The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

Abstract

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.

arXiv comment115 pages, 56 figures

References in corpus (144)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Entanglement Entropy and Quantum Field Theory
• Area laws for the entanglement entropy - a review
• Entanglement in quantum critical phenomena
• Efficient classical simulation of slightly entangled quantum computations
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Theory of ultracold Fermi gases
• Entropy and Area
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Efficient simulation of one-dimensional quantum many-body systems
• Real time evolution using the density matrix renormalization group
• Entanglement spectrum of a topological phase in one dimension
• Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Entanglement renormalization
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Quantum trajectories and open many-body quantum systems
• A class of quantum many-body states that can be efficiently simulated
• Time-dependent variational principle for quantum lattices
• Unifying time evolution and optimization with matrix product states
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm
• DMRG and periodic boundary conditions: a quantum information perspective
• Criticality, the area law, and the computational power of PEPS
• Spectral Gap and Exponential Decay of Correlations
• Classical simulation of quantum many-body systems with a tree tensor network
• Quench dynamics and non equilibrium phase diagram of the Bose-Hubbard model
• Area laws in a many-body localized state and its implications for topological order
• The iTEBD algorithm beyond unitary evolution
• Simulating quantum computation by contracting tensor networks
• Violation of the entropic area law for Fermions
• From density-matrix renormalization group to matrix product states
• Optimal control technique for Many Body Quantum Systems dynamics
• Entanglement Properties of the Harmonic Chain
• Entropy, entanglement, and area: analytical results for harmonic lattice systems
• Renormalization and tensor product states in spin chains and lattices
• Matrix product operator representations
• Spread of correlations in long-range interacting quantum systems
• Algorithms for entanglement renormalization
• Time-evolving a matrix product state with long-ranged interactions
• Simulation of strongly correlated fermions in two spatial dimensions with fermionic Projected Entangled-Pair States
• Parafermionic edge zero modes in Z_n-invariant spin chains
• The computational complexity of PEPS
• Atomic Quantum Simulation of U(N) and SU(N) Non-Abelian Lattice Gauge Theories
• Tensor network states and algorithms in the presence of a global U(1) symmetry
• PEPS as ground states: degeneracy and topology
• Tensor network decompositions in the presence of a global symmetry
• Continuous Matrix Product States for Quantum Fields
• Lattice gauge theories simulations in the quantum information era
• More is the Same; Phase Transitions and Mean Field Theories
• Non-abelian symmetries in tensor networks: a quantum symmetry space approach
• Tensor network states and geometry
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
• The Non-Abelian Density Matrix Renormalization Group Algorithm
• Density matrix renormalization group algorithms with a single center site
• The iPEPS algorithm, improved: fast full update and gauge fixing
• The mass spectrum of the Schwinger model with Matrix Product States
• Finite temperature dynamical DMRG and the Drude weight of spin-1/2 chains
• Relaxation of antiferromagnetic order in spin-1/2 chains following a quantum quench
• Matrix Product States for dynamical simulation of infinite chains
• A positive tensor network approach for simulating open quantum many-body systems
• Applying matrix product operators to model systems with long-range interactions
• Optical Abelian Lattice Gauge Theories
• A Strictly Single-Site DMRG Algorithm with Subspace Expansion
• Variational quantum Monte Carlo simulations with tensor-network states
• Real-Time Diagrammatic Monte Carlo for Nonequilibrium Quantum Transport
• Quantum data compression, quantum information generation, and the density-matrix renormalization group method
• Fermionic multi-scale entanglement renormalization ansatz
• Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm
• Density-matrix renormalization-group method in momentum space
• Real-time Dynamics in U(1) Lattice Gauge Theories with Tensor Networks
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Exact diagonalization: the Bose-Hubbard model as an example
• Perfect Sampling with Unitary Tensor Networks
• Entanglement entropy of two disjoint blocks in critical Ising models
• Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling
• Optimal control of complex atomic quantum systems
• Strings, Projected Entangled Pair States, and variational Monte Carlo methods
• Tensor Networks for Lattice Gauge Theories with continuous groups
• Implementing global Abelian symmetries in projected entangled-pair state algorithms
• Simultaneous dimerization and SU(4) symmetry breaking of 4-color fermions on the square lattice
• Simulating open quantum dynamics with time-dependent variational matrix product states: Towards microscopic correlation of environment dynamics and reduced system evolution
• Tensor network states and algorithms in the presence of a global SU(2) symmetry
• Non-Abelian string breaking phenomena with Matrix Product States
• Entanglement negativity in the critical Ising chain
• Entanglement renormalization and gauge symmetry
• Ground-State Properties of Quantum Many-Body Systems: Entangled-Plaquette States and Variational Monte Carlo
• Classical simulation versus universality in measurement based quantum computation
• Lattice Gauge Tensor Networks
• Reducing the numerical effort of finite-temperature density matrix renormalization group transport calculations
• Monte Carlo simulation with Tensor Network States
• Elementary excitations in gapped quantum spin systems
• Hybrid-space density matrix renormalization group study of the doped two-dimensional Hubbard model
• Optimal Persistent Currents for Interacting Bosons on a Ring with a Gauge Field
• Efficient Matrix Product State Method for periodic boundary conditions
• Calculus of continuous matrix product states
• Tensor Networks and Quantum Error Correction
• Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks
• Unconstrained Tree Tensor Network: An adaptive gauge picture for enhanced performance
• Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks
• Faster identification of optimal contraction sequences for tensor networks
• A class of highly entangled many-body states that can be efficiently simulated
• A two-dimensional algorithm of the density matrix renormalization group
• Entanglement entropy of two disjoint intervals in c=1 theories
• Optimized Decimation of Tensor Networks with Super-orthogonalization for Two-Dimensional Quantum Lattice Models
• Adiabatic quenches through an extended quantum critical region
• Simulation of time evolution with the MERA
• Computations in Quantum Tensor Networks
• Critical properties of homogeneous binary trees
• Characterizing the Haldane phase in quasi-one-dimensional spin-1 Heisenberg antiferromagnets
• Light Cone Matrix Product
• Dynamics of a quantum phase transition in the 1D Bose-Hubbard model: excitations and correlations induced by a quench
• The Tensor Network Theory Library
• Fourier transform of fermionic systems and the spectral tensor network
• Theory of Network Contractor Dynamics for Exploring Thermodynamic Properties of Two-dimensional Quantum Lattice Models
• A matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions
• Topological transitions from multipartite entanglement with tensor networks: a procedure for sharper and faster characterization
• Superfluid density and quasi-long-range order in the one-dimensional disordered Bose-Hubbard model
• Crossover from the classical to the quantum Kibble-Zurek scaling
• Application of the Density Matrix Renormalization Group in momentum space
• Entanglement structure of the Hubbard model in momentum space
• Global symmetries in tensor network states: symmetric tensors versus minimal bond dimension
• Matrix Product States Algorithms and Continuous Systems
• Concatenated tensor network states
• The area law and real-space renormalization
• Full characterization of the quantum linear-zigzag transition in atomic chains
• Heisenberg antiferromagnet on Cayley trees: low-energy spectrum and even/odd site imbalance
• Improved scaling of Time-Evolving Block-Decimation algorithm through Reduced-Rank Randomized Singular Value Decomposition
• Mutual information area laws for thermal free fermions
• Renormalization algorithm with graph enhancement
• Entanglement entropy and the complex plane of replicas
• A variational method based on weighted graph states
• Few-body systems capture many-body physics: tensor network approach
• Improved numerical methods for infinite spin chains with long-range interactions
• Variational Monte Carlo with the Multi-Scale Entanglement Renormalization Ansatz
• Ab-initio characterization of the quantum linear-zigzag transition using DMRG
• Variational Monte Carlo simulations using tensor-product projected states
• Probabilistic low-rank factorization accelerates tensor network simulations of critical quantum many-body ground states
• Density-of-states of many-body quantum systems from tensor networks
• Tensor networks-a new tool for old problems
• Tensor network methods with graph enhancement

