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arxiv: 2606.24803 · v1 · pith:MYG4CIFInew · submitted 2026-06-23 · ❄️ cond-mat.str-el · quant-ph

Introduction to matrix-product states and tensor networks

Gr\'egoire Misguich This is my paper

Pith reviewed 2026-06-25 22:06 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords matrix-product statestensor networksDMRGentanglementquantum many-body physicsPEPSopen quantum systems
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0 comments X

The pith

Matrix-product states represent low-entanglement quantum many-body systems efficiently using tensors with virtual indices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

These notes introduce the tensor-network formalism for quantum many-body physics, with matrix-product states as the central construction. They develop graphical notation, virtual indices, bond dimensions, gauge freedom, and canonical forms, showing how limited entanglement controls the efficiency of the representation. The text then presents the main algorithms for contractions, correlation functions, matrix-product operators, DMRG, and time evolution, followed by higher-dimensional generalizations and extensions to mixed states and open systems. A sympathetic reader would care because the methods replace exponential Hilbert-space storage with polynomial-cost tensor operations when entanglement is low.

Core claim

Matrix-product states factor a many-body wavefunction into a product of local tensors connected by auxiliary indices; the dimension of these indices bounds the entanglement that can be captured exactly, and standard linear-algebra operations such as QR and singular-value decompositions convert between equivalent representations while preserving the physical state.

What carries the argument

Matrix-product states (MPS), a chain of tensors each carrying one physical index for the local degree of freedom and two virtual indices that encode entanglement between neighboring sites.

If this is right

  • Expectation values of local operators are obtained by contracting a short segment of the MPS network with the corresponding matrix-product operator.
  • Ground-state search reduces to iterative local optimization of the MPS tensors under the DMRG sweep procedure.
  • Real-time evolution is performed by applying short-time operators while repeatedly truncating back to a chosen bond dimension.
  • Projected entangled-pair states generalize the same construction to two dimensions, with contraction performed approximately by boundary MPS methods.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The provision of Julia code examples indicates that the same tensor operations can be executed immediately in existing numerical libraries without new implementation.
  • The treatment of Lindblad dynamics shows the formalism extends from closed unitary evolution to open-system master equations with the same tensor objects.
  • Canonical-form choices imply that numerical stability can be improved by selecting a gauge that avoids ill-conditioned tensors during contractions.

Load-bearing premise

The reader already knows quantum mechanics and linear algebra well enough to follow tensor manipulations without extra explanation.

What would settle it

A concrete counter-example would be a quantum state whose entanglement entropy grows linearly with subsystem size yet is exactly reproduced by an MPS whose bond dimension stays bounded as system size increases.

Figures

Figures reproduced from arXiv: 2606.24803 by Gr\'egoire Misguich.

