Tailoring tensor network techniques to the quantics representation for highly inhomogeneous problems and few body problems

TL;DR

This paper adapts tensor network algorithms to the quantics representation, improving convergence and efficiency for solving high-dimensional PDEs with multiple length scales, demonstrated on large-scale problems.

Contribution

It introduces tailored tensor network methods inspired by multigrid techniques for the quantics representation, enhancing performance on inhomogeneous and few-body problems.

Findings

Faster and more robust convergence of tensor network algorithms.

Successful application to high-dimensional PDEs with up to 2^80 grid points.

Outperforms conventional approaches on complex inhomogeneous problems.

Abstract

Tensor network techniques are becoming increasingly popular tools to solve partial differential equations within the so-called quantics representation. Their popularity stems from the fact that their spatial resolution depends only logarithmically on the number of grid points, making them very tempting approaches in situations where two or more characteristic length scales are vastly different. A first generation of technique used ``out-of-the-box'' algorithms of the tensor network toolkit (e.g. the celebrated Density Matrix Product State (DMRG) algorithm) to solve these problems. These techniques were designed for situations (e.g. quantum magnetism) where the different degrees of freedom (e.g. spins) play equivalent roles. In the quantics representation, however, the different degrees of freedom correspond to the physics at different scales and therefore play inequivalent role. Here we show that by tailoring the tensor network algorithms to this particular case, in the spirit of the multigrid approach, we obtain faster and more robust convergence of the algorithms. We showcase the approach on linear (Poisson equation) and eigenvalue (Schr\"odinger equation) problems in two, three and four dimensions. Our simulations involve up to $2^{80}$ grid points and would represent, we argue, a very strong challenge for conventional approaches.