Tensor Network Renormalization

Abstract

We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement/correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.

Revised version, with improved numeric results and an additional appendix. Main text: 5 pages, 4 figures, Appendices: 9 pages, 12 figures

References in corpus (32)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Entanglement renormalization
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• A class of quantum many-body states that can be efficiently simulated
• Entanglement Renormalization and Holography
• Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
• Time Evolution of Entanglement Entropy from Black Hole Interiors
• Tensor renormalization group approach to 2D classical lattice models
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• Coarse-graining renormalization by higher-order singular value decomposition
• Algorithms for entanglement renormalization
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Tensor network states and geometry
• Second Renormalization of Tensor-Network States
• Entanglement renormalization for quantum fields
• Surface/State Correspondence as a Generalized Holography
• Renormalization of tensor-network states
• Entanglement renormalization, scale invariance, and quantum criticality
• Ground State and Finite Temperature Lanczos Methods
• Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
• Real Space Renormalization in Statistical Mechanics
• Coarse graining methods for spin net and spin foam models
• Exact holographic mapping and emergent space-time geometry
• Algorithms for tensor network renormalization
• Consistency Conditions for an AdS/MERA Correspondence
• Holographic Geometry of cMERA for Quantum Quenches and Finite Temperature
• Tensor Network and Black Hole
• Algorithms for entanglement renormalization: boundaries, impurities and interfaces
• Renormalization group contraction of tensor networks in three dimensions
• Boundary quantum critical phenomena with entanglement renormalization
• Phase Transitions of Ferromagnetic Potts Models on the Simple Cubic Lattice
• Entanglement, Tensor Networks and Black Hole Horizons

Cited by in corpus (270)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Tensor networks for complex quantum systems
• Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)
• Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT
• Recurrent Neural Network Wave Functions
• Unsupervised Generative Modeling Using Matrix Product States
• Complexity of Formation in Holography
• Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories
• Differentiable Programming Tensor Networks
• Hyper-optimized tensor network contraction
• Review on novel methods for lattice gauge theories
• Loop optimization for tensor network renormalization
• Neural Network Renormalization Group
• Einstein's Equations from Varying Complexity
• Tensor Network Algorithms: a Route Map
• A simple tensor network algorithm for two-dimensional steady states
• Faster Methods for Contracting Infinite 2D Tensor Networks
• Path-Integral Complexity for Perturbed CFTs
• Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
• Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
• Renormalization of tensor networks using graph independent local truncations
• Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions
• Decorated tensor network renormalization for lattice gauge theories and spin foam models
• Learning Relevant Features of Data with Multi-scale Tensor Networks
• Algorithms for tensor network renormalization
• Supervised Learning with Quantum-Inspired Tensor Networks
• Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories
• Path integral optimization as circuit complexity
• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• The Fluid Manifesto: Emergent symmetries, hydrodynamics, and black holes
• Holographic Spacetimes as Quantum Circuits of Path-Integrations
• Topological Complexity in AdS3/CFT2
• Tensor-Network Simulations of the Surface Code under Realistic Noise
• Quantum Information in Holographic Duality
• Anisotropic Tensor Renormalization Group
• Holographic Entanglement of Purification from Conformal Field Theories
• A measurement-based variational quantum eigensolver
• Entanglement renormalization and wavelets
• Tensor lattice field theory with applications to the renormalization group and quantum computing
• Mapping topological to conformal field theories through strange correlators
• Machine Learning Spatial Geometry from Entanglement Features
• Holographic Quantum Circuits from Splitting/Joining Local Quenches
• Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops
• Holographic subregion complexity under a thermal quench
• Renormalization group flows of Hamiltonians using tensor networks
• Topological conformal defects with tensor networks
• TensorNetwork: A Library for Physics and Machine Learning
• From Path Integrals to Tensor Networks for AdS/CFT
• Riemannian optimization of isometric tensor networks
• Principles and symmetries of complexity in quantum field theory
• Towards a phase diagram for spin foams
• Berezinskii-Kosterlitz-Thouless transition in lattice Schwinger model with one flavor of Wilson fermion
• Hypercuboidal renormalization in spin foam quantum gravity
• Tensor renormalization group study of the non-Abelian Higgs model in two dimensions
• TensorNetwork for Machine Learning
• Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation
• X-Symbols for Non-Abelian Symmetries in Tensor Networks
• Isometric Tensor Network representation of string-net liquids
• Tensor network analysis of critical coupling in two dimensional ${\varphi }^4$ theory
• Coarse graining flow of spin foam intertwiners
• Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity
• A gauge-invariant, rotor Hamiltonian from dual variables of 3D U(1) gauge theory
• Tensor network formulation of the massless Schwinger model
• Conformal data and renormalization group flow in critical quantum spin chains using periodic uniform matrix product states
• Conformal fields and operator product expansion in critical quantum spin chains
• A high-performance code for EPRL spin foam amplitudes
• Renormalization of symmetry restricted spin foam models with curvature in the asymptotic regime
• A tensor network quotient takes the vacuum to the thermal state
• Simulation of three-dimensional quantum systems with projected entangled-pair states
• Complex Quantum Networks: a Topical Review
• Fast counting with tensor networks
• Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices
• Variational Monte Carlo method for fermionic models combined with tensor networks and applications to the hole-doped two-dimensional Hubbard model
• Local scale transformations on the lattice with tensor network renormalization
• Calculation of fermionic Green functions with Grassmann higher-order tensor renormalization group
• Hamiltonian Renormalisation I: Derivation from Osterwalder-Schrader Reconstruction
• Bond-weighted Tensor Renormalization Group
• Model-Independent Learning of Quantum Phases of Matter with Quantum Convolutional Neural Networks
• Linear-time general decoding algorithm for the surface code
• Path-Integral Optimization from Hartle-Hawking Wave Function
• Entanglement Renormalization for Weakly Interacting Fields
• Efficient Simulation of Dynamics in Two-Dimensional Quantum Spin Systems with Isometric Tensor Networks
• Automatic Differentiation for Second Renormalization of Tensor Networks
• Tensor Renormalization Group with Randomized Singular Value Decomposition
• Calculation of higher-order moments by higher-order tensor renormalization group
• Mixed s-sourcery: Building many-body states using bubbles of Nothing
• Anomalies and entanglement renormalization
• Is Deep Learning a Renormalization Group Flow?
• Gauging tensor networks with belief propagation
• Tensor network simulation for the frustrated $J_1$-$J_2$ Ising model on the square lattice
• Finite-size and finite bond dimension effects of tensor network renormalization
• Holographic Path-Integral Optimization
• Boundary Tensor Renormalization Group
• Entanglement branching operator
• Complexity=Anything: Singularity Probes
• Variational autoencoder reconstruction of complex many-body physics
• Computing the renormalization group flow of two-dimensional ${\varphi }^4$ theory with tensor networks
• Renormalization Group Circuits for Weakly Interacting Continuum Field Theories
• A beginner's guide to non-abelian iPEPS for correlated fermions
• Tensor renormalization group and the volume independence in 2D U($N$) and SU($N$) gauge theories
• Horizon as Critical Phenomenon
• Hyper-optimized approximate contraction of tensor networks with arbitrary geometry
• A defect in holographic interpretations of tensor networks
• Resolving the Berezinskii-Kosterlitz-Thouless transition in the 2D XY model with tensor-network based level spectroscopy
• Phase Transition of the Ising Model on Fractal Lattice
• Accurate simulation of q-state clock model
• Gentle introduction to rigorous Renormalization Group: a worked fermionic example
• Complexity growth of rotating black holes with a probe string
• Non-Equilibrium Field Theory for Dynamics Starting from Arbitrary Athermal Initial Conditions
• Detecting and identifying 2D symmetry-protected topological, symmetry-breaking and intrinsic topological phases with modular matrices via tensor-network methods
• Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell
• Tensor networks for interpretable and efficient quantum-inspired machine learning
• Quantum information spreading in generalised dual-unitary circuits
• Partial lifting of degeneracy in the $J_1-J_2-J_3$ Ising antiferromagnet on the kagome lattice
• Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations
• Complexity and Emergence of Warped ${\text{AdS}}_3$ Space-time from Chiral Liouville Action
• Artificial out-of-plane Ising antiferromagnet on the kagome lattice with very small further neighbour couplings
• Tensor network formulation of two dimensional gravity
• Scale invariant transfer matrices and Hamiltionians
• Bayesian Renormalization
• Improved Quantum Computing with the Higher-order Trotter Decomposition
• Approximate encoding of quantum states using shallow circuits
• Tensor networks for quantum computing
• Tensor Networks for Lattice Gauge Theories beyond one dimension: a Roadmap
• Nuclear Structure from the In-Medium Similarity Renormalization Group
• Examples of symmetry-preserving truncations in tensor field theory
• Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories
• Nuclear norm regularized loop optimization for tensor network
• Efficient representation of long-range interactions in tensor network algorithms
• Circuit Complexity in Topological Quantum Field Theory
• Renormalization Group Circuits for Gapless States
