Tensor lattice field theory with applications to the renormalization group and quantum computing

TL;DR

This paper explores tensor reformulations of lattice field theories to improve renormalization group methods and facilitate quantum simulations, addressing limitations of traditional statistical sampling in complex models.

Contribution

It introduces tensor-based reformulations of lattice models, enabling exact coarse-graining, Hamiltonian derivation for quantum simulation, and discusses recent advances in tensor field theory applications.

Findings

Tensor reformulations provide exact coarse-graining formulas.

Hamiltonians suitable for quantum simulation are derived.

Progress in tensor field theory for various models is reviewed.

Abstract

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.