Abelian ones. We emphasize highly nontrivial topological properties of the excitations (fluxons and charges) in these non-Abelian discrete lattice gauge theories. We show that an implementation of these models may provide one with the realization of an ideal quantum computer, that is the computer that is protected from the noise and allows a full set of precise manipulations required for quantum computations. We suggest a few designs of the Josephson-junction arrays that provide the experimental implementations of these models and discuss the physical restrictions on the parameters of their junctions.