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arXiv:2602.07945

A quantum-inspired multi-level tensor-train monolithic space-time method for nonlinear PDEs

N. R. Rapaka, R. Peddinti, E. Tiunov, N. J. Faraj, A. N. Alkhooori, L. Aolita, Y. Addad, M. K. Riahi

math.NAcs.PFphysics.comp-phquant-phSubmitted Feb 8, 2026• Announced Feb 10, 2026
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Abstract

We propose a multilevel tensor-train (TT) framework for solving nonlinear partial differential equations (PDEs) in a global space-time formulation. While space-time TT solvers have demonstrated significant potential for compressed high-dimensional simulations, the literature contains few systematic comparisons with classical time-stepping methods, limited error convergence analyses, and little quantitative assessment of the impact of TT rounding on numerical accuracy. Likewise, existing studies fail to demonstrate performance across a diverse set of PDEs and parameter ranges. In practice, monolithic Newton iterations may stagnate or fail to converge in strongly nonlinear, stiff, or advection-dominated regimes, where poor initial guesses and severely ill-conditioned space-time Jacobians hinder robust convergence. We overcome this limitation by introducing a coarse-to-fine multilevel strategy fully embedded within the TT format. Each level refines both spatial and temporal resolutions while transferring the TT solution through low-rank prolongation operators, providing robust initializations for successive Newton solves. Residuals, Jacobians, and transfer operators are represented directly in TT and solved with the adaptive-rank DMRG algorithm. Numerical experiments for a selection of nonlinear PDEs including Fisher-KPP, viscous Burgers, sine-Gordon, and KdV cover diffusive, convective, and dispersive dynamics, demonstrating that the multilevel TT approach consistently converges where single-level space-time Newton iterations fail. In dynamic, advection-dominated (nonlinear) scenarios, multilevel TT surpasses single-level TT, achieving high accuracy with significantly reduced computational cost, specifically when high-fidelity numerical simulation is required.

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We propose a multilevel tensor-train (TT) framework for solving nonlinear partial differential equations (PDEs) in a global space-time formulation. While space-time TT solvers have demonstrated significant potential for compressed high-dimensional simulations, the literature contains few systematic comparisons with classical time-stepping methods, limited error convergence analyses, and little quantitative assessment of the impact of TT rounding on numerical accuracy. Likewise, existing studies fail to demonstrate performance across a diverse set of PDEs and parameter ranges. In practice, monolithic Newton iterations may stagnate or fail to converge in strongly nonlinear, stiff, or advection-dominated regimes, where poor initial guesses and severely ill-conditioned space-time Jacobians hinder robust convergence. We overcome this limitation by introducing a coarse-to-fine multilevel strategy fully embedded within the TT format. Each level refines both spatial and temporal resolutions while transferring the TT solution through low-rank prolongation operators, providing robust initializations for successive Newton solves. Residuals, Jacobians, and transfer operators are represented directly in TT and solved with the adaptive-rank DMRG algorithm. Numerical experiments for a selection of nonlinear PDEs including Fisher-KPP, viscous Burgers, sine-Gordon, and KdV cover diffusive, convective, and dispersive dynamics, demonstrating that the multilevel TT approach consistently converges where single-level space-time Newton iterations fail. In dynamic, advection-dominated (nonlinear) scenarios, multilevel TT surpasses single-level TT, achieving high accuracy with significantly reduced computational cost, specifically when high-fidelity numerical simulation is required.
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We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions.
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.
Quantum algorithms for partial differential equations (PDEs) face severe practical constraints on near-term hardware: limited qubit counts restrict spatial resolution to coarse grids, while circuit depth limitations prevent accurate long-time integration. These hardware bottlenecks confine quantum PDE solvers to low-fidelity regimes despite their theoretical potential for computational speedup. We introduce a multifidelity learning framework that corrects coarse quantum solutions to high-fidelity accuracy using sparse classical training data, facilitating the path toward practical quantum utility for scientific computing. The approach trains a low-fidelity surrogate on abundant quantum solver outputs, then learns correction mappings through a multifidelity neural architecture that balances linear and nonlinear transformations. Demonstrated on benchmark nonlinear PDEs including viscous Burgers equation and incompressible Navier-Stokes flows via quantum lattice Boltzmann methods, the framework successfully corrects coarse quantum predictions and achieves temporal extrapolation well beyond the classical training window. This strategy illustrates how one can reduce expensive high-fidelity simulation requirements while producing predictions that are competitive with classical accuracy. By bridging the gap between hardware-limited quantum simulations and application requirements, this work establishes a pathway for extracting computational value from current quantum devices in real-world scientific applications, advancing both algorithm development and practical deployment of near-term quantum computing for computational physics.
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.