In this section, we will show several representative TN Tensor network (TN) approaches for simulating the quantum lattice models in (d > 1) dimensions. We will mainly use the language of TN contractions. One may refer to several existing reviews (Schollwöck, Ann Phys 326:96–192, 2011; Verstraete et al., Adv Phys 57:143–224, 2008; Cirac and Verstraete, J Phys A Math Theor 42:504004, 2009; Orús, Ann Phys 349:117, 2014; Haegeman and Verstraete, Ann Rev Condens Matter Phys 8(1):355–406, 2017) for more exhaustive understanding on the TN simulations for quantum problems. We will focus on the algorithms based on PEPS Projected entangled pair state (PEPS) , and show the key roles that the 2D TN contraction algorithms presented in Sect. 3 play in the higher-dimensional cases.
In this section, we will show several representative TN Tensor network (TN) approaches for simulating the quantum lattice models in (d > 1) dimensions. We will mainly use the language of TN contractions. One may refer to several existing reviews (Schollwöck, Ann Phys 326:96–192, 2011; Verstraete et al., Adv Phys 57:143–224, 2008; Cirac and Verstraete, J Phys A Math Theor 42:504004, 2009; Orús, Ann Phys 349:117, 2014; Haegeman and Verstraete, Ann Rev Condens Matter Phys 8(1):355–406, 2017) for more exhaustive understanding on the TN simulations for quantum problems. We will focus on the algorithms based on PEPS Projected entangled pair state (PEPS) , and show the key roles that the 2D TN contraction algorithms presented in Sect. 3 play in the higher-dimensional cases.
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) .
This chapter is to introduce some basic definitions and concepts of TN Tensor network (TN) . We will show that the TN can be used to represent quantum many-body states, where we explain MPS Matrix product state (MPS) in 1D and PEPS Projected entangled pair state (PEPS) in 2D systems, as well as the generalizations to thermal states and operators. The quantum entanglement properties of the TN states including the area law of entanglement entropy will also be discussed. Finally, we will present several special TNs that can be exactly contracted, and demonstrate the difficulty of contracting TNs in general cases.
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.
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.
This work introduces an extension to the Tensor Network Quantum Virtual Machine (TNQVM) tool, enhancing the existing stack of ExaScale Tensor Network (ExaTN), ExaScale Accelerator (XACC), and TNQVM. It features a new plugin that enables efficient simulation of a Projected Entangled Pair State (PEPS), a 2D tensor network. To improve simulation efficiency for PEPS, we have implemented the snake boundary contraction algorithm. By integrating this capability into the existing stack, we enhance the overall functionality and versatility of the framework. We tested this new PEPS topology for a simple GHZ bell-pair generation quantum circuit and saw that its runtime is very close to that of the MPS topology. We estimate that the real potential of the PEPS topology becomes discernible when quantum circuits with multidimensional entanglement are simulated using tensor networks. In such cases, 1D tensor networks fail to represent or contract them efficiently.
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.