Efficient Computations in Quantum Field Theories Using Quantics Tensor Trains

Hiroshi Shinaoka^1*

¹Department of Physics, Saitama University, Tokiwa, Japan

* Presenter:Hiroshi Shinaoka, email:h.shinaoka@gmail.com

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. Particular attention is paid to quantum-inspired algorithms based on quantics/quantized tensor train (QTT) in many fields of science. Notable examples include the solutions to turbulence in Navier–Stokes equations [1] and the computation of Feynman diagrams [2,3].

In this talk, we demonstrate the strong potential of the QTT algorithms for solving quantum field theories. First, we briefly introduce quantics/quantized tensor train (QTT) representation [3,4] for compressing the space-time dependence of the correlation functions of quantum systems [5]. This method leverages length-scale separation to efficiently represent correlation functions, e.g., vertex functions and to solve diagrammatic equations in compressed form. Then, we will introduce “Quantics Tensor Cross Interpolation (QTCI)”[6], allowing us to construct a QTT representation of a function using adaptive sampling.

As applications of these technologies, we showcase the computation of Brillouin zone integrals [6], the integration of complex self-energy Feynman diagrams of a multiorbital electron-phonon model [7], the solution of nonequilibrium Dyson equation [8], and the self-consistent solution of the parquet equation [9]. If time permits, we will introduce our open-source C++ and Julia implementations [10].

[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] M. Murray, H. Shinaoka, and P. Werner. Nonequilibrium diagrammatic many-body simulations with quantics tensor trains. Phys. Rev. B 109, 165135 (2024).
[9] S. Roshap, M. Ritter, H. Shinaoka, J. von Delft, M. Wallerberger and A. Kauch, in preparation.
[10] Y. N. Fernández, M. K. Ritter, M. Jeannin, J.-W. Li, T. Kloss, O. Parcollet, J. von Delft, H. Shinaoka and X. Waintal, arXiv:2407.02454v1.

Keywords: Tensor networks, Quantum field theories, Quantics tensor train