Tensor Network Algorithms: a Route Map

Abstract

Tensor networks provide extremely powerful tools for the study of complex classical and quantum many-body problems. Over the last two decades, the increment in the number of techniques and applications has been relentless, and especially the last ten years have seen an explosion of new ideas and results that may be overwhelming for the newcomer. This short review introduces the basic ideas, the best established methods and some of the most significant algorithmic developments that are expanding the boundaries of the tensor network potential. The goal is to help the reader not only appreciate the many possibilities offered by tensor networks, but also find their way through state-of-the-art codes, their applicability and some avenues of ongoing progress.

Review paper. To appear (after edition) in Annu. Rev. Condens. Matter Phys. v2 fixes a problem with the placement of figures

References in corpus (38)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Real time evolution using the density matrix renormalization group
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• A class of quantum many-body states that can be efficiently simulated
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Area laws in quantum systems: mutual information and correlations
• Tensor renormalization group approach to 2D classical lattice models
• DMRG and periodic boundary conditions: a quantum information perspective
• Criticality, the area law, and the computational power of PEPS
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• From density-matrix renormalization group to matrix product states
• Entropy scaling and simulability by Matrix Product States
• Time-evolving a matrix product state with long-ranged interactions
• Tensor network states and algorithms in the presence of a global U(1) symmetry
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Continuous Matrix Product States for Quantum Fields
• Algorithms for finite Projected Entangled Pair States
• Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems
• Matrix Product States for dynamical simulation of infinite chains
• Variational quantum Monte Carlo simulations with tensor-network states
• The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
• Entanglement renormalization, scale invariance, and quantum criticality
• Time evolution algorithms for Matrix Product States and DMRG
• Generic Construction of Efficient Matrix Product Operators
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Chebyshev matrix product state approach for spectral functions
• Implementing global Abelian symmetries in projected entangled-pair state algorithms
• Quantum dynamics of thermalizing systems
• Monte Carlo simulation with Tensor Network States
• Density Matrix Renormalization Group in the Heisenberg Picture
• Tensor network techniques for the computation of dynamical observables in 1D quantum spin systems
• Simulation of time evolution with the MERA
• Sequentially generated states for the study of two dimensional systems
• Real-space renormalization yields finite correlations
• Infinite Matrix Product States vs Infinite Projected Entangled-Pair States on the Cylinder: a comparative study

Cited by in corpus (80)

