Quantum simulation of lattice gauge theories coupled to fermionic matter via anyonic regularization

The optimal regularization of infinite-dimensional degrees of freedom is a central open problem in the tractable simulation of lattice gauge theories on quantum computers. Here, we consider regularizing the gauge field by replacing the gauge group $G$ with a braided fusion category whose objects correspond to Wilson lines of the associated Chern-Simons theory $G_k$, with the level $k$ serving as the regularization parameter. We demonstrate how to couple these regularized $U(1)$ and $SU(2)$ gauge groups to fermionic matter using the framework of fusion surface models, which treats matter and gauge field excitations as interacting anyons. We then address the simulation of the Hamiltonians we construct on fault-tolerant quantum computers, providing explicit quantum circuit constructions for implementing the primitive gates in this model, namely, the $F$ and $R$ symbols of the $U(1)_k$ and $SU(2)_k$ anyon theories, which may be of independent interest.