The density-matrix renormalization group in the age of matrix product states

2010/08/20 by Ulrich Schollwöck, Ulrich Schollwoeck · 2 voices · 4,127 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Class (philosophy) #Density matrix renormalization group #Lattice (music) #Matrix product state #Physics of Superconductivity and Magnetism #Product (mathematics) #Quantum #Quantum many-body systems #Realization (probability) #Renormalization #Renormalization group #cond-mat.str-el

paper · pdf · doi:10.1016/j.aop.2010.09.012

published in Annals of Physics 326(1), 96-192 (Elsevier BV) · 118 pages, 63 figures; minor changes in this final version: typos eliminated and a few references added

arxiv published 2010/08/20 · openalex publication_date 2010/10/02 · arxiv created 2011/01/03 · arxiv updated 2011/01/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

The density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms.

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• A Practical Framework for Simulating Time-Resolved Spectroscopy Based on a Real-time Dyson Expansion
• Density‐matrix renormalization group study of many‐body localization in floquet eigenstates
• Universal Rephasing Dynamics after a Quantum Quench via Sudden Coupling of Two Initially Independent Condensates
• Structure of matrix product locally purifiable density operators
• Phase-space methods for many-body quantum optics
• Field-induced quasi-bound state within the two-magnon continuum of a square-lattice Heisenberg antiferromagnet
• Tensor network solvers for ultra-large tight-binding Hamiltonians: algorithms and applications
• Modelling Emotional Memory in Children with Tensor Networks
• Frustrated polaritons
• Accelerating two-dimensional tensor network contractions using QR-decompositions
• Theory of the Matchgate Commutant
• Chebyshev matrix product state approach for spectral functions
• Gapless Symmetry-Protected Topological Order
• Longitudinal static optical properties of hydrogen chains: Finite field extrapolations of matrix product state calculations
• Tree tensor network hierarchical equations of motion based on time-dependent variational principle for efficient open quantum dynamics in structured thermal environments
• Contemporary tensor network approaches to gapless and topological phases in an extended Bose-Hubbard ladder
• Cluster-Based Haldane State in an Edge-Shared Tetrahedral Spin-Cluster Chain: Fedotovite K2Cu3O(SO4)3
• trainsum -- A Python package for quantics tensor trains
• Matrix product states approaches to operator spreading in ergodic quantum systems
• NMR relaxation in the spin-1 Heisenberg chain
• Reentrant topological phase transition in a bridging model between Kitaev and Haldane chains
• Entanglement negativity after a global quantum quench
• Matrix product ansatz for Fermi fields in one dimension
• Second-order functional renormalization group approach to one-dimensional systems in real and momentum space
• QuantumToolbox.jl: An efficient Julia framework for simulating open quantum systems
• An Optimally Accurate Lanczos Algorithm in the Matrix Product State Representation
• A Unified Variational Framework for Quantum Excited States
• Matrix product state formulation of frequency-space dynamics at finite temperatures
• Out-of-equilibrium dynamics with matrix product states
• Neural-Network Approach to Dissipative Quantum Many-Body Dynamics
• A tensor-train multidimensional inverse Laplace transform
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• Solving Classical and Quantum Spin Glasses with Deep Boltzmann Quantum States
• Ergodicity breaking in matrix-product-state effective Hamiltonians
• Low T-count preparation of nuclear eigenstates with tensor networks
• Tunable many-body burst in isolated quantum systems
• Topological phases of coupled Su-Schrieffer-Heeger wires
• Characterization of Topological Phases of Spin-1/2 Frustrated Ferromagnetic–Antiferromagnetic Alternating Heisenberg Chains by Entanglement Spectrum
• Complexity of controlling quantum many-body dynamics
• Robust techniques for polarization and detection of nuclear spin ensembles
