Multiscale Space-Time Ansatz for Correlation Functions of Quantum Systems Based on Quantics Tensor Trains

Abstract

The 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 multiscale space-time ansatz for correlation functions of quantum systems based on quantics tensor trains (QTTs), “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 (also known as matrix product states). We numerically verify the ansatz for various equilibrium and nonequilibrium systems and demonstrate compression ratios 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.