Figure 1
Figure 1. Figure 1: Examples of TNs: a) MPS for a chain of length 8, b) binary tree TN for a system of 8 sites, [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Bell state |ψ⟩ = √ 1 2 (|00⟩ + |11⟩) viewed as a two-index tensor, as the output of a quantum circuit, and as the corresponding TN. H is the Hadamard gate (matrix) H = √ 1 2  1 1 1 −1  and CNOT = |0⟩ ⟨0| ⊗ I + |1⟩ ⟨1| ⊗ X, where X is the Pauli X matrix. avoiding operations that would be exponential in N. 3 1.1.2.2 Time evolution Computing time evolution is also an important domain where TNs are very … view at source ↗
Figure 3
Figure 3. Figure 3: Graphical representation of a few tensor examples and associated operations. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: AKLT state for a spin-1 chain. a) The Hilbert space is enlarged by replacing each spin-1 (green [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Construction of an MPS from a vector representation by successive QR decompositions. At [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Partial contractions of a left-canonical MPS and a right-canonical MPS. For each value of [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Transformation of an MPS into left-canonical form by successive QR decompositions. At each [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Top: mixed-canonical form of a six-site MPS with the orthogonality center at site [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Expectation value of a single-site observable [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: MPS representation of the scalar product [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Graphical definition of the transfer matrix for a translationally invariant MPS. The physical [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: TN representation of the two-point correlation function of the type [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: A Hamiltonian (or some generic operator) viewed either as a single tensor or factorized as a [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: MPO representations of a two-qubit gate U acting on neighboring sites (top) or on non￾neighboring sites (bottom). The diagonal tensors (gray) simply propagate the virtual index from site i to site j while acting as the identity in the physical space. 2.6.5 Finite-state automaton picture We describe a general method to construct an MPO representation of a Hamiltonian based on the finite￾state automaton pic… view at source ↗
Figure 15
Figure 15. Figure 15: Exact multiplication of an MPO (bond dimension [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Schematic representation of the zip-up algorithm. The MPO and MPS tensors are contracted [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Energy expectation value E = ⟨ψ| H |ψ⟩ / ⟨ψ| |ψ⟩ as a TN. The state |ψ⟩ is in an MPS form (tensors A[i] ) and the Hamiltonian is expressed as an MPO (orange tensors). E is iteratively minimized in the DMRG algorithm by optimizing one or two tensors of the MPS at a time while keeping the others fixed (see [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: a) Expectation value ⟨ψ| H |ψ⟩ as a quadratic function of the tensors A[i] and A[i+1]. The vector v contains the coefficients of the product A[i]A[i+1]. The vector v has dimension d 2χ 2 , where d is the physical dimension of the local Hilbert space and χ is the bond dimension of the MPS. b) Matrix H of the effective Hamiltonian for the optimization of the tensors at sites i and i + 1. H has dimension d 2… view at source ↗
Figure 19
Figure 19. Figure 19: Time-evolution of an MPS using the TEBD algorithm. a) The time-evolution operator is [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Magnetization profile for the domain wall quench in the XX spin- [PITH_FULL_IMAGE:figures/full_fig_p032_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Mapping a Hamiltonian with nearest-neighbor interactions on a [PITH_FULL_IMAGE:figures/full_fig_p036_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Data from code example #11 for a 7 × 6 lattice. The correlation considered is the connected density-density correlation between the two sites that are the farthest apart on the lattice, at opposite corners (see code). The left panel shows the relative error on the ground-state energy and on the con￾nected density-density correlation as a function of the MPS bond dimension. The right panel shows the entang… view at source ↗
Figure 23
Figure 23. Figure 23: A PEPS representation of a quantum many-body state on a 2d lattice, with a subsystem [PITH_FULL_IMAGE:figures/full_fig_p037_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: PEPS representation of the toric-code ground state. a) In the toric code each vertex [PITH_FULL_IMAGE:figures/full_fig_p040_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Contraction of a finite PEPS using double-layer tensors. The norm [PITH_FULL_IMAGE:figures/full_fig_p041_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Left: three representations of a density matrix. The matrix [PITH_FULL_IMAGE:figures/full_fig_p044_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Diagrammatic representation of the vectorization procedure for a quantum channel in Kraus [PITH_FULL_IMAGE:figures/full_fig_p046_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Brick-wall random circuit with Haar-random 2-qubit gates (blue) and single-qubit depolarizing [PITH_FULL_IMAGE:figures/full_fig_p047_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: FXEB for random brick-wall circuits with N = 8 qubits as a function of the circuit depth D. The two curves correspond to the ideal case, p = 0, and to a single-qubit depolarizing channel with error probability p = 0.05 applied after each unitary layer (see [PITH_FULL_IMAGE:figures/full_fig_p047_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Left: Operator-space entanglement entropy (OSEE) across the central bond plotted as a [PITH_FULL_IMAGE:figures/full_fig_p050_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Example of image compression by keeping only the 30 largest singular values of a grayscale [PITH_FULL_IMAGE:figures/full_fig_p052_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Graphical representation of the singular value decomposition (SVD) of a matrix [PITH_FULL_IMAGE:figures/full_fig_p053_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Graphical representation of the QR decomposition. The matrix [PITH_FULL_IMAGE:figures/full_fig_p053_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Graphical representation of the map Γ[m,n] , which contracts a boundary matrix X with the MPS tensors on the block from site m to site n, leaving the physical indices of the block “open” (Eq. (105)). The block is called injective if the map Γ[m,n] is injective. Injectivity implies that the space spanned by the states Γ[m,n](X) has dimension χ 2 . Generic MPS are injective in the sense that non-injective M… view at source ↗
Figure 35
Figure 35. Figure 35: Parent Hamiltonian of an injective MPS. Each local term [PITH_FULL_IMAGE:figures/full_fig_p056_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: MPS representation of the AKLT valence-bond solid state (see also Fig. [PITH_FULL_IMAGE:figures/full_fig_p058_36.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. These lecture notes introduce tensor-network methods in quantum many-body physics with emphasis on matrix-product states (MPS). They develop graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR/SVD decompositions, and the role of entanglement; then present MPS algorithms including contractions, correlation functions, matrix-product operators, DMRG, and time evolution; briefly cover PEPS and approximate contraction; and discuss tensor-network representations of mixed states, quantum channels, and Lindblad dynamics with applications to thermal states and open systems. The presentation includes short Julia/ITensor code examples and was prepared for the 9th Les Houches Summer School on Computational Physics: Open Quantum Systems.

