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

The optimal regularization of infinite-dimensional gauge-field degrees of freedom is a central open problem in the 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 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 regularized Hamiltonian, in the Kogut-Susskind formulation, on fault-tolerant quantum computers. We provide explicit quantum circuit constructions for implementing the primitive gates in this model, the $F$ and $R$ symbols, for $U(1)_k$ and $SU(2)_k$ anyon theories.