This paper explores a novel framework for analyzing and manipulating tensor networks, centered around the geometric properties of phase fields. Traditional tensor network analysis often overlooks the inherent geometric structure of these networks, hindering effective manipulation and understanding. We propose a "phase-field gradient" – a dynamically evolving field representing the phase relationships within a tensor network – that explicitly incorporates differential geometry to provide a more precise and insightful method for analyzing and manipulating network topology. This approach allows for the direct manipulation of network structure, opening avenues for new research in areas such as graph theory, optimization, and machine learning. The paper details the mathematical foundations of the phase-field gradient, provides a concrete implementation demonstrating its utility, and discusses potential applications and future directions.This paper explores a novel framework for analyzing and manipulating tensor networks, centered around the geometric properties of phase fields. Traditional tensor network analysis often overlooks the inherent geometric structure of these networks, hindering effective manipulation and understanding. We propose a "phase-field gradient" – a dynamically evolving field representing the phase relationships within a tensor network – that explicitly incorporates differential geometry to provide a more precise and insightful method for analyzing and manipulating network topology. This approach allows for the direct manipulation of network structure, opening avenues for new research in areas such as graph theory, optimization, and machine learning. The paper details the mathematical foundations of the phase-field gradient, provides a concrete implementation demonstrating its utility, and discusses potential applications and future directions.

Sep 4, 20260 citations

Survey of Tensor Networks

Samuel Desrosiers, Glen Evenbly, Thomas E. Baker

Bulletin of the American Physical SocietyMar 5, 20190 citations

Tensor Network States and Geometry

Glen Evenbly, Guifré Vidal

arXiv:1106.1082Journal of Statistical PhysicsJun 10, 2011266 citations

ExaTN: Scalable GPU-Accelerated High-Performance Processing of General Tensor Networks at Exascale

Dmitry I. Lyakh, Thien Nguyen, Daniel Claudino +2 more

We present ExaTN (Exascale Tensor Networks), a scalable GPU-accelerated C++ library which can express and process tensor networks on shared- as well as distributed-memory high-performance computing platforms, including those equipped with GPU accelerators. Specifically, ExaTN provides the ability to build, transform, and numerically evaluate tensor networks with arbitrary graph structures and complexity. It also provides algorithmic primitives for the optimization of tensor factors inside a given tensor network in order to find an extremum of a chosen tensor network functional, which is one of the key numerical procedures in quantum many-body theory and quantum-inspired machine learning. Numerical primitives exposed by ExaTN provide the foundation for composing rather complex tensor network algorithms. We enumerate multiple application domains which can benefit from the capabilities of our library, including condensed matter physics, quantum chemistry, quantum circuit simulations, as well as quantum and classical machine learning, for some of which we provide preliminary demonstrations and performance benchmarks just to emphasize a broad utility of our library.We present ExaTN (Exascale Tensor Networks), a scalable GPU-accelerated C++ library which can express and process tensor networks on shared- as well as distributed-memory high-performance computing platforms, including those equipped with GPU accelerators. Specifically, ExaTN provides the ability to build, transform, and numerically evaluate tensor networks with arbitrary graph structures and complexity. It also provides algorithmic primitives for the optimization of tensor factors inside a given tensor network in order to find an extremum of a chosen tensor network functional, which is one of the key numerical procedures in quantum many-body theory and quantum-inspired machine learning. Numerical primitives exposed by ExaTN provide the foundation for composing rather complex tensor network algorithms. We enumerate multiple application domains which can benefit from the capabilities of our library, including condensed matter physics, quantum chemistry, quantum circuit simulations, as well as quantum and classical machine learning, for some of which we provide preliminary demonstrations and performance benchmarks just to emphasize a broad utility of our library.

Frontiers in Applied Mathematics and StatisticsJul 6, 202219 citations

Lectures on Quantum Tensor Networks

Jacob Biamonte

Situated as a language between computer science, quantum physics and mathematics, tensor network theory has steadily grown in popularity and can now be found in applications ranging across the entire field of quantum information processing. This book aims to present the best contemporary practices in the use of tensor networks as a reasoning tool, placing quantum states, operators and processes on the same compositional footing. The book has 7 parts and over 40 subsections which took shape in over a decade of teaching. In addition to covering the foundations, the book covers important applications such as matrix product states, open quantum systems and entanglement $-$ all cast into the diagrammatic tensor network language. The intended audience includes those in quantum information science wishing to learn about tensor networks. It includes scientists who have employed tensor networks in their modeling codes who have interest in the tools graphical reasoning capacity. The audience further includes the graduate student researcher, whom with some effort, should find this book accessible. I would appreciate it if you emailed me about any mistakes or typos you find.Situated as a language between computer science, quantum physics and mathematics, tensor network theory has steadily grown in popularity and can now be found in applications ranging across the entire field of quantum information processing. This book aims to present the best contemporary practices in the use of tensor networks as a reasoning tool, placing quantum states, operators and processes on the same compositional footing. The book has 7 parts and over 40 subsections which took shape in over a decade of teaching. In addition to covering the foundations, the book covers important applications such as matrix product states, open quantum systems and entanglement $-$ all cast into the diagrammatic tensor network language. The intended audience includes those in quantum information science wishing to learn about tensor networks. It includes scientists who have employed tensor networks in their modeling codes who have interest in the tools graphical reasoning capacity. The audience further includes the graduate student researcher, whom with some effort, should find this book accessible. I would appreciate it if you emailed me about any mistakes or typos you find.

arXiv:1912.10049Dec 20, 201924 citations

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all cast into the diagrammatic tensor network language. The intended audience includes those in quantum information science wishing to learn about tensor networks. It includes scientists who have employed tensor networks in their modeling codes who have interest in the tools graphical reasoning capacity. The audience further includes the graduate student researcher, whom with some effort, should find this book accessible. I would appreciate it if you emailed me about any mistakes or typos you find.