Significance. If the explanations are accurate and the code examples function as described, the notes supply a practical, self-contained introduction to established tensor-network techniques together with reproducible code. This combination can serve as a useful teaching resource for researchers entering the field of computational condensed-matter physics, particularly those attending summer schools focused on open quantum systems.

minor comments (2)
  1. The notes assume background in quantum mechanics and linear algebra; adding a short prerequisites paragraph near the beginning would help readers assess readiness.
  2. Code snippets are described as short and based on ITensor/ITensorMPS/TensorMixedStates; confirming that all examples run without modification on current package versions would strengthen the reproducibility claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the lecture notes and for recommending acceptance. We appreciate the recognition of the notes as a practical, self-contained introduction with reproducible code examples.

Circularity Check

0 steps flagged

Expository notes with no novel derivations or claims

full rationale

This is a set of lecture notes introducing established tensor-network methods (MPS, PEPS, MPO, DMRG, time evolution, mixed-state representations) with standard graphical notation, decompositions, and ITensor code examples. No new predictions, uniqueness theorems, fitted parameters, or derivations are advanced. All material is standard background material presented for pedagogical purposes at a summer school, with no load-bearing steps that could reduce to self-definition, self-citation chains, or renaming of results. The document is therefore self-contained against external benchmarks and exhibits no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As an introductory tutorial on existing techniques, the paper introduces no new free parameters, axioms, or invented entities.

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Works this paper leans on

124 extracted references · 95 canonical work pages · 1 internal anchor

  1. [1]

    Misguich

    G. Misguich. Introduction to matrix-product states and tensor networks – Julia code examples. url: https://github.com/gregoire-misguich/Introduction-to-matrix-product-states- and-tensor-networks

  2. [2]

    Fishman, S

    M. Fishman, S. White, and E. Stoudenmire. The ITensor Software Library for Tensor Network Calculations. SciPost Phys. Codebases(2022), 004. doi: 10.21468/SciPostPhysCodeb.4

    work page doi:10.21468/scipostphyscodeb.4 2022

  3. [3]

    Houdayer and G

    J. Houdayer and G. Misguich.TensorMixedStates: A Julia Library for Simulating Pure and Mixed Quantum States Using Matrix Product States. SciPost Phys. Codebases (2026), 072. doi: 10 . 21468/SciPostPhysCodeb.72. url: https://github.com/jerhoud/TensorMixedStates.jl

    2026

  4. [4]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio.Colloquium: Area Laws for the Entanglement Entropy. Rev. Mod. Phys.82.1 (2010), 277–306.doi: 10.1103/RevModPhys.82.277

    work page doi:10.1103/revmodphys.82.277 2010

  5. [5]

    Corboz, R

    P. Corboz, R. Orús, B. Bauer, and G. Vidal. Simulation of Strongly Correlated Fermions in Two Spatial Dimensions with Fermionic Projected Entangled-Pair States. Phys. Rev. B 81.16 (2010), 165104. doi: 10.1103/PhysRevB.81.165104

    work page doi:10.1103/physrevb.81.165104 2010

  6. [6]

    Mortier, L

    Q. Mortier, L. Devos, L. Burgelman, B. Vanhecke, N. Bultinck, F. Verstraete, J. Haegeman, and L. Vanderstraeten.Fermionic Tensor Network Methods. SciPost Phys.18.1 (2025), 012.doi: 10.21468/SciPostPhys.18.1.012

    work page doi:10.21468/scipostphys.18.1.012 2025

  7. [7]

    The Density Matrix Renormalization Group in Quantum Chemistry

    G.K.-L.ChanandS.Sharma. The Density Matrix Renormalization Group in Quantum Chemistry. Annu. Rev. Phys. Chem.62 (2011), 465–481.doi: 10.1146/annurev-physchem-032210-103338

    work page doi:10.1146/annurev-physchem-032210-103338 2011

  8. [8]

    Szalay, M

    S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, and Ö. Legeza.Tensor Prod- uct Methods and Entanglement Optimization for Ab Initio Quantum Chemistry. Int. J. Quantum Chem. 115.19 (2015), 1342–1391.doi: 10.1002/qua.24898

    work page doi:10.1002/qua.24898 2015

  9. [9]

    G. Vidal. Classical Simulation of Infinite-Size Quantum Lattice Systems in One Spatial Dimension. Phys. Rev. Lett.98.7 (2007), 070201.doi: 10.1103/PhysRevLett.98.070201

    work page doi:10.1103/physrevlett.98.070201 2007

  10. [10]

