Learning Feynman Diagrams with Tensor Trains

TL;DR

This paper introduces a tensor train approach to efficiently compute high-order Feynman diagram expansions in quantum many-body problems, surpassing traditional Quantum Monte Carlo methods in precision and speed.

Contribution

The authors develop a tensor train method using tensor cross interpolation to accurately sum all Feynman diagrams, enabling high-precision, real-time evolution calculations in complex quantum systems.

Findings

Outperforms diagrammatic Quantum Monte Carlo in precision and speed

Achieves convergence rates of 1/N^2 or faster

Effectively handles strongly oscillatory integrals and sign problems

Abstract

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.