  We discuss recent progress in Tensor Lattice Field Theory and economical,
symmetry preserving, truncations suitable for quantum computations or
simulations. We focus on spin and gauge models with continuous Abelian
symmetries such as the Abelian Higgs model and emphasize noise-robust
implementations of Gauss's law. We discuss recent progress concerning the
comparison between field digitizations and character expansions, symmetry
breaking in tensor language, wave-packet preparation and possible new
implementations of Abelian models using Rydberg atoms.