    I. P. McCulloch. Infinite Size Density Matrix Renormalization Group, Revisited. 2008. arXiv: 0804.2509 [cond-mat]

    Pith/arXiv arXiv 2008

  11. [11]

    Jordan, R

    J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac.Classical Simulation of Infinite-Size Quantum Lattice Systems in Two Spatial Dimensions. Phys. Rev. Lett.101.25 (2008), 250602.doi: 10.1103/PhysRevLett.101.250602

    work page doi:10.1103/physrevlett.101.250602 2008

  12. [12]

    Verstraete and J

    F. Verstraete and J. I. Cirac.Continuous Matrix Product States for Quantum Fields. Phys. Rev. Lett. 104.19 (2010), 190405.doi: 10.1103/PhysRevLett.104.190405

    work page doi:10.1103/physrevlett.104.190405 2010

  13. [13]

    A. E. Feiguin, E. Rezayi, C. Nayak, and S. Das Sarma.Density Matrix Renormalization Group Study of Incompressible Fractional Quantum Hall States. Phys. Rev. Lett.100.16 (2008), 166803. doi: 10.1103/PhysRevLett.100.166803. 58

    work page doi:10.1103/physrevlett.100.166803 2008

  14. [14]

    M. P. Zaletel and R. S. K. Mong.Exact Matrix Product States for Quantum Hall Wave Functions. Phys. Rev. B86.24 (2012), 245305.doi: 10.1103/PhysRevB.86.245305

    work page doi:10.1103/physrevb.86.245305 2012

  15. [15]

    Topological Characterization of Fractional Quantum Hall Ground States from Microscopic Hamiltonians

    M.P.Zaletel,R.S.K.Mong,andF.Pollmann. Topological Characterization of Fractional Quantum Hall Ground States from Microscopic Hamiltonians. Phys. Rev. Lett.110.23 (2013), 236801.doi: 10.1103/PhysRevLett.110.236801

    work page doi:10.1103/physrevlett.110.236801 2013

  16. [16]

    Östlund and S

    S. Östlund and S. Rommer.Thermodynamic Limit of Density Matrix Renormalization. Phys. Rev. Lett. 75.19 (1995), 3537–3540.doi: 10.1103/PhysRevLett.75.3537

    work page doi:10.1103/physrevlett.75.3537 1995

  17. [17]

    I. V. Oseledets. Tensor-Train Decomposition. SIAM J. Sci. Comput.33 (2011), 2295. doi: 10. 1137/090752286

    2011

  18. [18]

    Stoudenmire and S

    E. Stoudenmire and S. R. White. Studying Two-Dimensional Systems with the Density Matrix Renormalization Group. Annu. Rev. Condens. Matter Phys.3.1 (2012), 111–128.doi: 10.1146/ annurev-conmatphys-020911-125018

    2012

  19. [19]

    Nishino, Y

    T. Nishino, Y. Hieida, K. Okunishi, N. Maeshima, Y. Akutsu, and A. Gendiar.Two-Dimensional Tensor Product Variational Formulation.Prog. Theor. Phys.105.3 (2001), 409–417.doi: 10.1143/ PTP.105.409

    2001

  20. [20]

    Verstraete and J

    F. Verstraete and J. I. Cirac. Renormalization Algorithms for Quantum-Many Body Systems in Two and Higher Dimensions. 2004. arXiv:cond-mat/0407066

    Pith/arXiv arXiv 2004

  21. [21]

    G. Vidal. Entanglement Renormalization. Phys. Rev. Lett.99.22 (2007), 220405. doi: 10.1103/ PhysRevLett.99.220405

    2007

  22. [22]

    G. Vidal. Class of Quantum Many-Body States That Can Be Efficiently Simulated. Phys. Rev. Lett. 101.11 (2008), 110501.doi: 10.1103/PhysRevLett.101.110501

    work page doi:10.1103/physrevlett.101.110501 2008

  23. [23]

    Evenbly and G

    G. Evenbly and G. Vidal. Algorithms for Entanglement Renormalization. Phys. Rev. B 79.14 (2009), 144108. doi: 10.1103/PhysRevB.79.144108

    work page doi:10.1103/physrevb.79.144108 2009

  24. [24]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek.Geometric and Renormalized Entropy in Conformal Field Theory. Nuclear Physics B424.3 (1994), 443–467.doi: 10.1016/0550-3213(94)90402-2

    work page doi:10.1016/0550-3213(94)90402-2 1994

  25. [25]