• Boson condensation and instability in the tensor network representation of string-net states
• Matrix product state renormalization
• Tensor Renormalization Group Algorithms with a Projective Truncation Method
• Bond-weighting method for the Grassmann tensor renormalization group
• Block belief propagation algorithm for two-dimensional tensor networks
• Tensor network representations from the geometry of entangled states
• Boundary conformal spectrum and surface critical behaviors of the classical spin systems: a tensor network renormalization study
• Operator-algebraic construction of gauge theories and Jones' actions of Thompson's groups
• Scaling dimensions from linearized tensor renormalization group transformations
• Tensor RG approach to high-temperature fixed point
• Differentiable Programming of Isometric Tensor Networks
• Surface growth scheme for bulk reconstruction and tensor network
• Tensor renormalization group in bosonic field theory
• Implementation of rigorous renormalization group method for ground space and low-energy states of local Hamiltonians
• Emergence of the XY-like phase in the deformed spin-3/2 AKLT systems
• Boundary theories of critical matchgate tensor networks
• Variational corner transfer matrix renormalization group method for classical statistical models
• Q-curvature and Path Integral Complexity
• Tensor networks as conformal transformations
• Spin-1/2 XXZ Heisenberg chain in a longitudinal magnetic field
• Discrete Gravity on Random Tensor Network and Holographic Rényi Entropy
• Network Renormalization
• Critical line of the triangular Ising antiferromagnet in a field from a $C_3$-symmetric corner transfer matrix algorithm
• Triad second renormalization group
• Entanglement entropy and correlations in loop quantum gravity
• Collective Monte Carlo updates through tensor network renormalization
• Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices
• Tensor Network Based Finite-Size Scaling for Two-Dimensional Classical Models
• Lifetime-based Optimization for Simulating Quantum Circuits on a New Sunway Supercomputer
• The structure of fixed-point tensor network states characterizes patterns of long-range entanglement
• Holographic complexity of the disk subregion in (2+1)-dimensional gapped systems
• Renormalization group and approximate error correction
• Holographic unitary renormalization group for correlated electrons -- I: a tensor network approach
• Matrix product state fixed points of non-Hermitian transfer matrices
• GrassmannTN: a Python package for Grassmann tensor network computations
• The Inverse of Exact Renormalization Group Flows as Statistical Inference
• To be unitary-invariant or not?: a simple but non-trivial proposal for the complexity between states in quantum mechanics/field theory
• Partonic distribution functions and amplitudes using tensor network methods
• Tensor network calculation of the logarithmic correction exponent in the XY model
• Multi-impurity method for the bond-weighted tensor renormalization group
• Efficient tensor network algorithm for layered systems
• Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models
• Tensor renormalization group study of (1+1)-dimensional U(1) gauge-Higgs model at $\theta =\pi$ with Lüscher's admissibility condition
• Algorithms for Tensor Network Contraction Ordering
• Tensor Network Computations That Capture Strict Variationality, Volume Law Behavior, and the Efficient Representation of Neural Network States
• Sign problem in tensor network contraction
• Tensor Renormalization Group Centered About a Core Tensor
• Improved coarse-graining methods on two dimensional tensor networks including fermions
• Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point
• Magic entanglement renormalization for quantum fields
• Generalization of the tensor renormalization group approach to 3-D or higher dimensions
• Stack operation of tensor networks
• Singular-Value-Decomposition Analysis of Associative Memory in a Neural Network
• Higher-order tensor renormalization group approach to lattice glass model
• Holographic analysis of boundary correlation functions for the hyperbolic-lattice Ising model
• Capturing long-range memory structures with tree-geometry process tensors
• Statistical Mechanics Approach to the Holographic Renormalization Group: Bethe Lattice Ising Model and p-adic AdS/CFT
• Tensor Network Renormalization Study on the Crossover in Classical Heisenberg and $\mathrm{R}{\mathrm{P}}^2$ Models in Two Dimensions
• Properties of the Hamiltonian Renormalisation and its application to quantum mechanics on the circle
• Tensor network method for reversible classical computation
• Combination of Tensor Network States and Green's function Monte Carlo
• Phase boundary location with information-theoretic entropy in tensor renormalization group flows
• Phase transitions of a 2D deformed-AKLT model
• Méthodes de calcul avec réseaux de tenseurs en physique (Basic tensor network computations in physics)
• Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach
• Evaluating the Jones polynomial with tensor networks
• Deriving the non-perturbative gravitational dual of quantum Liouville theory from BCFT operator algebra
• Loop-TNR analysis of CP(1) model with theta term
• Shortcut Matrix Product States and its applications
• Loop Series Expansions for Tensor Networks
• Three-dimensional real space renormalization group with well-controlled approximations
