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The Journal of The Math3ma Institute

From Rotations and Groups to Tensor Networks

Edwin Miles Stoudenmire

Published Jan 1, 2023• 0 citations
Publisher

Abstract

Even the simplest mathematical concepts can have surprisingly deep struc-ture and hidden connections to other topics. In this article we explore rotations of two-dimensional objects. After viewing rotations first concretely as small matrices, then more abstractly as members of a "group", we consider how rotations transform simple functions (x, y, x -y , etc.) and their products. Transforming products of three or more functions is best captured by introducing multi-dimensional generalizations of vectors and matrices known as tensors and chaining these tensors together. The resulting object known as a tensor train (or matrix product state) is one of the most powerful computational tools for applications such as modeling fluid flows, understanding properties of metals and crystals, predicting financial markets, and simulating quantum computers.

DOI
10.55409/math3ma2023022

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Even the simplest mathematical concepts can have surprisingly deep struc-ture and hidden connections to other topics. In this article we explore rotations of two-dimensional objects. After viewing rotations first concretely as small matrices, then more abstractly as members of a "group", we consider how rotations transform simple functions (x, y, x -y , etc.) and their products. Transforming products of three or more functions is best captured by introducing multi-dimensional generalizations of vectors and matrices known as tensors and chaining these tensors together. The resulting object known as a tensor train (or matrix product state) is one of the most powerful computational tools for applications such as modeling fluid flows, understanding properties of metals and crystals, predicting financial markets, and simulating quantum computers.
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 is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algorithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the "quantics" representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross-Pitaevskii) within the "quantics" representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements.
The study of quantum systems involving two or more qubits necessitates the development of mathematical tools which capture the interactions between the qubits. This leads to the construction of objects called tensors which are elements of a vector space called a tensor product space. This material is essential since tensors are required for defining the means of data representation in quantum computation. The tensor product is the result of combining two vectors in a particular way designed to capture the interaction between the various parts of the individual vectors. The results of finding tensor products reside in a vector space called the tensor product space, the elements of which are called tensors. The formal construction of a tensor product space is somewhat abstract and can be difficult for the student to understand.
In this informal course, I intend to give an introduction to Tensor Networks and their use for numerical simulations.The class is targeted towards PhD students and researchers with no background on the subject. The goal is for the audience to be able to concretely use TNs in their research at the end of the course.The first session will start with a quick global introduction to tensor networks, and then focus on the one-dimensional Matrix Product States (Tensor Trains).We will go through their properties, strengths and limitations, and give a few explicit examples. We will then study a few concrete time-evolution and groundstate-search algorithms for finite and infinite systems.
The tensor train (TT) model is widely used to approximate high-dimensional tensors, enabling efficient handling of data that may exceed available memory. TT helps address the curse of dimensionality in applications such as system identification and dynamic programming. In some applications, TT is known as a ``matrix product state" (MPS). The last product is also known under the acronym MPO--MPS. The proposed algorithms achieve an improved trade-off between computational efficiency and accuracy compared to state-of-the-art methods.
Transfer matrices and matrix product operators play a ubiquitous role in the field of many-body physics. This review gives an idiosyncratic overview of applications, exact results, and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and new results. Topics discussed are exact solutions of transfer matrices in equilibrium and nonequilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.