    Vidal, J

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev.Entanglement in Quantum Critical Phenomena. Phys. Rev. Lett.90.22 (2003), 227902.doi: 10.1103/PhysRevLett.90.227902

    work page doi:10.1103/physrevlett.90.227902 2003

  26. [26]

    Calabrese and J

    P. Calabrese and J. Cardy. Entanglement Entropy and Quantum Field Theory. J. Stat. Mech. Theory Exp.2004.06 (2004), P06002.doi: 10.1088/1742-5468/2004/06/P06002

    work page doi:10.1088/1742-5468/2004/06/p06002 2004

  27. [27]

    Entanglement Renormalization and Holography.Physical Review D86.6(2012),065007

    B.Swingle. Entanglement Renormalization and Holography.Physical Review D86.6(2012),065007. doi: 10.1103/PhysRevD.86.065007

    work page doi:10.1103/physrevd.86.065007 2012

  28. [28]

    Holographic derivation of entanglement entropy from the anti-de

    S. Ryu and T. Takayanagi. Holographic Derivation of Entanglement Entropy from AdS/CFT. Phys. Rev. Lett.96.18 (2006), 181602.doi: 10.1103/PhysRevLett.96.181602

    work page doi:10.1103/physrevlett.96.181602 2006

  29. [29]

    S. R. White. Density Matrix Formulation for Quantum Renormalization Groups. Phys. Rev. Lett. 69.19 (1992), 2863–2866.doi: 10.1103/PhysRevLett.69.2863

    work page doi:10.1103/physrevlett.69.2863 1992

  30. [30]

    S. R. White.Density-Matrix Algorithms for Quantum Renormalization Groups.Phys. Rev. B48.14 (1993), 10345–10356.doi: 10.1103/PhysRevB.48.10345

    work page doi:10.1103/physrevb.48.10345 1993

  31. [31]

    Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326 (2011) 96–192.doi:10.1016/j.aop.2010.09.012

    U. Schollwöck. The Density-Matrix Renormalization Group in the Age of Matrix Product States. Ann. Phys.January 2011 Special Issue 326.1 (2011), 96–192.doi: 10.1016/j.aop.2010.09.012

    work page doi:10.1016/j.aop.2010.09.012 2011

  32. [32]

    S. R. White and A. E. Feiguin.Real-Time Evolution Using the Density Matrix Renormalization Group. Phys. Rev. Lett.93.7 (2004), 076401.doi: 10.1103/PhysRevLett.93.076401

    work page doi:10.1103/physrevlett.93.076401 2004

  33. [33]

    Time-Dependent Density-Matrix Renormalization- Group Using Adaptive Effective Hilbert Spaces.J

    A.J.Daley,C.Kollath,U.Schollwöck,andG.Vidal. Time-Dependent Density-Matrix Renormalization- Group Using Adaptive Effective Hilbert Spaces.J. Stat. Mech. Theory Exp.2004.04 (2004), P04005. doi: 10.1088/1742-5468/2004/04/P04005

    work page doi:10.1088/1742-5468/2004/04/p04005 2004

  34. [34]

    Real-Time Dynamics in Spin-1/2 Chains with Adaptive Time-Dependent Density Matrix Renormalization Group.Phys

    D.Gobert,C.Kollath,U.Schollwöck,andG.Schütz. Real-Time Dynamics in Spin-1/2 Chains with Adaptive Time-Dependent Density Matrix Renormalization Group.Phys. Rev. E71.3(2005),036102. doi: 10.1103/PhysRevE.71.036102

    work page doi:10.1103/physreve.71.036102 2005

  35. [35]

    G. Vidal. Efficient Classical Simulation of Slightly Entangled Quantum Computations. Phys. Rev. Lett. 91.14 (2003), 147902.doi: 10.1103/PhysRevLett.91.147902

    work page doi:10.1103/physrevlett.91.147902 2003

  36. [36]

    G. Vidal. Efficient Simulation of One-Dimensional Quantum Many-Body Systems.Phys. Rev. Lett. 93.4 (2004), 040502.doi: 10.1103/PhysRevLett.93.040502. 59

    work page doi:10.1103/physrevlett.93.040502 2004

  37. [37]

    Haegeman, J

    J. Haegeman, J. I. Cirac, T. J. Osborne, I. Pižorn, H. Verschelde, and F. Verstraete. Time- Dependent Variational Principle for Quantum Lattices. Phys. Rev. Lett. 107.7 (2011), 070601. doi: 10.1103/PhysRevLett.107.070601

    work page doi:10.1103/physrevlett.107.070601 2011

  38. [38]