• Rotations, Negative Eigenvalues, and Newton Method in Tensor Network Renormalization Group
• Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction
• Finite-size scaling analysis of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states
• Emergent geometry and path integral optimization for a Lifshitz action
• A Monte Carlo algorithm for spin foam intertwiners
• Entanglement Entropy of the Two-Dimensional +-J Ising Model on the Nishimori Line
• Reduced tensor network formulation for non-Abelian gauge theories in arbitrary dimensions
• 3D Tensor Renormalisation Group at High Temperatures
• Boundary CFT and tensor network approach to surface critical phenomena of the tricritical 3-state Potts model
• Complex Langevin analysis of 2D U(1) gauge theory on a torus with a $\theta$ term
• Grassmann tensor renormalization group approach to $(1+1)$-dimensional two-color lattice QCD at finite density
• Transfer Matrix and Lattice Dilatation Operator for High-Quality Fixed Points in Tensor Network Renormalization Group
• Tensor-Ring Decomposition with Index-Splitting
• Controlling quantum many-body systems using reduced-order modelling
• Exact Renormalization of Wave Functionals yields Continuous MERA
• Quantum subspace expansion approach for simulating dynamical response functions of Kitaev spin liquids
• Initial tensor construction and dependence of the tensor renormalization group on initial tensors
• Loop update for infinite projected entangled-pair states in two spatial dimensions
• The landscape of software for tensor computations
• Tensor Renormalization Group for interacting quantum fields
• Two Dimensional Isometric Tensor Networks on an Infinite Strip
• All-mode Renormalization for Tensor Network with Stochastic Noise
• Measurement of entanglement entropy in the two-dimensional Potts model using wavelet analysis
• Tensor network simulations for nonorientable surfaces
• Spectroscopy with the tensor renormalization group method
• Exploiting the Hermitian symmetry in tensor network algorithms
• Finite-size scaling on the torus with periodic projected entangled-pair states
• Phase structure analysis of 2d lattice CP(1) model with $\theta$ term using tensor renormalization group method
• Tensor Renormalization Group Calculations of Partition-Function Ratios
• Accuracy and Efficiency of Simplified Tensor Network Codes
• p-adic CFT is a holographic tensor network
• Error metric for non-trace-preserving quantum operations
• A Practical Introduction to Tensor Network Renormalization with TNRKit.jl
• What kind of "complexity" is dual to holographic complexity?
• Essential difference between 2D and 3D from the perspective of real-space renormalization group
• Tensor network renormalization approach to antiferromagnetic 6-state clock model on the Union Jack lattice
• Tensor networks for $p$-spin models
• The Tangent Space to the Manifold of Critical Classical Hamiltonians Representable by Tensor Networks
• Entanglement Renormalization of the class of Continuous Matrix Product States
• Positive bias makes tensor-network contraction tractable
• Lattice-reflection symmetry in tensor-network renormalization group with entanglement filtering in two and three dimensions
• Tensor renormalization group approach to critical phenomena via symmetry-twisted partition functions
• Implicitly disentangled renormalization
• Variational boundary based tensor network renormalization group
• Virasoro and Kac-Moody algebra in generic tensor network representations of 2d critical lattice partition functions
• Accelerating two-dimensional tensor network contractions using QR decompositions
• Forward-mode automatic differentiation for the tensor renormalization group and its relation to the impurity method
• Fast Tensor Disentangling Algorithm
• Efficient 2D Tensor Network Simulation of Quantum Systems
• Neuralized Fermionic Tensor Networks for Quantum Many-Body Systems
• Information Compression and Performance Evaluation of Tic-Tac-Toe's Evaluation Function Using Singular Value Decomposition
• Graphical Representations and Worm Algorithms for the O($N$) Spin Model
• Complex path simulations of geometrically frustrated ladders
• Bucket Renormalization for Approximate Inference
• Entanglement scaling and criticality of infinite-size quantum many-body systems in continuous space addressed by a tensor network approach
• Electric Field Propagation Through Singular Value Decomposition
• Minimizing couplings in renormalization by preserving short-range mutual information
• Two-dimensional $J_1$-$J_2$ clock model: Enhanced symmetries, emergent orders, and Landau-incompatible transitions
• Holographic Aspects of Dynamical Mean-Field Theory
• Holographic Space-time and Quantum Information
• Noise-tolerant Detection of $Z_N$ Topological Orders in Quantum Many-body States
• Tensor renormalization group study of cold and dense QCD in the strong coupling limit
• Lattice and PT symmetries in tensor-network renormalization group: Case study of a hard-square lattice gas model
• Lubkin-Page typicality bounds for Type~II von~Neumann factors
• Weak coupling theory of nickel-based 327 superconductors
• Visual configuration segmentation of quantum states for phase identification in many-body systems
• Universal spectrum structure at nonequilibrium critical points in the (1+1)-dimensional directed percolation
• Note on Localized Objects as Constrained States of Holographic Variables