Solving the Gross-Pitaevskii equation on multiple different scales using the quantics tensor train representation

TL;DR

This paper introduces a tensor train-based solver for the time-dependent Gross-Pitaevskii equation, enabling high-precision solutions across seven orders of magnitude in length scales on a standard laptop.

Contribution

The authors develop a quantics tensor train approach that efficiently handles non-linear PDEs across multiple scales, surpassing traditional methods in computational efficiency.

Findings

Resolved phenomena across seven orders of magnitude in one dimension.

Solved two-dimensional Gross-Pitaevskii equations on grids exceeding a trillion points.

Achieved high-precision solutions within one hour on a single laptop core.

Abstract

Solving partial differential equations of highly featured problems represents a formidable challenge, where reaching high precision across multiple length scales can require a prohibitive amount of computer memory or computing time. However, the solutions to physics problems typically have structures operating on different length scales, and as a result exhibit a high degree of compressibility. Here, we use the quantics tensor train representation to build a solver for the time-dependent Gross-Pitaevskii equation. We demonstrate that the quantics approach generalizes well to the presence of the non-linear term in the equation. We show that we can resolve phenomena across length scales separated by seven orders of magnitude in one dimension within one hour on a single core in a laptop, greatly surpassing the capabilities of more naive methods. We illustrate our methodology with various modulated optical trap potentials presenting features at vastly different length scales, including solutions to the Gross-Pitaevskii equation on two-dimensional grids above a trillion points ($2^{20} \times 2^{20}$). This quantum-inspired methodology can be readily extended to other partial differential equations combining spatial and temporal evolutions, providing a powerful method to solve highly featured differential equations at unprecedented length scales.