We present a tensor network model (TNM) for forecasting nonlinear and chaotic dynamics, bridging quantum many-body methods with classical complex systems. The TNM leverages hierarchical tensor contractions to encode non-Markovian temporal correlations and multiscale structures, enabling compact and interpretable representations of chaotic flows. Using the Lorenz and Rössler systems as benchmarks, we show that the TNM accurately reconstructs short-term trajectories and faithfully captures the attractor geometry. The model enables robust short-term forecasting beyond several Lyapunov times, offering a meaningful horizon for data-driven prediction under chaos. Inhomogeneous parametrization of weight tensors improves convergence and robustness compared to homogeneous parametrization, while scaling with bond dimension reveals saturation beyond modest values, consistent with the low intrinsic dimensionality of the chaotic attractor. This work establishes tensor networks as a universal paradigm for data-driven modeling of complex dynamical systems, offering physically motivated control of model expressivity and opening pathways toward applications in climate systems and hybrid quantum-classical simulations.
We present a tensor network model (TNM) for forecasting nonlinear and chaotic dynamics, bridging quantum many-body methods with classical complex systems. The TNM leverages hierarchical tensor contractions to encode non-Markovian temporal correlations and multiscale structures, enabling compact and interpretable representations of chaotic flows. Using the Lorenz and Rössler systems as benchmarks, we show that the TNM accurately reconstructs short-term trajectories and faithfully captures the attractor geometry. The model enables robust short-term forecasting beyond several Lyapunov times, offering a meaningful horizon for data-driven prediction under chaos. Inhomogeneous parametrization of weight tensors improves convergence and robustness compared to homogeneous parametrization, while scaling with bond dimension reveals saturation beyond modest values, consistent with the low intrinsic dimensionality of the chaotic attractor. This work establishes tensor networks as a universal paradigm for data-driven modeling of complex dynamical systems, offering physically motivated control of model expressivity and opening pathways toward applications in climate systems and hybrid quantum-classical simulations.
This paper investigates the application of tensor networks to model chaotic dynamical systems. We present a novel method for representing and manipulating chaotic data using tensor networks, enabling the simulation of their evolution and the generation of predictive capabilities. The core mechanism centers around leveraging the inherent properties of tensor networks to capture the complex dynamics of chaotic systems, offering a powerful framework for studying and potentially predicting their behavior. The study encompasses the design of a tensor network representation, the implementation of a simulation algorithm, and an analysis of the resulting dynamical behavior. The potential of this approach to modeling complex systems and generating predictive capabilities is discussed.
The prediction of complex dynamics remains an open problem across many domains of physics, where nonlinearities and multiscale interactions severely limit the reliability of conventional forecasting methods. Quantum Reservoir Computing has emerged as a promising paradigm for information processing by exploiting the nonlinear dynamical evolution of quantum systems to generate expressive feature representations of input signals. Here, we introduce a hybrid quantum-classical reservoir architecture capable of handling multivariate time series through quantum evolution combined with classical memory enhancement. Our model employs a five-qubit transverse-field Ising Hamiltonian with input-modulated dynamics and temporal multiplexing. We apply this framework to two paradigmatic models of chaotic behavior in fluid dynamics, where multiscale dynamics and nonlinearities play a dominant role: a low-dimensional truncation of the two-dimensional Navier-Stokes equations and the Lorenz-63 system. By systematically scanning the quantum system's parameter space, we identify regions that maximize forecasting performance. The observed robustness and reliable performances for both dynamical systems suggest that this hybrid quantum approach offers a flexible platform for modeling complex nonlinear time series.