• Quantum simulation of baryon scattering in SU(2) lattice gauge theory
• Quantum simulation of the Schwinger model: A study of feasibility
• Haldane Phase in the Hubbard Model at 2/3-Filling for the Organic Molecular CompoundMo3S7(dmit)3
• Branch-and-Bound Tensor Networks for Exact Ground-State Characterization
• Tuning energy transport using interacting vibrational modes
• Excitation spectrum and density matrix renormalization group iterations
• Quantum many-body systems out of equilibrium
• Nonperturbative calculation of phonon effects on spin squeezing
• From Thermal Rectifiers to Thermoelectric Devices
• Autonomous Quantum Simulation through Large Language Model Agents
• Matrix product operator representations for the local conserved quantities of the spin-1/2 XYZ chain
• Real-time dynamics in the one-dimensional Hubbard model
• Two-bath spin-boson model: Phase diagram and critical properties
• Nonequilibrium dynamics of bosonic Mott insulators in an electric field
• Characterization of a Topological Mott Insulator in One Dimension
• Quantifying Pauli Errors in Single-Photon Resource-State Generation
• DAMPyF: a Python implementation of the DAMPF method for the simulation of open-system dynamics
• Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays
• Matrix product solutions of boundary driven quantum chains
• LogosQ: A High-Performance and Type-Safe Quantum Computing Library in Rust
• Quantum mechanical NMR simulation algorithm for protein-size spin systems
• Ferromagnetism of magnetic impurities coupled indirectly via conduction electrons: Insights from various theoretical approaches
• Vortex and Meissner phases of strongly interacting bosons on a two-leg ladder
• Exact Overlaps in the Kondo Problem
• Thermometry of ultracold atoms via nonequilibrium work distributions
• Information scrambling in chaotic systems with dissipation
• Minimally entangled typical thermal states algorithm with Trotter gates
• Theoretical methods to treat a single dissipative bosonic mode coupled globally to an interacting many-body system
• EAQGA: A Quantum-Enhanced Genetic Algorithm with Novel Entanglement-Aware Crossovers
• Quantum State Design and Emergent Confinement Mechanism in Measured Tensor Network States
• Tensor network states in time-bin quantum optics
• Dynamical mean-field theory on the real-frequency axis: p−d hybridization and atomic physics in SrMnO3
• Systematic construction of density functionals based on matrix product state computations
• A general automatic method for optimal construction of matrix product operators using bipartite graph theory
• Order in Quantum Compass and Orbital egModels
• Speed of quantum information spreading in chaotic systems
• Infinite-Size Density Matrix Renormalization Group with Parallel Hida’s Algorithm
• Towards quantum machine learning with tensor networks
• What Limits the Simulation of Quantum Computers?
• Quantum probing beyond pure dephasing
• Selected applications of typicality to real-time dynamics of quantum many-body systems
• Matrix Product State for Higher-Order Tensor Compression and Classification
• Phase diagram of theJ1−J2Heisenberg model on the kagome lattice
• Efficient Excitations and Spectra within a Perturbative Renormalization Approach
• Bethe-Boltzmann hydrodynamics and spin transport in the XXZ chain
• Stability of encoded macroscopic quantum superpositions
• Quantitative analysis of interaction effects in generalized Aubry-André-Harper models
• Phase diagram of 1+1D Abelian-Higgs model and its critical point
• Spin many-body phases in standard and topological waveguide QED simulators
• Emerging quantum computing algorithms for quantum chemistry
• Chebyshev expansion for impurity models using matrix product states
• Lower bounds for ground states of condensed matter systems
• Quantum chaos and thermalization in isolated systems of interacting particles
• Hilbert space renormalization for the many-electron problem
• Optical conductivity of the Hubbard chain away from half filling
• Mott-insulating phases of the Bose-Hubbard model on quasi-1D ladder lattices
