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Studies in systems, decision and control

Towards a Tensor Network Representation of Complex Systems

Pouria Mistani, Samira Pakravan, Frédéric Gibou

Published Dec 11, 2018• 2 citations
Publisher
DOI
10.1007/978-3-319-98923-5_4

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This book serves as an introductory yet thorough guide to tensor networks and their applications in quantum computation and quantum information, designed for advanced undergraduate and graduate-level readers. In Part I, foundational topics are covered, including tensor structures and network representations like Matrix Product States (MPS) and Tree Tensor Networks (TTN). These preliminaries provide readers with the core mathematical tools and concepts necessary for quantum physics and quantum computing applications, bridging the gap between multi-linear algebra and complex quantum systems. Part II explores practical applications of tensor networks in simulating quantum dynamics, with a particular focus on the efficiency they offer for systems of high computational complexity. Key topics include Hamiltonian dynamics, quantum annealing, open system dynamics, and optimization strategies using TN frameworks. A final chapter addresses the emerging role of “quantum magic” in tensor networks. It delves into non-stabilizer states and their contribution to quantum computational power beyond classical simulability, featuring methods such as stabilizer-enhanced MPS and the Clifford-dressed TDVP. In presenting tensor networks as tools for understanding quantum complexity, this book aims to foster a deeper collaboration between the many-body physics and quantum computing communities, inviting a broader audience to engage with these recent developments.
This paper explores a novel approach to representing and quantifying uncertainty in complex systems using tensor networks. Traditional methods often struggle with the intricate correlations inherent in these systems, leading to inaccurate or computationally expensive uncertainty estimates. We propose leveraging the power of tensor networks to directly represent the joint distribution of variables, capturing these correlations efficiently. This approach offers the potential for significantly improved accuracy and scalability compared to conventional probabilistic representations. The core idea is to encode the uncertainty as a distributed representation within a tensor network, enabling efficient computation of correlations and ultimately, more precise uncertainty quantification. We demonstrate the conceptual framework and outline the key elements for developing a practical implementation of this method.
Tensor networks were developed in the context of many-body physics as compressed representations of multiparticle quantum states. These representations mitigate the exponential complexity of many-body systems by capturing only the most relevant dependencies. Due to the formal similarity between quantum entanglement and statistical correlations, tensor networks have recently been integrated in machine learning, operating both as alternative learning architectures and as decompositions of components of neural networks. The expectation is that the theoretical understanding of tensor networks developed within quantum many-body physics leads to novel methods that offer advantages in terms of computational efficiency, explainability, or privacy. Here we review the use of tensor networks in the context of machine learning, providing a critical assessment of the state of the art, the potential advantages, and the challenges that must be overcome.
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.