Tensor networks can compactly represent high-dimensional data in the form of low-rank tensor decompositions, which helps capture complex structures and correlations within the data. However, their inherently multilinear nature limits their potential as universal approximators. To address this limitation, tensor networks are combined with classical neural network layers, introducing nonlinear elements to enhance the model’s expressive power. The Hybrid Tensor Networks (HTN) system encompasses two prevalent tensor network configurations, Matrix Product States and Tree Tensor Networks, which facilitates the adaptable design of models customized for particular machine learning applications. This framework has been validated on classification and image reconstruction tasks, demonstrating its potential to handle complex data structures and its applicability within the machine learning field. Employing this approach, the goal is to establish a foundation for future research on hybrid quantum-classical models, thereby opening up avenues for the realm of quantum machine learning.
The accurate simulation of complex systems, encompassing fields like materials science, quantum chemistry, and finance, often presents a formidable computational challenge. Traditional methods frequently struggle to scale effectively, particularly when dealing with high-dimensionality or intricate correlations. This paper introduces a novel simulation approach leveraging tensor networks (TN) in conjunction with quantum-inspired sampling techniques. The core idea is to approximate the dynamics of these complex systems by exploiting the inherent structure of tensor networks and employing sampling strategies inspired by quantum measurement processes. We demonstrate that this hybrid approach offers a pathway towards scalable simulations, achieving high accuracy while mitigating the computational bottlenecks associated with conventional methods. The key innovations lie in the adaptive construction of TNs based on the system's correlation structure and the utilization of quantum-inspired sampling methods to efficiently capture the relevant dynamics. This work presents a theoretical framework and outlines a methodology for implementing such simulations, paving the way for advancements in the modeling and understanding of complex systems.
Accurately simulating chaotic quantum many-body dynamics remains a major computational challenge for classical methods, due to the rapid buildup and spatial spreading of entanglement during the evolution. This raises the question of whether machine learning can provide an effective alternative for predicting quantum dynamics. We address this question by formulating quantum dynamics as a time-series forecasting problem, using a few-qubit PXP chain, realizable with Rydberg-atom arrays, as a benchmark. By varying the initial state, the system spans dynamical regimes ranging from ergodic behavior to quantum many-body scarring, providing a controlled setting for testing forecasting models across qualitatively different dynamics. We compare two contrasting architectures: an expressive nonlinear Transformer and DLinear, a simple linear forecasting model. The Transformer accurately predicts dynamics in the more ergodic regime, but its performance progressively deteriorates as the initial state approaches the scarred limit. In contrast, DLinear remains accurate across the entire family of initial states, with its main deviations consisting of small high-frequency oscillations that have little effect on the overall prediction error. Remarkably, these results show that observables generated by complex quantum many-body dynamics can be forecast with high accuracy through a simple linear mapping from past to future observations. This reveals that the complexity of the underlying quantum evolution need not translate into an equally complex forecasting problem.
Predicting the properties of correlated materials and molecules often requires a many-body description that goes beyond single-particle and mean-field approximations. Tensor networks address this challenge by exploiting the entanglement structure of quantum states to construct compact, systematically improvable representations. Although developed largely for idealized model Hamiltonians, these methods are increasingly being integrated into material-specific calculations. This Perspective introduces their underlying principles and examines how network geometry, bond-dimension truncation, and approximate contraction determine the accuracy and computational reach of these approaches. We survey applications across regimes of increasing complexity, ranging from ground-state properties of quantum magnets and itinerant-fermion systems to excitation spectra, nonequilibrium dynamics, finite-temperature and open systems, and molecular electronic structure. We then examine four routes connecting these methods to real materials, including experimentally constrained models, downfolded Hamiltonians, dynamical mean-field theory with tensor network impurity solvers, and quantum embedding. Throughout, we evaluate the maturity of existing capabilities and their principal limitations, synthesizing this assessment into a practical method-selection guide. We argue that reliable materials predictions require quantifying uncertainties introduced at each stage, from constructing material-specific Hamiltonians to extracting experimentally accessible observables. Finally, we identify the algorithmic advances and reproducible computational practices needed to establish tensor networks as a routine tool for predictive materials science and quantum chemistry.