• Simulating Quantum Circuits with Tree Tensor Networks using Density-Matrix Renormalization Group Algorithm
• Mutual Information Scaling for Tensor Network Machine Learning
• Fast computation of dissipative quantum systems with ensemble rank truncation
• Entanglement spreading after local fermionic excitations in the XXZ chain
• Braiding Non-Abelian Quasiholes in Fractional Quantum Hall States
• Connecting entanglement in time and space: Improving the folding algorithm
• Operator entanglement entropy of the time evolution operator in chaotic systems
• Bosonic vs. Fermionic Matter in Quantum Simulations of 2+1D Gauge Theories
• Equilibration and order in quantum Floquet matter
• Quantum Criticality and Spin Liquid Phase in the Shastry-Sutherland model
• Generation of photonic tensor network states with Circuit QED
• Interaction distance in the extended XXZ model
• Scaling Hypothesis for Matrix Product States
• Towards a generalized hydrodynamics description of Rényi entropies in integrable systems
• Thermal drag in electronic conductors
• Probing the Bond Order Wave Phase Transitions of the Ionic Hubbard Model by Superlattice Modulation Spectroscopy
• Delayed coherent quantum feedback from a scattering theory and a matrix product state perspective
• Efficiency of fermionic quantum distillation
• Quantum phases of strongly interacting bosons on a two-leg Haldane ladder
• Topological order in the Kitaev/Majorana chain in the presence of disorder and interactions
• Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamics
• Light scattering and dissipative dynamics of many fermionic atoms in an optical lattice
• Quantum phase transition as an interplay of Kitaev and Ising interactions
• Anomalous quantum information scrambling for Z3 parafermion chains
• Finding the phase diagram of strongly correlated disordered bosons using quantum quenches
• Chiral spin condensation in a one-dimensional optical lattice
• Machine Learning Detection of Bell Nonlocality in Quantum Many-Body Systems
• Post-Ehrenfest many-body quantum interferences in ultracold atoms far out of equilibrium
• Robustness of gauge-invariant dynamics against defects in ultracold-atom gauge theories
• Bounding the finite-size error of quantum many-body dynamics simulations
• Generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions
• Non-Abelian string breaking phenomena with matrix product states
• Adaptive Lanczos-vector method for dynamic properties within the density matrix renormalization group
• Entanglement revivals as a probe of scrambling in finite quantum systems
• Adaptive cluster approximation for reduced density-matrix functional theory
• Emergent Kitaev materials in synthetic Fermi-Hubbard bilayers
• Topological Transitions from Multipartite Entanglement with Tensor Networks: A Procedure for Sharper and Faster Characterization
• Unsupervised Generative Modeling Using Matrix Product States
• かがわ遠隔医療ネットワークから日本版EHRの実現へ (総特集 ITによる医療現場の具体的変革)
• Tree tensor network approach to simulating Shor's algorithm
• Confinement: A real-time visualization
• Fine-grained dynamics of entanglement in non-integrable quenches far across the Ising quantum critical point
• Synthetic gauge fields in synthetic dimensions: interactions and chiral edge modes
• Efficient tomography of a quantum many-body system
• Dynamical quantum phase transitions in collapse and revival oscillations of a quenched superfluid
• Quantifying Complexity in Quantum Phase Transitions via Mutual Information Complex Networks
• Quasi-flat-band physics in a two-leg ladder model and its relation to magic-angle twisted bilayer graphene
• Unitary circuits of finite depth and infinite width from quantum channels
• Two Phases Inside the Bose Condensation Dome of Yb2Si2O7
• Dicke Transition in Open Many-Body Systems Determined by Fluctuation Effects
• Density functionals and Kohn-Sham potentials with minimal wavefunction preparations on a quantum computer
• Practical variational tomography for critical one-dimensional systems
• Page curve like dynamics in Interacting Quantum Systems
