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Solving the Gross-Pitaevskii equation with quantic tensor trains: Ground states and nonlinear dynamics

Qian Can Chen, I-Kang Liu, Jheng Wei Li, Chia Min Chung

Published Aug 10, 2026• 0 citations
Publisher

Abstract

© Q.-C. Chen et al. We develop a tensor network framework based on the quantic tensor train (QTT) format to efficiently solve the Gross-Pitaevskii equation (GPE), which governs Bose-Einstein condensates under mean-field theory. For sufficiently regular functions, the QTT approach enables exponentially fine grids with computational cost that grows only linearly. By adapting the time-dependent variational principle (TDVP) and gradient descent methods, we accurately handle the GPE’s nonlinearities within the QTT structure. Our approach enables high-resolution simulations with drastically reduced computational cost on highly refined grids. We benchmark the ground states and dynamics of BECs, including vortex lattice formation and breathing modes, demonstrating superior performance over regular-grid-based methods and stable long-time evolution due to saturating bond dimensions. This establishes QTT as a powerful tool for nonlinear quantum simulations.

DOI
10.25405/ncl.34552800

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© Q.-C. Chen et al. We develop a tensor network framework based on the quantic tensor train (QTT) format to efficiently solve the Gross-Pitaevskii equation (GPE), which governs Bose-Einstein condensates under mean-field theory. For sufficiently regular functions, the QTT approach enables exponentially fine grids with computational cost that grows only linearly. By adapting the time-dependent variational principle (TDVP) and gradient descent methods, we accurately handle the GPE’s nonlinearities within the QTT structure. Our approach enables high-resolution simulations with drastically reduced computational cost on highly refined grids. We benchmark the ground states and dynamics of BECs, including vortex lattice formation and breathing modes, demonstrating superior performance over regular-grid-based methods and stable long-time evolution due to saturating bond dimensions. This establishes QTT as a powerful tool for nonlinear quantum simulations.
We develop a tensor network framework based on the quantic tensor train (QTT) format to efficiently solve the Gross-Pitaevskii equation (GPE), which governs Bose-Einstein condensates under mean-field theory. For sufficiently regular functions, the QTT approach enables exponentially fine grids with computational cost that grows only linearly. By adapting the time-dependent variational principle (TDVP) and gradient descent methods, we accurately handle the GPE’s nonlinearities within the QTT structure. Our approach enables high-resolution simulations with drastically reduced computational cost on highly refined grids. We benchmark the ground states and dynamics of BECs, including vortex lattice formation and breathing modes, demonstrating superior performance over regular-grid-based methods and stable long-time evolution due to saturating bond dimensions. This establishes QTT as a powerful tool for nonlinear quantum simulations.
We present a quantum-inspired solver for the one-dimensional Gross-Pitaevskii equation in the Quantics Tensor-Train (QTT) representation. By evolving the system entirely within a low-rank tensor manifold, the method sidesteps the memory and runtime barriers that limit conventional finite-difference and spectral schemes. Two complementary algorithms are developed: an imaginary-time projector that drives the condensate toward its variational ground state and a rank-adapted fourth-order Runge-Kutta integrator for real-time dynamics. The framework captures a broad range of physical scenarios - including barrier-confined condensates, quasi-random potentials, long-range dipolar interactions, and multicomponent spinor dynamics - without leaving the compressed representation. Relative to standard discretizations, the QTT approach achieves an exponential reduction in computational resources while retaining quantitative accuracy, thereby extending the practicable regime of Gross-Pitaevskii simulations on classical hardware. These results position tensor networks as a practical bridge between high-performance classical computing and prospective quantum hardware for the numerical treatment of nonlinear Schrodinger-type partial differential equations.
Solving partial differential equations across multiple length scales represents a formidable challenge where reaching high precision 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 tensor network solver for the time-dependent Gross-Pitaevskii equation. We demonstrate that the quantics approach generalizes well to the presence of the nonlinear 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 × 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.
Abstract This paper introduces a novel numerical method based on an energy-minimizing normalized residual network (EM-NormResNet) to compute the ground-state solution of Bose–Einstein condensates at zero or low temperatures. Starting from the three-dimensional Gross–Pitaevskii equation (GPE), we reduce it to the 1D and 2D GPEs because of the radial symmetry and cylindrical symmetry. The ground-state solution is formulated by minimizing the energy functional under constraints, which is directly solved using the EM-NormResNet approach. The paper provides detailed solutions for the ground states in 1D, 2D (with radial symmetry), and 3D (with cylindrical symmetry). We use the Thomas–Fermi approximation as the target function to pre-train the neural network. Then, the formal network is trained using the energy minimization method. In contrast to traditional numerical methods, our neural network approach introduces two key innovations: (i) a novel normalization technique designed for high-dimensional systems within an energy-based loss function; (ii) improved training efficiency and model robustness by incorporating gradient stabilization techniques into residual networks. Extensive numerical experiments validate the method’s accuracy across different spatial dimensions.
Accurately solving high-dimensional partial differential equations (PDEs) remains a central challenge in computational mathematics. Traditional numerical methods, while effective in low-dimensional settings or on coarse grids, often struggle to deliver the precision required in practical applications. Recent machine learning-based approaches offer flexibility but frequently fall short in terms of accuracy and reliability, particularly in industrial contexts. In this work, we explore a quantum-inspired method based on quantized tensor trains (QTT), enabling efficient and accurate solutions to PDEs in a variety of challenging scenarios. Through several representative examples, we show that the QTT approach can achieve logarithmic scaling in memory and computational cost for linear PDEs when the relevant QTT ranks remain moderate. We also develop QTT space-time formulations that treat time as an additional dimension, allowing the full temporal evolution to be represented and solved globally rather than through sequential time stepping. For the nonlinear Burgers equation, we study both time-stepping and a frozen-coefficient space-time Picard scheme in QTT form, and report empirical convergence behavior on smooth one-dimensional viscous test problems. Additionally, we present a proof-of-concept data-driven workflow within the quantum-inspired framework, in which sampled source data are interpolated into QTT form and then incorporated directly into the structured PDE solver.
Abstract The Gross–Pitaevskii equation and its generalisations to dissipative and dipolar gases have been very useful in describing dynamics of cold atomic gases, as well as polaritons and other nonlinear systems. For some of these applications the numerically accessible grid spacing can become a limiting factor, especially in describing turbulent dynamics and short-range effects of dipole-dipole interactions. We explore the application of tensor networks to these systems, where (in analogy to related work in fluid and plasma dynamics), they allow for physically motivated data compression that makes simulations possible on large spatial grids which would be unfeasible with direct numerical simulations. Analysing different non-equilibrium cases involving vortex formation, we find that these methods are particularly efficient, especially in combination with a matrix product operator representation of the quantum Fourier transform, which enables a spectral approach to calculation of both equilibrium states and time-dependent dynamics. The efficiency of these methods has interesting physical implications for the structure in the states that are generated by these dynamics, and provides a path to describe cold gas experiments that are challenging for existing methods.