Cited by in corpus (103)

• Quantum Computing for High-Energy Physics: State of the Art and Challenges. Summary of the QC4HEP Working Group
• Review on novel methods for lattice gauge theories
• Quantum Simulating Nature's Fundamental Fields
• Quantifying nonstabilizerness of matrix product states
• Stabilizer entropies and nonstabilizerness monotones
• Unsupervised phase discovery with deep anomaly detection
• One decade of quantum optimal control in the chopped random basis
• Many-body magic via Pauli-Markov chains -- from criticality to gauge theories
• Lattice Quantum Electrodynamics in (3+1)-dimensions at finite density with Tensor Networks
• Tensor lattice field theory with applications to the renormalization group and quantum computing
• Nonstabilizerness via matrix product states in the Pauli basis
• Entanglement generation in $(1+1)D$ QED scattering processes
• Time Dependent Variational Principle for Tree Tensor Networks
• Efficient Tensor Network ansatz for high-dimensional quantum many-body problems
• Two-dimensional quantum-link lattice Quantum Electrodynamics at finite density
• Critical behaviors of non-stabilizerness in quantum spin chains
• Simulating 2+1d ${\mathbb{Z}}_3$ lattice gauge theory with iPEPS
• Efficient mapping for Anderson impurity problems with matrix product states
• Non-equilibrium phase transitions in $(1+1)$-dimensional quantum cellular automata with controllable quantum correlations
• Probing thermalization through spectral analysis with matrix product operators
• Charge and statistics of lattice quasiholes from density measurements: a Tree Tensor Network study
• Simulating (2+1)D SU(2) Yang-Mills Lattice Gauge Theory at finite density with tensor networks
• Entanglement entropy production in Quantum Neural Networks
• Digital quantum simulation of a (1+1)D SU(2) lattice gauge theory with ion qudits
• Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency
• Adaptive-weighted tree tensor networks for disordered quantum many-body systems
• Real-time scattering in the lattice Schwinger model
• Time-dependent variational principle with controlled bond expansion for matrix product states
• Superfluid to Mott transition in a Bose-Hubbard ring: Persistent currents and defect formation
• Optimal sampling of dynamical large deviations in two dimensions via tensor networks
• Augmenting Density Matrix Renormalization Group with Disentanglers
• Supersolid-Superfluid phase separation in the extended Bose-Hubbard model
• Topological phase diagrams of exactly solvable non-Hermitian interacting Kitaev chains
• Tensor Networks for Lattice Gauge Theories beyond one dimension: a Roadmap
• An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation
• Classical algorithms for many-body quantum systems at finite energies
• Entanglement in the Quantum Game of Life
• Surrogate models for quantum spin systems based on reduced order modeling
• Quantum Annealing for Neural Network optimization problems: a new approach via Tensor Network simulations
• Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories
• Strong zero modes in integrable quantum circuits
• Neural Projected Quantum Dynamics: a systematic study
• Anticoncentration and state design of random tensor networks
• Bias-Field Digitized Counterdiabatic Quantum Algorithm for Higher-Order Binary Optimization
• Ground state search by local and sequential updates of neural network quantum states
• Matrix Product States with Backflow correlations
• Entanglement of formation of mixed many-body quantum states via Tree Tensor Operators
• From spin chains to real-time thermal field theory using tensor networks
• Approximating the long time average of the density operator: Diagonal ensemble
• Converting long-range entanglement into mixture: tensor-network approach to local equilibration
• From Tree Tensor Network to Multiscale Entanglement Renormalization Ansatz
• Many-body entropies and entanglement from polynomially-many local measurements
• Computing time-periodic steady-state currents via the time evolution of tensor network states
• Bulk Localised Transport States in Infinite and Finite Quasicrystals via Magnetic Aperiodicity
• Topological phases of an interacting Majorana Benalcazar-Bernevig-Hughes model
• On the descriptive power of Neural-Networks as constrained Tensor Networks with exponentially large bond dimension
• Dynamical Ginzburg criterion for the quantum-classical crossover of the Kibble-Zurek mechanism
• Detection of Berezinskii-Kosterlitz-Thouless transition via Generative Adversarial Networks
• Hybrid Tree Tensor Networks for quantum simulation
• Constrained dynamics and confinement in the two-dimensional quantum Ising model
• 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
• Roughening dynamics of interfaces in the two-dimensional quantum Ising model
• Fractional resonances and prethermal states in Floquet systems
• Squashed entanglement in one-dimensional quantum matter
• Drude weight increase by orbital and repulsive interactions in fermionic ladders
• Parton Distribution Functions in the Schwinger model from Tensor Network States
• Neural network enhanced hybrid quantum many-body dynamical distributions
• Digital quantum simulation of lattice fermion theories with local encoding
• Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach
• Quantum circuit compilation with quantum computers
• Loop Series Expansions for Tensor Networks
• Non-equilibrium boundary driven quantum systems: models, methods and properties
• Simulating real-time molecular electron dynamics efficiently using the time-dependent density matrix renormalization group
• Emergence of non-Abelian SU(2) invariance in Abelian frustrated fermionic ladders
• Low-temperature Gibbs states with tensor networks
• Optimizing Radiotherapy Plans for Cancer Treatment with Tensor Networks
• Finite-temperature Rydberg arrays: quantum phases and entanglement characterization
• Cons-training Tensor Networks: Embedding and Optimization Over Discrete Linear Constraints
• Many-body correlations in one-dimensional optical lattices with alkaline-earth(-like) atoms
• Approximate inverse measurement channel for shallow shadows
• Prethermalization of quantum systems interacting with non-equilibrium environments
• Stable many-body resonances in open quantum systems
• Efficient optimization and conceptual barriers in variational finite Projected Entangled-Pair States
• Optimisation of ultrafast singlet fission in 1D rings towards unit efficiency
• Dynamical quantum phase transition and thermal equilibrium in the lattice Thirring model
• Clifford-Dressed Variational Principles for Precise Loschmidt Echoes
• The Cytnx Library for Tensor Networks
• Efficient tensor network simulation of multi-emitter non-Markovian systems
• Entanglement transitions in a boundary-driven open quantum many-body system
• Simulating Quantum Circuits with Tree Tensor Networks using Density-Matrix Renormalization Group Algorithm
• Variational optimization of projected entangled-pair states on the triangular lattice
• Variational approach to open quantum systems with long-range competing interactions
• A Synthesis of Hidden Subgroup Quantum Algorithms and Quantum Chemical Dynamics
• Matrix product state simulations of quantum quenches and transport in Coulomb blockaded superconducting devices
• Optimal sampling of tensor networks targeting wave function's fast decaying tails
• Process Tensor Approaches to Non-Markovian Quantum Dynamics
• Dynamical cluster-based strategy for improving tensor network algorithms in quantum circuit simulations
• Adaptive quantum dynamics with the time-dependent variational Monte Carlo method
• Self-interacting processes via Doob conditioning
• Excited states from local effective Hamiltonians of matrix product states and their entanglement spectrum transition
• Numerically efficient unitary evolution for Hamiltonians beyond nearest-neighbors
• Phase structure of the one-dimensional ${\mathbb{Z}}_2$ lattice gauge theory with second nearest-neighbor interactions