Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2202.02138 · Frontiers in Applied Mathematics and Statistics

A Practical Guide to the Numerical Implementation of Tensor Networks I: Contractions, Decompositions, and Gauge Freedom

Glen Evenbly

Published Jun 1, 2022• 23 citations
arXivHTMLPDF

Abstract

We present an overview of the key ideas and skills necessary to begin implementing tensor network methods numerically, which is intended to facilitate the practical application of tensor network methods for researchers that are already versed with their theoretical foundations. These skills include an introduction to the contraction of tensor networks, to optimal tensor decompositions, and to the manipulation of gauge degrees of freedom in tensor networks. The topics presented are of key importance to many common tensor network algorithms such as DMRG, TEBD, TRG, PEPS, and MERA.

DOI
10.3389/fams.2022.806549

Related papers

Two-Dimensional Tensor Networks and Contraction Algorithms

Shi-Ju Ran, Emanuele Tirrito, Cheng Peng +4 more

In this section, we will first demonstrate in Sect. 3.1 that many important physical problems can be transformed to 2D TNs Tensor network (TN) , and the central tasks become to compute the corresponding TN contractions. From Sects. 3.2 to 3.5, we will then present several paradigm contraction algorithms of 2D TNs including TRG Tensor renormalization group (TRG) , TEBD Time-evolving block decimation (TEBD) , and CTMRG Corner transfer matrix renormalization group (CTMRG) . Relations to other distinguished algorithms and the exactly contractible TNs will also be discussed.

Lecture notes in physics•Jan 1, 2020•4 citations

Tensor Network Techniques for Quantum Computation

Mario Collura, Guglielmo Lami, Nishan Ranabhat +1 more

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.

arXiv:2503.04423•Dec 1, 2024•7 citations

Tensor Networks in a Nutshell

Jacob Biamonte, Ville Bergholm

Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits, channels, protocols, open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions, the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae, whereas the second is Penrose's tensor contraction algorithm, returning the number of $3$-edge-colorings of $3$-regular planar graphs.

arXiv:1708.00006•Jul 31, 2017•113 citations

Introduction to matrix-product states and tensor networks

Grégoire Misguich

These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.

cond-mat.str-elCross-listedarXiv:2606.24803•Jun 24, 2026•0 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.

arXiv:1912.10049•Dec 20, 2019•24 citations

DMRG Approach to Optimizing Two-Dimensional Tensor Networks

Katharine Hyatt, Edwin Miles Stoudenmire

Tensor network algorithms have been remarkably successful solving a variety of problems in quantum many-body physics. However, algorithms to optimize two-dimensional tensor networks known as PEPS lack many of the aspects that make the seminal density matrix renormalization group (DMRG) algorithm so powerful for optimizing one-dimensional tensor networks known as matrix product states. We implement a framework for optimizing two-dimensional PEPS tensor networks which includes all of steps that make DMRG so successful for optimizing one-dimension tensor networks. We present results for several 2D spin models and discuss possible extensions and applications.

arXiv:1908.08833•Aug 23, 2019•16 citations
We present an overview of the key ideas and skills necessary to begin implementing tensor network methods numerically, which is intended to facilitate the practical application of tensor network methods for researchers that are already versed with their theoretical foundations. These skills include an introduction to the contraction of tensor networks, to optimal tensor decompositions, and to the manipulation of gauge degrees of freedom in tensor networks. The topics presented are of key importance to many common tensor network algorithms such as DMRG, TEBD, TRG, PEPS, and MERA.
In this section, we will first demonstrate in Sect. 3.1 that many important physical problems can be transformed to 2D TNs Tensor network (TN) , and the central tasks become to compute the corresponding TN contractions. From Sects. 3.2 to 3.5, we will then present several paradigm contraction algorithms of 2D TNs including TRG Tensor renormalization group (TRG) , TEBD Time-evolving block decimation (TEBD) , and CTMRG Corner transfer matrix renormalization group (CTMRG) . Relations to other distinguished algorithms and the exactly contractible TNs will also be discussed.
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.
Tensor network methods are taking a central role in modern quantum physics and beyond. They can provide an efficient approximation to certain classes of quantum states, and the associated graphical language makes it easy to describe and pictorially reason about quantum circuits, channels, protocols, open systems and more. Our goal is to explain tensor networks and some associated methods as quickly and as painlessly as possible. Beginning with the key definitions, the graphical tensor network language is presented through examples. We then provide an introduction to matrix product states. We conclude the tutorial with tensor contractions evaluating combinatorial counting problems. The first one counts the number of solutions for Boolean formulae, whereas the second is Penrose's tensor contraction algorithm, returning the number of 333-edge-colorings of 333-regular planar graphs.
These notes provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS). They develop the basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation. The main MPS algorithms are then introduced, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods. The notes also briefly discuss projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction. Finally, tensor-network representations of mixed states, quantum channels, and Lindblad dynamics are presented, with applications to thermal states and open quantum systems. The presentation is accompanied by short Julia code examples based on ITensor, ITensorMPS, and TensorMixedStates. These notes were written for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems, held in June 2026.
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.
Tensor network algorithms have been remarkably successful solving a variety of problems in quantum many-body physics. However, algorithms to optimize two-dimensional tensor networks known as PEPS lack many of the aspects that make the seminal density matrix renormalization group (DMRG) algorithm so powerful for optimizing one-dimensional tensor networks known as matrix product states. We implement a framework for optimizing two-dimensional PEPS tensor networks which includes all of steps that make DMRG so successful for optimizing one-dimension tensor networks. We present results for several 2D spin models and discuss possible extensions and applications.