Multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains

TL;DR

This paper introduces a multiscale tensor train approach using quantics tensor trains to efficiently approximate and compute high-dimensional quantum correlation functions, significantly reducing computational complexity.

Contribution

It proposes a novel multiscale space-time ansatz based on quantics tensor trains for quantum correlation functions, enabling efficient high-dimensional tensor decompositions.

Findings

Achieved several orders of magnitude data compression.

Demonstrated stability and efficiency in solving Dyson and Bethe-Salpeter equations.

Validated the approach on various equilibrium and nonequilibrium quantum systems.

Abstract

Correlation functions of quantum systems -- central objects in quantum field theories -- are defined in high-dimensional space-time domains. Their numerical treatment thus suffers from the curse of dimensionality, which hinders the application of sophisticated many-body theories to interesting problems. Here, we propose a multi-scale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains (QTT), ``qubits'' describing exponentially different length scales. The ansatz then assumes a separation of length scales by decomposing the resulting high-dimensional tensors into tensor trains (known also as matrix product states). We numerically verify the ansatz for various equilibrium and nonequilibrium systems and demonstrate compression rates of several orders of magnitude for challenging cases. Essential building blocks of diagrammatic equations, such as convolutions or Fourier transforms are formulated in the compressed form. We numerically demonstrate the stability and efficiency of the proposed methods for the Dyson and Bethe-Salpeter equations. {The QTT representation} provides a unified framework for implementing efficient computations of quantum field theories.