• Crossover of correlation functions near a quantum impurity in a Tomonaga-Luttinger liquid
• Symmetry fractionalization in the topological phase of the spin-12J1−J2triangular Heisenberg model
• Abelian topological order of ν=2/5 and 3/7 fractional quantum Hall states in lattice models
• Phase diagram of thespin−12triangularJ1−J2Heisenberg model on a three-leg cylinder
• Self-assembling tensor networks and holography in disordered spin chains
• Pure density functional for strong correlation and the thermodynamic limit from machine learning
• Nonequilibrium dynamics in pumped Mott insulators
• DMRG study of FQHE systems in the open cylinder geometry
• Tensor network strategies for calculating biexcitons and trions in monolayer two-dimensional materials beyond the ground state
• Interaction quantum quenches in the one-dimensional Fermi-Hubbard model with spin imbalance
• Classification of trivial spin-1 tensor network states on a square lattice
• Spin-Orbit-Coupled Correlated Metal Phase in Kondo Lattices: An Implementation with Alkaline-Earth Atoms
• Hydrodynamical phase transition for domain-wall melting in the XY chain
• Kondo dynamics in one-dimensional doped ferromagnetic insulators
• Density-Dependent Synthetic Gauge Fields Using Periodically Modulated Interactions
• Efficient simulation of moiré materials using the density matrix renormalization group
• Phase-induced transport in atomic gases: From superfluid to Mott insulator
• Simulating open quantum dynamics with time-dependent variational matrix product states: Towards microscopic correlation of environment dynamics and reduced system evolution
• Comparative Benchmarking of Utility-Scale Quantum Emulators
• Chiral Floquet Phases of Many-Body Localized Bosons
• Incommensurate spiral magnetic order on anisotropic triangular lattice: Dynamical mean-field study in a spin-rotating frame
• One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures
• Realizing string breaking dynamics in a Z₂ lattice gauge theory on quantum hardware
• Entanglement of exact excited states of Affleck-Kennedy-Lieb-Tasaki models: Exact results, many-body scars, and violation of the strong eigenstate thermalization hypothesis
• Entanglement spreading after a geometric quench in quantum spin chains
• Connecting Topological Anderson and Mott Insulators in Disordered Interacting Fermionic Systems
• Synthetic-gauge-field-induced resonances and Fulde-Ferrell-Larkin-Ovchinnikov states in a one-dimensional optical lattice
• Efficient Simulation of Finite-Temperature Open Quantum Systems
• Real-time decay of a highly excited charge carrier in the one-dimensional Holstein model
• The mass spectrum of the Schwinger model with matrix product states
• Phase diagram of the isotropic spin-32model on thez=3Bethe lattice
• From classical to quantum criticality
• Nonequilibrium critical dynamics in the quantum chiral clock model
• Taming convergence in the determinant approach for x-ray excitation spectra
• Simulating fermions in spin-dependent potentials with spin models on an energy lattice
• Thermal tensor renormalization group simulations of square-lattice quantum spin models
• Cold atoms meet lattice gauge theory
• Entanglement and Disordered-Enhanced Topological Phase in the Kitaev Chain
• Validating quantum-classical programming models with tensor network simulations
• Probing many-body dynamics on a 51-atom quantum simulator
• Finite-temperature dynamics and thermal intraband magnon scattering in Haldane spin-one chains
• Quantum dynamics of the square-lattice Heisenberg model
• Fermionic Orbital Optimization in Tensor Network States
• Restoring phase coherence in a one-dimensional superconductor using power-law electron hopping
• Kibble Zurek mechanism of topological defect formation in quantum field theory with matrix product states
• Real-time scattering of interacting quasiparticles in quantum spin chains
• Corner transfer matrices for 2D strongly coupled many-body Floquet systems
• Invariance of Topological Indices Under Hilbert Space Truncation
• Impact of nonlocal correlations over different energy scales: A dynamical vertex approximation study
