  We use tensor network techniques to obtain high order perturbative
diagrammatic expansions for the quantum many-body problem at very high
precision. The approach is based on a tensor train parsimonious representation
of the sum of all Feynman diagrams, obtained in a controlled and accurate way
with the tensor cross interpolation algorithm. It yields the full time
evolution of physical quantities in the presence of any arbitrary time
dependent interaction. Our benchmarks on the Anderson quantum impurity problem,
within the real time non-equilibrium Schwinger-Keldysh formalism, demonstrate
that this technique supersedes diagrammatic Quantum Monte Carlo by orders of
magnitude in precision and speed, with convergence rates $1/N^2$ or faster,
where N is the number of function evaluations. The method also works in
parameter regimes characterized by strongly oscillatory integrals in high
dimension, which suffer from a catastrophic sign problem in Quantum
Monte-Carlo. Finally, we also present two exploratory studies showing that the
technique generalizes to more complex situations: a double quantum dot and a
single impurity embedded in a two dimensional lattice.
