Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.
Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.
Tensor networks (TNs) have become one of the most essential building blocks for various fields of theoretical physics such as condensed matter theory, statistical mechanics, quantum information, and quantum gravity. This review provides a unified description of a series of developments in the TN from the statistical mechanics side. In particular, we begin with the variational principle for the transfer matrix of the 2D Ising model, which naturally leads us to the matrix product state (MPS) and the corner transfer matrix (CTM). We then explain how the CTM can be evolved to such MPS-based approaches as density matrix renormalization group (DMRG) and infinite time-evolved block decimation. We also elucidate that the finite-size DMRG played an intrinsic role for incorporating various quantum information concepts in subsequent developments in the TN. After surveying higher-dimensional generalizations like tensor product states or projected entangled pair states, we describe tensor renormalization groups (TRGs), which are a fusion of TNs and Kadanoff-Wilson type real-space renormalization groups, focusing on their fixed point structures. We then discuss how the difficulty in TRGs for critical systems can be overcome in the tensor network renormalization and the multi-scale entanglement renormalization ansatz.
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 chapter is aimed at understanding TN algorithms from the perspective of MLA Multi-linear algebra (MLA) . In Sect. 5.1, we start from a simple example with a 1D TN Tensor network (TN) stripe, which can be “contracted” by solving the eigenvalue decomposition of matrices. This relates to several important MPS techniques such as canonicalization (Orús and Vidal, Phys Rev B 78:155117, 2008) that enables to implement optimal truncations of the bond dimensions of MPSs Matrix product state (MPS) (Sect. 5.1.1). In Sect. 5.2, we discuss about super-orthogonalization (Ran et al., Phys Rev B 86:134429, 2012) inspired by Tucker decomposition (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324–1342, 2000) in MLA, which is also a higher-dimensional generalization of canonicalization; it is proposed to implement optimal truncations of the iPEPSs defined on trees. In Sect. 5.3.1, we explain based on the rank-1 decomposition (De Lathauwer et al., SIAM J Matrix Anal Appl 21:1253–1278, 2000) that super-orthogonalization in fact provides the “loopless approximation” of the iPEPSs Infinite projected entangled pair state (iPEPS) on regular lattices (Ran et al., Phys Rev B 88:064407, 2013); it explains how the approximations in the simple update algorithm works for the ground-state simulations on 2D regular lattices (Jiang et al., Phys Rev Lett 101:090603, 2008). In Sect. 5.4, we will discuss tensor ring decomposition (TRD) Tensor ring decomposition (TRD) (Ran, Phys Rev E 93:053310, 2016), which is a rank-N generalization of the rank-1 decomposition. TRD naturally provides a unified description of iDMRG (White, Phys Rev Lett 69:2863, 1992; Phys Rev B 48:10345–10356, 1993; McCulloch, Infinite size density matrix renormalization group, revisited, 2008. arXiv:0804.2509) Infinite density matrix renormalization group (iDMRG) , iTEBD (Vidal, Phys Rev Lett 98:070201, 2007) Infinite time-evolving block decimation (iTEBD) , and CTMRG Corner transfer matrix renormalization group (CTMRG) (Orús and Vidal, Phys Rev B 80:094403, 2009; Fishman et al., Faster methods for contracting infinite 2D tensor networks, 2017. arXiv:1711.05881) when considering the contractions of 2D TNs Tensor network (TN) .
Exact many-body quantum problems are known to be computationally hard due to the exponential scaling of the numerical resources required. Since the advent of the Density Matrix Renormalization Group, it became clear that a successful strategy to work around this obstacle was to develop numerical methods based on the well-known theoretical renormalization group. In recent years, it was realized that quantum states engineered via numerical renormalization allow a variational representation in terms of a tensor network picture. The discovery provided a further boost to the effectiveness of these techniques, not only due to the increased flexibility and manipulability, but also because tensor network states embed a direct interface to the entanglement they carry, so that one can directly address many-body quantum correlations within these variational ansatz states. This lead to the application of several numerical tools, originally developed in the field of quantum-information, to approach condensed matter problems.
Renormalization group theory of tensor network states provides a powerful tool for studying quantum many-body problems and a new paradigm for understanding entangled structures of complex systems. In recent decades the theory has rapidly evolved into a universal framework and language employed by researchers in fields ranging from condensed matter theory to machine learning. This book presents a pedagogical and comprehensive introduction to this field for the first time. After an introductory survey on the major advances in tensor network algorithms and their applications, it introduces step-by-step the tensor network representations of quantum states and the tensor-network renormalization group methods developed over the past three decades. Basic statistical and condensed matter physics models are used to demonstrate how the tensor network renormalization works. An accessible primer for scientists and engineers, this book would also be ideal as a reference text for a graduate course in this area.
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.