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arXiv:2507.03134

Quantics Tensor Train for solving Gross-Pitaevskii equation

Aleix Bou-Comas, Marcin Płodzień, Luca Tagliacozzo, Juan José García‐Ripoll

Published Jul 3, 2025• 0 citations
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Abstract

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.

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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.
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.
© 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.
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.
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.
We develop a dynamical tensor-train method for the Wigner--Fokker--Planck equation arising in continuous-variable open quantum systems. The Wigner function is evolved directly in tensor-train form using a projector-splitting integrator, while the nonlocal Wigner pseudo-differential operator is treated through low-rank representations of the potential difference and matrix-free exponential actions. The resulting method avoids full tensor-product computations in high-dimensional phase space. Numerical experiments in one, two, and three spatial dimensions demonstrate the accuracy and scalability of the method, including simulations in a six-dimensional phase space.
We present a quantum-inspired tensor-network framework for solving advection-diffusion-reaction (ADR) partial differential equations. Discretized solution fields are encoded as matrix product states (MPS), while differential operators are represented as matrix product operators (MPOs). Time integration is performed entirely in tensor-network form using explicit Euler updates with controlled truncation. The method is evaluated on one- and two-dimensional ADR problems and compared with high-accuracy Runge-Kutta reference solutions. Numerical results show that the proposed representation remains compact, stable, and accurate across a range of dynamical regimes. The solver captures both local solution profiles and global observables while maintaining small bond dimensions throughout the simulation. These results highlight the potential of tensor networks as efficient structure-preserving tools for PDE simulation in multiple spatial dimensions.