    Haegeman, C

    J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete.Unifying Time Evo- lution and Optimization with Matrix Product States. Phys. Rev. B 94.16 (2016), 165116. doi: 10.1103/PhysRevB.94.165116

    work page doi:10.1103/physrevb.94.165116 2016

  39. [39]

    M. P. Zaletel, R. S. K. Mong, C. Karrasch, J. E. Moore, and F. Pollmann. Time-Evolving a Matrix Product State with Long-Ranged Interactions. Phys. Rev. B 91.16 (2015), 165112. doi: 10.1103/PhysRevB.91.165112

    work page doi:10.1103/physrevb.91.165112 2015

  40. [40]

    Y. Zhou, E. M. Stoudenmire, and X. Waintal.What Limits the Simulation of Quantum Comput- ers?: Phys. Rev. X10.4 (2020), 041038.doi: 10.1103/PhysRevX.10.041038

    work page doi:10.1103/physrevx.10.041038 2020

  41. [41]

    Ayral, T

    T. Ayral, T. Louvet, Y. Zhou, C. Lambert, E. M. Stoudenmire, and X. Waintal.Density-Matrix Renormalization Group Algorithm for Simulating Quantum Circuits with a Finite Fidelity. PRX Quantum 4.2 (2023), 020304.doi: 10.1103/PRXQuantum.4.020304

    work page doi:10.1103/prxquantum.4.020304 2023

  42. [42]

    Cerezo-Roquebrún, A

    S. Cerezo-Roquebrún, A. Bou-Comas, J. T. Schneider, E. López, L. Tagliacozzo, and S. Carignano. Spatio-Temporal Tensor-Network Approaches to out-of-Equilibrium Dynamics Bridging Open and Closed Systems. 2025. arXiv:2502.20214 [quant-ph]

    arXiv 2025

  43. [43]

    Carignano, G

    S. Carignano, G. Lami, J. D. Nardis, and L. Tagliacozzo.Overcoming the Entanglement Barrier with Sampled Tensor Networks. 2025. arXiv:2505.09714 [quant-ph]

    arXiv 2025

  44. [44]

    Verstraete, J

    F. Verstraete, J. J. García-Ripoll, and J. I. Cirac.Matrix Product Density Operators: Simulation of Finite-Temperature and Dissipative Systems. Phys. Rev. Lett.93.20 (2004), 207204. doi: 10. 1103/PhysRevLett.93.207204

    2004

  45. [45]

    Zwolak and G

    M. Zwolak and G. Vidal.Mixed-State Dynamics in One-Dimensional Quantum Lattice Systems: A Time-Dependent Superoperator Renormalization Algorithm. Phys. Rev. Lett.93.20 (2004), 207205. doi: 10.1103/PhysRevLett.93.207205

    work page doi:10.1103/physrevlett.93.207205 2004

  46. [46]

    Keeling, E

    J. Keeling, E. M. Stoudenmire, M.-C. Bañuls, and D. R. Reichman.Process Tensor Approaches to Non-Markovian Quantum Dynamics. 2025. arXiv:2509.07661 [quant-ph]

    Pith/arXiv arXiv 2025

  47. [47]

    Pan and P

    F. Pan and P. Zhang. Simulation of Quantum Circuits Using the Big-Batch Tensor Network Method. Phys. Rev. Lett.128.3 (2022), 030501.doi: 10.1103/PhysRevLett.128.030501

    work page doi:10.1103/physrevlett.128.030501 2022

  48. [48]

    Tindall, M

    J. Tindall, M. Fishman, E. M. Stoudenmire, and D. Sels.Efficient Tensor Network Simulation of IBM’s Eagle Kicked Ising Experiment. PRX Quantum 5.1 (2024), 010308. doi: 10 . 1103 / PRXQuantum.5.010308

    2024

  49. [49]

    Classification of Gapped Symmetric Phases in One-Dimensional Spin Systems

    X.Chen,Z.-C.Gu,andX.-G.Wen. Classification of Gapped Symmetric Phases in One-Dimensional Spin Systems. Phys. Rev. B83.3 (2011), 035107.doi: 10.1103/PhysRevB.83.035107

    work page doi:10.1103/physrevb.83.035107 2011

  50. [50]

    Tanguy, J

    X. Chen, Z.-C. Gu, and X.-G. Wen.Complete Classification of One-Dimensional Gapped Quantum Phases in Interacting Spin Systems. Phys. Rev. B84.23 (2011), 235128.doi: 10.1103/PhysRevB. 84.235128

    work page doi:10.1103/physrevb 2011

  51. [51]

