Tensor networks are a popular and computationally efficient approach to simulate general quantum systems on classical computers and, in a broader sense, a framework for dealing with high-dimensional numerical problems. This paper presents a broad but not exhaustive literature review of state-of-the-art applications of tensor networks and related topics across many research domains, including: machine learning, mathematical optimization, materials science, quantum chemistry and quantum circuit simulation. This review aims to clarify which classes of relevant applications have been proposed for which class of tensor networks, as well as to highlight main application results and limitations compared with other classical or quantum simulation methods. We intend this review to be a high-level tour on tensor network applications that is easy to read by non-experts, so basic technical details of tensor networks are summarized.
Tensor networks are a popular and computationally efficient approach to simulate general quantum systems on classical computers and, in a broader sense, a framework for dealing with high-dimensional numerical problems. This paper presents a broad but not exhaustive literature review of state-of-the-art applications of tensor networks and related topics across many research domains, including: machine learning, mathematical optimization, materials science, quantum chemistry and quantum circuit simulation. This review aims to clarify which classes of relevant applications have been proposed for which class of tensor networks, as well as to highlight main application results and limitations compared with other classical or quantum simulation methods. We intend this review to be a high-level tour on tensor network applications that is easy to read by non-experts, so basic technical details of tensor networks are summarized.
Tensor network has been a promising numerical tool for computational problems across science and AI. For their emerging and fast development especially in the intersection between AI and science, this paper tries to present a compact review, regarding both their applications and its own recent technical development including open-source tools. Specifically, we make the observations that tensor network plays a functional role in matrix compression and representation, information fusion, as well as quantum-inspired algorithms, which can be generally regarded as Science4AI in our survey. On the other hand, there is an emerging line of research in tensor network in AI4Science especially like learning quantum many-body physics by using e.g. neural network quantum state. Importantly, we unify tensorization methodologies across classical and modern architectures, and particularly show how tensorization bridges low-order parameter spaces to high-dimensional representations without exponential parameter growth, and further point out their potential use in scientific computing. We conclude the paper with outlook for future trends.
Tensor network has been a promising numerical tool for computational problems across science and AI. For their emerging and fast development especially in the intersection between AI and science, this paper tries to present a compact review, regarding both their applications and its own recent technical development including open-source tools. Specifically, we make the observations that tensor network plays a functional role in matrix compression and representation, information fusion, as well as quantum-inspired algorithms, which can be generally regarded as Science4AI in our survey. On the other hand, there is an emerging line of research in tensor network in AI4Science especially like learning quantum many-body physics by using e.g. neural network quantum state. Importantly, we unify tensorization methodologies across classical and modern architectures, and particularly show how tensorization bridges low-order parameter spaces to high-dimensional representations without exponential parameter growth, and further point out their potential use in scientific computing. We conclude the paper with outlook for future trends.
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.
In the effort to develop useful quantum computers, simulating quantum machines with conventional classical computing resources is a key capability. Such simulations will always face limits, preventing the emulation of quantum computers at substantial scale; however, by pushing the envelope through optimal choices of algorithms and hardware, the value of simulator tools can be maximized. This work reviews state-of-the-art numerical simulation methods, i.e., classical algorithms that emulate quantum computer evolution under specific operations. We focus on the mainstream state-vector and tensor-network paradigms, while briefly mentioning alternative methods. Moreover, we review the diverse applications of simulation across different facets of quantum computer development, including understanding the fundamental differences between quantum and classical computations, exploring algorithmic design for quantum advantage, predicting quantum processor performance at the design stage, and efficiently characterizing fabricated devices for rapid iterations. This review complements recent surveys of current tools and implementations; here, we aim to provide readers with an essential understanding of the theoretical basis of classical simulation methods, a detailed discussion of their advantages and limitations, and an overview of the demands and challenges arising from practical use cases.
This paper explores the application of tensor network methods for simulating molecular dynamics on quantum computers. Traditional molecular dynamics simulations are computationally expensive, particularly for complex systems exhibiting strong quantum correlations. Tensor networks provide a powerful framework for representing and manipulating high-dimensional quantum states, potentially offering a pathway to efficiently simulate molecular dynamics by directly leveraging quantum computational resources. This work details the theoretical framework, outlining how tensor network techniques, specifically Matrix Product States (MPS) and Tree Tensor Networks (TTN), can be adapted for this purpose. We present a simplified model of a diatomic molecule to illustrate the core concept: the construction of a tensor network representation of the molecular wavefunction, followed by the application of time evolution operators to simulate dynamics. The focus is on demonstrating the potential of this approach rather than a fully optimized simulation, highlighting key challenges and future research directions. The efficiency gains compared to classical methods are discussed, acknowledging the significant technological hurdles involved in realizing this approach on near-term quantum devices. We investigate the scalability of the method, considering the limitations imposed by the bond dimension of the tensor network. The ultimate goal is to demonstrate how tensor networks can provide a viable strategy for tackling computationally intractable molecular dynamics problems, paving the way for a new era of quantum-enhanced molecular simulations.
We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimensional physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer’s companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.