Machine learning inspired analysis of the dyson equation via quantics tensor cross interpolation

Abstract

Confronting many-body problems in quantum field theory entails managing vastamounts of data, thus, being confronted with the challenge of balancingcomputational feasibility and accuracy of functional dependencies. This thesis elaborates on quantum tensor cross interpolation (QTCI), an innovative approach that merges two effective methods for handling tensors inmulti-dimensional space-time: the quantics representation and tensor cross interpolation (TCI). QTCI benefits from both methods having distinct strategies in addressing numerical challenges. While the first method focuses on separating various length scales, TCI uses the restructured data to construct matrix productstates (MPS) and compressing them, while maintaining an acceptable error. Inthis thesis, QTCI was used to solve the Dyson equation for the Hubbard modelin one-, two-, and three-dimensional k-space featuring a self-energy inspired by aself-energy deep in the Mott phase. It was found that the method is able to reliably and efficiently compress vast amounts of data while retaining an acceptable error. The computational effort increases with dimension, complexity of the function and accuracy which limits the applicability of QTCI depending on the computational resources available. Within the framework of this thesis, it was demonstrated that the maximum bond dimension Dmax, a measurement quantifying the complexity of functions compressed by QTCI, is linked to boththe inverse temperature β and dimensionality n of the system by a universalpower-law Dmax(β , n) = An · β^((n−1)0.253) with a dimension dependent factor An.