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• Nonstabilizerness via matrix product states in the Pauli basis
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• Non-stabilizerness versus entanglement in matrix product states
• Augmenting Density Matrix Renormalization Group with Clifford Circuits
• Sublinear scaling in non-Markovian open quantum systems simulations
• Nonstabilizerness in U(1) lattice gauge theory
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• Real-time scattering in the lattice Schwinger model
• Optimal sampling of dynamical large deviations in two dimensions via tensor networks
• Adaptive variational quantum minimally entangled typical thermal states for finite temperature simulations
• Hilbert space delocalization under random unitary circuits
• On temporal entropy and the complexity of computing the expectation value of local operators after a quench
• Tensor networks for quantum computing
• Tensor Networks for Lattice Gauge Theories beyond one dimension: a Roadmap
• Atomic Quantum Technologies for Quantum Matter and Fundamental Physics Applications
• Generating function for projected entangled-pair states
• Bias-Field Digitized Counterdiabatic Quantum Algorithm for Higher-Order Binary Optimization
• Neural Projected Quantum Dynamics: a systematic study
• Quantum Complexity Fluctuations from Nuclear and Hypernuclear Forces
• Converting long-range entanglement into mixture: tensor-network approach to local equilibration
• Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices
• Fate of entanglement in magnetism under Lindbladian or non-Markovian dynamics and conditions for their transition to Landau-Lifshitz-Gilbert classical dynamics
• Nonstabilizerness dynamics in many-body localized systems
• Entanglement and confinement in lattice gauge theory tensor networks
• Fermionic wave packet scattering: a quantum computing approach
• Holographic scattering and non-minimal RT surfaces
• Simulation of Quantum Computers: Review and Acceleration Opportunities
• Ab-initio tree-tensor-network digital twin for quantum computer benchmarking in 2D
• Dynamical heterogeneity and large deviations in the open quantum East glass model from tensor networks
• Fermionic Magic Resources of Quantum Many-Body Systems
• On the Magnetization of the $120^{\circ }$ order of the Spin-1/2 Triangular Lattice Heisenberg Model: a DMRG revisit
• Parton Distribution Functions in the Schwinger model from Tensor Network States
• Extending Non-Perturbative Simulation Techniques for Open-Quantum Systems to Excited-State Proton Transfer and Ultrafast Non-Adiabatic Dynamics
• Charge and spin gaps of the ionic Hubbard model with density-dependent hopping
• The Green's function Monte Carlo combined with projected entangled pair state approach to the frustrated $J_1$-$J_2$ Heisenberg model
• Computing quantum magic of state vectors
• Impact and Interplay of Quantum Coherence and Dissipative Dynamics for Isotope Effects in Excited-State Intramolecular Proton Transfer
• Quantum circuit compilation with quantum computers
• Approximately-symmetric neural networks for quantum spin liquids
• Limitations of tensor network approaches for optimization and sampling: A comparison to quantum and classical Ising machines
• Low-temperature Gibbs states with tensor networks
• The resource theory of tensor networks
• Single-layer tensor network approach for three-dimensional quantum systems
• Spectral gaps of two- and three-dimensional many-body quantum systems in the thermodynamic limit
• The Role of Quantum Computing in Advancing Scientific High-Performance Computing: A perspective from the ADAC Institute
• Meson thermalization with a hot medium in the open Schwinger model
• Many-body correlations in one-dimensional optical lattices with alkaline-earth(-like) atoms
• Stabilizer-Accelerated Quantum Many-Body Ground-State Estimation
• Efficient and systematic calculation of arbitrary observables for the matrix product state excitation ansatz
• Spontaneous symmetry breaking in a $SO(3)$ non-Abelian lattice gauge theory in $2+1$D with quantum algorithms
• Survey on Computational Applications of Tensor Network Simulations
• Dynamical quantum phase transition and thermal equilibrium in the lattice Thirring model
• Hyperoptimized approximate contraction of tensor networks for rugged-energy-landscape spin glasses on periodic square and cubic lattices
• Efficient tensor network simulation of multi-emitter non-Markovian systems
• Positive bias makes tensor-network contraction tractable
• Revisiting the symmetry-resolved entanglement for non-invertible symmetries in $1+1$d conformal field theories
• Spectroscopy and complex-time correlations using minimally entangled typical thermal states
• Coexistence of insulating phases in confined fermionic chains with a Wannier-Stark potential
• Simulating Quantum Circuits with Tree Tensor Networks using Density-Matrix Renormalization Group Algorithm
• Optimal sampling of tensor networks targeting wave function's fast decaying tails
• Unconventional correlated metallic behavior due to interorbital Coulomb interaction
• Tensor Network Estimation of Distribution Algorithms
• A hybrid method integrating Green's function Monte Carlo and projected entangled pair states
• The Software Landscape for the Density Matrix Renormalization Group
• The ITransverse.jl library for transverse tensor network contractions
• Numerically exact quantum dynamics with tensor networks: Predicting the decoherence of interacting spin systems
• Les Houches Lecture Notes on Tensor Networks
• Phase structure of the one-dimensional ${\mathbb{Z}}_2$ lattice gauge theory with second nearest-neighbor interactions
• Time evolution of controlled many-body quantum systems with matrix product operators
• Process Tensor Approaches to Non-Markovian Quantum Dynamics
• Limits of Clifford Disentangling in Tensor Network States
• Excited states from local effective Hamiltonians of matrix product states and their entanglement spectrum transition
• Self-interacting processes via Doob conditioning
• Variational Tensor Network Simulation of Gaussian Boson Sampling and Beyond
• Variational matrix product states for combinatorial optimization
• Holographic Aspects of Dynamical Mean-Field Theory