Physics Department - Exploiting Hidden Low-rank Structures in Quantum Field Theories

Abstract
Tensor networks are a powerful tool for compressing wave functions and density matrices of quantum systems in physics. Recent developments have shown that tensor network techniques can efficiently compress many functions beyond these traditional objects. Notable examples include the solutions to turbulence in Navier–Stokes equations [1] and the computation of Feynman diagrams [2,3]. These advancements have heralded a new era in the use of tensor networks for expediting the resolution of various complex equations in physics. This talk will provide an overview of our work utilizing tensor networks for computations based on quantum field theories.

First, we will introduce the Quantics/quantized Tensor Train (QTT) representation [3,4] for compressing the space-time dependence of correlation functions in quantum systems [5], leveraging inherent length-scale separation for efficient representation.

Second, we will present a robust tool named "Quantics Tensor Cross Interpolation" [6], which learns a quantics low-rank representation of a given function. Applications include the computation of Brillouin zone integrals [6] and integration of complex self-energy Feynman diagrams for multiorbital electron-phonon impurity models [7].

Finally, we will introduce new algorithms [8] and open-source libraries [9] for tensor cross interpolation.

[1] N. Gourianov et al., Nat. Comput. Sci. 2, 30 (2022).
[2] Y. N. Fernandez et al., PRX 12, 041018 (2022).
[3] I. V. Oseledets, Dokl. Math. 80, 653 (2009).
[4] B. N. Khoromskij, Constr. Approx. 34, 257 (2011).
[5] H. Shinaoka et al., PRX 13, 021015 (2023).
[6] M. K. Ritter, Y. N. Fernández, M. Wallerberger, J. von Delft, H. Shinaoka and X. Waintal, PRL 132, 056501 (2024).
[7] H. Ishida, N. Okada, S. Hoshino, H. Shinaoka, arXiv:2405.06440v2.
[8] Y. N. Fernández, M. K. Ritter, M. Jeannin, J.-W. Li, T. Kloss, T. Louvet, S. Terasaki, O. Parcollet, J. von Delft, H. Shinaoka and X. Waintal, arXiv:2407.02454v1.
[9] https://tensor4all.org