• Chebyshev expansion of spectral functions using restricted Boltzmann machines
• Fast scrambling without appealing to holographic duality
• Tree-tensor-network classifiers for machine learning: From quantum inspired to quantum assisted
• Many-body molecule formation at a domain wall in a one-dimensional strongly interacting ultracold Fermi gas
• Unraveling Dicke Superradiant Decay with Separable Coherent Spin States
• Generation of chiral solitons in antiferromagnetic chains by a quantum quench
• Average eigenstate entanglement entropy of the XY chain in a transverse field and its universality for translationally invariant quadratic fermionic models
• Continuous Matrix Product Operator Approach to Finite Temperature Quantum States
• Efficient Reconstruction of CAS-CI-Type Wave Functions for a DMRG State Using Quantum Information Theory and a Genetic Algorithm
• Quantum stabilization of photonic spatial correlations
• Phenomenology of anomalous transport in disordered one-dimensional systems
• Error estimates for extrapolations with matrix-product states
• Block-Lanczos Density-Matrix Renormalization-Group Approach to Spin Transport in Heisenberg Chains Coupled to Leads
• Detecting topological order with ribbon operators
• Front dynamics in the XY chain after local excitations
• Algorithms for Entanglement Renormalization: Boundaries, Impurities and Interfaces
• Quantum quenches in 1 + 1 dimensional conformal field theories
• 461 Fe の高温における転位挙動の動的観察(レール・棒・線・焼入性・クラッド鋼・その他, 性質, 日本鉄鋼協会 第 98 回(秋季)講演大会)
• Suppressing decoherence in quantum plasmonic systems by the spectral-hole-burning effect
• Symmetry-protected non-Fermi liquids, kagome spin liquids, and the chiral Kondo lattice model
• Nonequilibrium Kondo effect in a magnetic field: auxiliary master equation approach
• Phases of one-dimensional SU(N) cold atomic Fermi gases—From molecular Luttinger liquids to topological phases
• Intrinsic decoherence in isolated quantum systems
• Tree tensor networks and entanglement spectra
• Fluctuations and non-Hermiticity in the stochastic approach to quantum spins
• Quantum Machine Learning Tensor Network States
• Optimization at the boundary of the tensor network variety
• Universal non-equilibrium dynamics of pure states and density-dependent thermalization in Sachdev-Ye-Kitaev model
• Symmetrizing the Constraints -- Density Matrix Renormalization Group for Constrained Lattice Models
• Magnetic phase separation in a frustrated ferrimagnetic chain under a magnetic field
• A tensor network approach to sensing quantum light-matter interactions
• Spectral densities of a dispersive dielectric sphere in the modified Langevin noise formalism
• Observing the emergence of a velocity hierarchy in matter waves
• Quantum Error Management in Practice: A Cross-Stack Benchmark
• Dynamics of many-body localization in the presence of particle loss
• Quantum critical probing and simulation of colored quantum noise
• Entangled Dynamics in Macroscopic Quantum Tunneling of Bose-Einstein Condensates
• Symmetry-protected topological order identified via Gutzwiller-guided density-matrix-renormalization-group: SO(n) spin chains
• A Quantum-Inspired Algorithm for Wave Simulation Using Tensor Networks
• Cross-talk in superconducting qubit lattices with tunable couplers -- comparing transmon and fluxonium architectures
• Steady-state and quench-dependent relaxation of a quantum dot coupled to one-dimensional leads
• Inhomogeneous entanglement structure in monoaxial chiral ferromagnetic quantum spin chain
• Fractional Chern Insulator and Quantum Anomalous Hall Crystal in Twisted MoTe₂
• Estimating the best separable approximation of non-pure spin-squeezed states
• Finite-temperature real-time properties of magnetic polarons in two-dimensional quantum antiferromagnets
• Magnetic polarons at finite temperature: One-hole spectroscopy study
• Parton Distribution Functions in the Schwinger model from Tensor Network States
• Variational optimization algorithms for uniform matrix product states
• Self-dual S₃-invariant quantum chains
