  We discuss the successes and limitations of statistical sampling for a
sequence of models studied in the context of lattice QCD and emphasize the need
for new methods to deal with finite-density and real-time evolution. We show
that these lattice models can be reformulated using tensorial methods where the
field integrations in the path-integral formalism are replaced by discrete
sums. These formulations involve various types of duality and provide exact
coarse-graining formulas which can be combined with truncations to obtain
practical implementations of the Wilson renormalization group program. Tensor
reformulations are naturally discrete and provide manageable transfer matrices.
Combining truncations with the time continuum limit, we derive Hamiltonians
suitable to perform quantum simulation experiments, for instance using cold
atoms, or to be programmed on existing quantum computers. We review recent
progress concerning the tensor field theory treatment of non-compact scalar
models, supersymmetric models, economical four-dimensional algorithms,
noise-robust enforcement of Gauss's law, symmetry preserving truncations and
topological considerations. We discuss connections with other tensor network
approaches.