    Schuch, D

    N. Schuch, D. Pérez-García, and I. Cirac. Classifying Quantum Phases Using Matrix Product States and Projected Entangled Pair States. Phys. Rev. B 84.16 (2011), 165139. doi: 10.1103/ PhysRevB.84.165139

    2011

  52. [52]

    Exact Solution of a 1D Asymmetric Exclusion Model Using a Matrix Formulation

    B.Derrida,M.R.Evans,V.Hakim,andV.Pasquier. Exact Solution of a 1D Asymmetric Exclusion Model Using a Matrix Formulation. J. Phys. A Math. Gen.26.7 (1993), 1493.doi: 10.1088/0305- 4470/26/7/011

    work page doi:10.1088/0305- 1993

  53. [53]

    K. Mallick. Some Exact Results for the Exclusion Process. J. Stat. Mech. Theory Exp.2011.01 (2011), P01024. doi: 10.1088/1742-5468/2011/01/P01024

    work page doi:10.1088/1742-5468/2011/01/p01024 2011

  54. [54]

    T. Nishino. Density Matrix Renormalization Group Method for 2D Classical Models. J. Phys. Soc. Jpn. 64.10 (1995), 3598–3601.doi: 10.1143/JPSJ.64.3598

    work page doi:10.1143/jpsj.64.3598 1995

  55. [55]

    Orús and G

    R. Orús and G. Vidal. Infinite Time-Evolving Block Decimation Algorithm beyond Unitary Evo- lution. Phys. Rev. B78.15 (2008), 155117.doi: 10.1103/PhysRevB.78.155117

    work page doi:10.1103/physrevb.78.155117 2008

  56. [56]

    Oseledets and E

    I. Oseledets and E. Tyrtyshnikov. TT-cross Approximation for Multidimensional Arrays. Linear Algebra Appl.432.1 (2010), 70–88.doi: 10.1016/j.laa.2009.07.024. 60

    work page doi:10.1016/j.laa.2009.07.024 2010

  57. [57]

    Núñez Fernández, M

    Y. Núñez Fernández, M. K. Ritter, M. Jeannin, J.-W. Li, T. Kloss, T. Louvet, S. Terasaki, O. Par- collet, J. von Delft, H. Shinaoka, et al.Learning Tensor Networks with Tensor Cross Interpolation: New Algorithms and Libraries. SciPost Phys.18.3 (2025), 104.doi: 10.21468/SciPostPhys.18. 3.104

    work page doi:10.21468/scipostphys.18 2025

  58. [58]

    M. K. Ritter, Y. Núñez Fernández, M. Wallerberger, J. von Delft, H. Shinaoka, and X. Waintal. Quantics Tensor Cross Interpolation for High-Resolution Parsimonious Representations of Multi- variate Functions.Phys. Rev. Lett.132.5(2024),056501. doi: 10.1103/PhysRevLett.132.056501

    work page doi:10.1103/physrevlett.132.056501 2024

  59. [59]

    Waintal, C.-H

    X. Waintal, C.-H. Huang, and C. W. Groth.Who Can Compete with Quantum Computers? Lecture Notes on Quantum Inspired Tensor Networks Computational Techniques. 2026. arXiv:2601.03035 [quant-ph]

    arXiv 2026

  60. [60]

    R. Orús. A Practical Introduction to Tensor Networks: Matrix Product States and Projected En- tangled Pair States. Ann. Phys.349 (2014), 117–158.doi: 10.1016/j.aop.2014.06.013

    work page doi:10.1016/j.aop.2014.06.013 2014

  61. [61]

    J. C. Bridgeman and C. T. Chubb.Hand-Waving and Interpretive Dance: An Introductory Course on Tensor Networks. J. Phys. A Math. Theor.50.22 (2017), 223001. doi: 10.1088/1751-8121/ aa6dc3

    work page doi:10.1088/1751-8121/ 2017

  62. [62]

    Silvi, F

    P. Silvi, F. Tschirsich, M. Gerster, J. Jünemann, D. Jaschke, M. Rizzi, and S. Montangero.The Tensor Networks Anthology: Simulation Techniques for Many-Body Quantum Lattice Systems. SciPost Phys. Lect.(2019), 008. doi: 10.21468/SciPostPhysLectNotes.8

    work page doi:10.21468/scipostphyslectnotes.8 2019

  63. [63]

    R. Orús. Tensor Networks for Complex Quantum Systems. Nat. Rev. Phys.1.9 (2019), 538–550. doi: 10.1038/s42254-019-0086-7

    work page doi:10.1038/s42254-019-0086-7 2019

  64. [64]

