Tensor Trains Efficiently Solve Complex Physics Problems Across Scales

The efficient solution of partial differential equations remains a significant computational challenge, particularly when modelling physical systems exhibiting features across a wide range of length scales. Traditional numerical methods often struggle with the memory and time requirements needed to achieve high precision in such scenarios. Marcel Niedermeier, from Aalto University, and Adrien Moulinas, from Univ. Grenoble Alpes, alongside Thibaud Louvet, Jose L. Lado and Xavier Waintal, detail a novel approach in their paper, ‘Solving the Gross-Pitaevskii equation on multiple different scales using the quantics tensor train representation’. They demonstrate a method utilising the ‘quantics tensor train’ – a hierarchical, low-rank approximation technique – to solve the time-dependent Gross-Pitaevskii equation, a fundamental equation in Bose-Einstein condensation and nonlinear optics, with unprecedented efficiency and resolution across seven orders of magnitude in one dimension, even on standard laptop hardware.

The pursuit of efficient numerical solutions to partial differential equations remains a persistent challenge across numerous scientific disciplines, demanding substantial computational resources and innovative algorithmic approaches. Traditional methods often struggle with the computational demands of high-resolution simulations, particularly when dealing with problems exhibiting features across a wide range of length scales, hindering practical application and limiting the scope of investigation. However, many physical systems possess inherent structures that allow for a compressed representation of their solutions, suggesting opportunities for more efficient algorithms and a paradigm shift in computational modelling.

Recent advances in quantum computing have unexpectedly spurred innovation in classical computational techniques, revealing connections between seemingly disparate fields and fostering cross-disciplinary collaboration. The development of methods for representing many-body quantum systems, such as tensor networks, has revealed their applicability to a broader range of mathematical problems, offering a powerful toolkit for tackling complex simulations. These tensor network methods offer a means of representing high-dimensional data with significantly reduced computational cost, by exploiting underlying correlations and redundancies and enabling simulations previously considered intractable. This has led to algorithms capable of solving certain problems with a computational time scaling as low as logarithmic with respect to the number of discretisation points, representing a substantial improvement over conventional methods.

The core principle behind these improvements lies in the ability to compress the information needed to represent the solution, moving beyond traditional grid-based approaches and embracing more efficient data structures. Rather than storing the value of the solution at every point in the discretised domain, tensor network methods represent it as a network of interconnected tensors, capturing the essential information while minimising redundancy. A tensor is a multi-dimensional array, generalising scalars, vectors and matrices. This representation allows for efficient calculations of derivatives and integrals, and can significantly reduce the memory requirements, enabling simulations on standard hardware. The tensor cross-interpolation (TCI) algorithm further enhances this efficiency by providing a means of transforming initial conditions and operators into the tensor network formalism, streamlining the computational process.

This work focuses on applying the Quantics Tensor Train (QTT) representation to solve the time-dependent Gross-Pitaevskii equation, a fundamental equation in the study of Bose-Einstein condensates and other quantum systems, offering a versatile platform for investigating complex physical phenomena. The Gross-Pitaevskii equation is particularly challenging due to its non-linear term, which introduces complexities in the numerical solution and demands robust algorithms capable of handling these intricacies. By demonstrating the effectiveness of the QTT approach in handling this non-linearity, this research aims to provide a powerful tool for simulating complex physical phenomena across multiple length scales, opening new avenues for scientific discovery.

The methodology presented achieves a significant improvement in computational efficiency, enabling the resolution of features separated by seven orders of magnitude in one dimension within a reasonable timeframe on standard hardware, surpassing the limitations of conventional methods. This capability opens new possibilities for simulating complex systems with unprecedented detail, allowing researchers to investigate phenomena previously inaccessible due to computational constraints. The approach is readily extensible to other partial differential equations, offering a versatile solution for a wide range of scientific and engineering applications, solidifying its position as a valuable tool for computational modelling.

Researchers now demonstrate a method leveraging tensor train (TT) representations to solve the time-dependent Gross-Pitaevskii equation, offering a versatile platform for investigating complex physical phenomena. This approach exploits the inherent compressibility present in solutions to physical problems, enabling simulations across a broad range of length scales.

The methodology centres on representing the wavefunction of the Bose-Einstein condensate (BEC) using the tensor train format, offering a compact and efficient representation of the system. This technique decomposes high-dimensional tensors into a network of lower-dimensional matrices, significantly reducing computational demands and memory requirements, enabling simulations on standard hardware. Crucially, the method demonstrates robust performance even in the presence of the non-linear term inherent in the Gross-Pitaevskii equation.

Results indicate a substantial improvement in computational efficiency, enabling researchers to investigate complex physical phenomena with unprecedented detail and accuracy. Researchers achieve solutions resolving phenomena across length scales differing by seven orders of magnitude within one hour on a standard laptop processor, surpassing the capabilities of conventional numerical methods. The method proves effective in simulating BECs within complex modulated optical trap potentials, including solutions on two-dimensional grids exceeding one trillion points, demonstrating the scalability and versatility of the approach.