• Vector-spin-chirality order in a dimerized frustrated spin-12chain
• Rotor Jackiw-Rebbi Model: A Cold-Atom Approach to Chiral Symmetry Restoration and Charge Confinement
• Experimental Implementation of a Qubit-Efficient Variational Quantum Eigensolver with Analog Error Mitigation on a Superconducting Quantum Processor
• Successive randomized compression: A randomized algorithm for the compressed MPO-MPS product
• Comparative Analysis of Classical and Quantum-Inspired Solvers: A Preliminary Study on the Weighted Max-Cut Problem
• Gross–Neveu–Wilson model and correlated symmetry-protected topological phases
• Quantum Entanglement in Neural Network States
• Scalable projected entangled-pair state representation of random quantum circuit states
• Fast and direct preparation of a genuine lattice BEC via the quantum Mpemba effect
• Independent e- and m-anyon confinement in the parallel field toric code on non-square lattices
• The XYZ chain with Dzyaloshinsky–Moriya interactions: from spin–orbit-coupled lattice bosons to interacting Kitaev chains
• Multireference Perturbation Theory with Cholesky Decomposition for the Density Matrix Renormalization Group. [europepmc]
• Efficient representation of quantum many-body states with deep neural networks. [europepmc]
• A simple tensor network algorithm for two-dimensional steady states. [europepmc]
• Simulating quantum light propagation through atomic ensembles using matrix product states. [europepmc]
• Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamics. [europepmc]
• Efficient non-Markovian quantum dynamics using time-evolving matrix product operators. [europepmc]
• Density-Matrix Embedding Theory Study of the One-Dimensional Hubbard-Holstein Model. [europepmc]
• Tensor network simulation of multi-environmental open quantum dynamics via machine learning and entanglement renormalisation. [europepmc]
• A molecular movie of ultrafast singlet fission. [europepmc]
• One-dimensional flat bands in twisted bilayer germanium selenide. [europepmc]
• Impact of Vibrational Modes in the Plasmonic Purcell Effect of Organic Molecules. [europepmc]
• A general charge transport picture for organic semiconductors with nonlocal electron-phonon couplings. [europepmc]
• Observing non-ergodicity due to kinetic constraints in tilted Fermi-Hubbard chains. [europepmc]
• Spin-Pure Stochastic-CASSCF via GUGA-FCIQMC Applied to Iron-Sulfur Clusters. [europepmc]
• Cold atoms meet lattice gauge theory. [europepmc]
• Prethermalization in one-dimensional quantum many-body systems with confinement. [europepmc]
• Magnetically mediated hole pairing in fermionic ladders of ultracold atoms. [europepmc]
• Resolution of Electronic States in Heisenberg Cluster Models within the Unitary Group Approach. [europepmc]
• Evidence for the utility of quantum computing before fault tolerance. [europepmc]
• Flexible DMRG-Based Framework for Anharmonic Vibrational Calculations. [europepmc]
• Wavefunction matching for solving quantum many-body problems. [europepmc]
• Parallel Implementation of the Density Matrix Renormalization Group Method Achieving a Quarter petaFLOPS Performance on a Single DGX-H100 GPU Node. [europepmc]
• Comparison of Matrix Product State and Multiconfiguration Time-Dependent Hartree Methods for Nonadiabatic Dynamics of Exciton Dissociation. [europepmc]
• Polaritonic Chemistry Using the Density Matrix Renormalization Group Method. [europepmc]
• Nonadiabatic Field: A Conceptually Novel Approach for Nonadiabatic Quantum Molecular Dynamics. [europepmc]

Discussions

• So I have the following example of I think a left-canonical MPS of a PBC spin chain that is not normalized. For a single site, it looks like this: $$A = \begin{bmatrix} \frac{1}{\sqrt{2}}| 0 \rangle & [stackexchange, 0 points]
• I have been reading Schollwöck's review on Matrix Product states, and I understand the general principle and motivation behind them. However I cannot wrap my head around the construction of the matric [stackexchange, 0 points]

Related

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• Density Matrix Renormalization: A Review of the Method and its Applications