    J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete.Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, Theorems. Rev. Mod. Phys.93.4 (2021), 045003. doi: 10.1103/RevModPhys.93.045003

    work page doi:10.1103/revmodphys.93.045003 2021

  65. [65]

    M. C. Bañuls. Tensor Network Algorithms: A Route Map. Annu. Rev. Condens. Matter Phys.14 (2023), 173–191.doi: 10.1146/annurev-conmatphys-040721-022705

    work page doi:10.1146/annurev-conmatphys-040721-022705 2023

  66. [66]

    B. V.-D. Cuiper, W. Wiesiolek, and F. Verstraete.Les Houches Lecture Notes on Tensor Networks

  67. [67]

    arXiv: 2512.24390 [cond-mat.str-el]

    Pith/arXiv arXiv

  68. [68]

    Paeckel, T

    S. Paeckel, T. Köhler, A. Swoboda, S. R. Manmana, U. Schollwöck, and C. Hubig.Time-Evolution Methods for Matrix-Product States. Ann. Phys.411 (2019), 167998.doi: 10.1016/j.aop.2019. 167998

    work page doi:10.1016/j.aop.2019 2019

  69. [69]

    von Delft

    J. von Delft. Lecture Course on Tensor Networks, Ludwig Maximilian University Munich (2025). url: https://www.theorie.physik.uni-muenchen.de/lsvondelft/teams-groups/teaching/ tensornetworks1

    2025

  70. [70]

    J. Gray. quimb: a python library for quantum information and many-body calculations. J. Open Source Softw.3.29 (2018), 819.doi: 10.21105/joss.00819

    work page doi:10.21105/joss.00819 2018

  71. [71]

    Hauschild and F

    J. Hauschild and F. Pollmann.Efficient Numerical Simulations with Tensor Networks: Tensor Net- work Python (TeNPy).SciPost Phys. Lect.(2018), 005.doi: 10.21468/SciPostPhysLectNotes.5

    work page doi:10.21468/scipostphyslectnotes.5 2018

  72. [72]

    url: https://tenpy.readthedocs.io

    TenPy: Tensor Network Python Library. url: https://tenpy.readthedocs.io

  73. [73]

    url: https://tensornetwork.org/

    tensornetwork.org. url: https://tensornetwork.org/

  74. [74]

    G. Evenbly. tensor.net. url: https://tensor.net/

  75. [75]

    Pippan, S

    P. Pippan, S. R. White, and H. G. Evertz.Efficient Matrix-Product State Method for Periodic Boundary Conditions. Phys. Rev. B81.8 (2010), 081103.doi: 10.1103/PhysRevB.81.081103

    work page doi:10.1103/physrevb.81.081103 2010

  76. [76]

    Perez-Garcia, F

    D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac.Matrix Product State Representations. Quantum Info. Comput.7.5 (2007), 401–430.doi: 10.5555/2011832.2011833

    work page doi:10.5555/2011832.2011833 2007

  77. [77]

    Affleck, T.Kennedy, E.H.Lieb, andH

    I. Affleck, T.Kennedy, E.H.Lieb, andH. Tasaki.Rigorous Results on Valence-Bond Ground States in Antiferromagnets. Phys. Rev. Lett.59.7 (1987), 799–802.doi: 10.1103/PhysRevLett.59.799

    work page doi:10.1103/physrevlett.59.799 1987

  78. [78]

    Simulation of Two-Dimensional Quantum Systems Using a Tree Tensor Network That Exploits the Entropic Area Law

    L.Tagliacozzo,G.Evenbly,andG.Vidal. Simulation of Two-Dimensional Quantum Systems Using a Tree Tensor Network That Exploits the Entropic Area Law. Phys. Rev. B80.23 (2009), 235127. doi: 10.1103/PhysRevB.80.235127

    work page doi:10.1103/physrevb.80.235127 2009

  79. [79]

    Investigating neutron polarizabilities through compton scattering on3He

    N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac.Computational Complexity of Projected Entangled Pair States. Phys. Rev. Lett.98.14 (2007), 140506. doi: 10.1103/PhysRevLett.98. 140506. 61

    work page doi:10.1103/physrevlett.98 2007

  80. [80]

    G. M. Crosswhite and D. Bacon. Finite Automata for Caching in Matrix Product Algorithms. Phys. Rev. A78.1 (2008), 012356.doi: 10.1103/PhysRevA.78.012356

    work page doi:10.1103/physreva.78.012356 2008

Showing first 80 references.