Chiral gauge theories underlie the Standard Model of particle physics, yet their nonperturbative formulation remains one of the outstanding challenges in theoretical physics. I will describe how recent developments in understanding anomalies on the lattice, "qubit" regularizations of lattice QFTs, and tensor network methods have made it possible, for the first time, to nonperturbatively define and numerically simulate 1+1-dimensional Abelian chiral gauge theories as lattice models of qudits. We find that tensor network methods work surprisingly well, allowing us to compute the central charge, scaling dimensions, and low-energy quasiparticle dispersion relations in excellent agreement with analytic predictions for the famous 3-4-5-0 model. This paves the way for both classical and quantum simulations of chiral observables relevant to the nonequilibrium dynamics of chiral gauge theories, such as those arising in electroweak baryogenesis.