Entanglement renormalization and gauge symmetry

Abstract

A lattice gauge theory is described by a redundantly large vector space that is subject to local constraints, and can be regarded as the low energy limit of an extended lattice model with a local symmetry. We propose a numerical coarse-graining scheme to produce low energy, effective descriptions of lattice models with a local symmetry, such that the local symmetry is exactly preserved during coarse-graining. Our approach results in a variational ansatz for the ground state(s) and low energy excitations of such models and, by extension, of lattice gauge theories. This ansatz incorporates the local symmetry in its structure, and exploits it to obtain a significant reduction of computational costs. We test the approach in the context of the toric code with a magnetic field, equivalent to Z2 lattice gauge theory, for lattices with up to 16 x 16 sites (16^2 x 2 = 512 spins) on a torus. We reproduce the well-known ground state phase diagram of the model, consisting of a deconfined and spin polarized phases separated by a continuous quantum phase transition, and obtain accurate estimates of energy gaps, ground state fidelities, Wilson loops, and several other quantities.

reviewed version as published in PRB; this version includes a new section about the accuracy of the results several corrections and added citations

References in corpus (68)

• Fault-tolerant quantum computation by anyons
• The density-matrix renormalization group
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Topological quantum memory
• Critical Phenomena and Renormalization-Group Theory
• String-net condensation: A physical mechanism for topological phases
• Entanglement renormalization
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• Topological orders and Edge excitations in FQH states
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• A class of quantum many-body states that can be efficiently simulated
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Ground state overlap and quantum phase transitions
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• QCD forces and heavy quark bound states
• Fidelity, dynamic structure factor, and susceptibility in critical phenomena
• Quantum critical scaling of the geometric tensors
• Corner Transfer Matrix Renormalization Group Method
• Full nonperturbative QCD simulations with 2+1 flavors of improved staggered quarks
• Simulation of two dimensional quantum systems on an infinite lattice revisited: corner transfer matrix for tensor contraction
• Topological quantum order: stability under local perturbations
• Algorithms for entanglement renormalization
• Simulation of strongly correlated fermions in two spatial dimensions with fermionic Projected Entangled-Pair States
• Tensor network decompositions in the presence of a global symmetry
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• Second Renormalization of Tensor-Network States
• The phase diagram of neutral quark matter: Self-consistent treatment of quark masses
• Large N expansion for frustrated and doped quantum antiferromagnets
• Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension
• The Non-Abelian Density Matrix Renormalization Group Algorithm
• Fermionic Projected Entangled Pair States
• Ground-State Fidelity and Bipartite Entanglement in the Bose-Hubbard Model
• Low-energy effective theory of the toric code model in a parallel field
• Entanglement entropy in gauge theories and the holographic principle for electric strings
• Fidelity susceptibility, scaling, and universality in quantum critical phenomena
• Frustrated antiferromagnets with entanglement renormalization: ground state of the spin-1/2 Heisenberg model on a kagome lattice
• Topological multicritical point in the Toric Code and 3D gauge Higgs Models
• Renormalization of tensor-network states
• Ground state fidelity from tensor network representations
• Entanglement renormalization and topological order
• Quantum fidelity and quantum phase transitions in matrix product states
• Entanglement renormalization, scale invariance, and quantum criticality
• Fermionic multi-scale entanglement renormalization ansatz
• Entanglement renormalization in two spatial dimensions
• Quantum phase transitions and quantum fidelity in free fermion graphs
• Exploring frustrated spin-systems using Projected Entangled Pair States (PEPS)
• Contraction of fermionic operator circuits and the simulation of strongly correlated fermions
• Multi-scale Entanglement Renormalization Ansatz in Two Dimensions: Quantum Ising Model
• Self-duality and bound states of the toric code model in a transverse field
• Adiabatic Preparation of Topological Order
• Unitary circuits for strongly correlated fermions
• Quantum MERA Channels
• Exact entanglement renormalization for string-net models
• Detecting Dual Superconductivity in the Ground State of Gauge Theory
• Unified approach to Quantum and Classical Dualities
• Vertical Density Matrix Algorithm: A Higher-Dimensional Numerical Renormalization Scheme based on the Tensor Product State Ansatz
• The Density Matrix Renormalization Group Method applied to Interaction Round a Face Hamiltonians
• Tensor product representation of topological ordered phase: necessary symmetry conditions
• The Spectrum of the 2+1 Dimensional Gauge Ising Model
• Determination of the chiral condensate from 2+1-flavor lattice QCD
• Finite-Size Scaling in the transverse Ising Model on a Square Lattice
• Ground-state properties of the spin-1/2 antiferromagnetic Heisenberg model on the triangular lattice: A variational study based on entangled-plaquette states
• Matrix product representation of gauge invariant states in a Z_2 lattice gauge theory
• Symmetry breaking and criticality in tensor-product states
• Numerical Study of Spin-1/2 XXZ Model on Square Lattice from Tensor Product States
• Homogeneous MERA states: an information theoretical analysis
• Direct numerical computation of disorder parameters

Cited by in corpus (78)

• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Tensor networks for complex quantum systems
• Simulating Lattice Gauge Theories within Quantum Technologies
• Studying Two Dimensional Systems With the Density Matrix Renormalization Group
• Remarks on entanglement entropy for gauge fields
• Can One Trust Quantum Simulators?
• Lattice gauge theories simulations in the quantum information era
• Simulations of non-Abelian gauge theories with optical lattices
• The mass spectrum of the Schwinger model with Matrix Product States
• Review on novel methods for lattice gauge theories
• Tensor networks for Lattice Gauge Theories and Atomic Quantum Simulation
• Optical Abelian Lattice Gauge Theories
• Lecture Notes of Tensor Network Contractions
• Matrix product states for gauge field theories
• Anyons and matrix product operator algebras
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• Gauging quantum states: from global to local symmetries in many-body systems
• The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
• Tensor Networks for Lattice Gauge Theories with continuous groups
• Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
• Non-Abelian string breaking phenomena with Matrix Product States
• Entanglement negativity in the critical Ising chain
• Coarse graining methods for spin net and spin foam models
• Decorated tensor network renormalization for lattice gauge theories and spin foam models
• Efficient Basis Formulation for (1+1)-Dimensional SU(2) Lattice Gauge Theory: Spectral calculations with matrix product states
• Lattice Gauge Tensor Networks
• Confinement and lack of thermalization after quenches in the bosonic Schwinger model
• Hamiltonian simulation of the Schwinger model at finite temperature
• Deeply inelastic scattering structure functions on a hybrid quantum computer
• Fusion basis for lattice gauge theory and loop quantum gravity
• 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
• Entanglement entropy of two disjoint intervals in c=1 theories
• Finite-density phase diagram of a (1+1)-d non-abelian lattice gauge theory with tensor networks
• Breakdown of a perturbed Z_N topological phase
• Confinement and string breaking for QED${}_2$ in the Hamiltonian picture
• Towards a phase diagram for spin foams
• Toolbox for Abelian lattice gauge theories with synthetic matter
• Algorithms for entanglement renormalization: boundaries, impurities and interfaces
• Phase Diagram and Conformal String Excitations of Square Ice using Gauge Invariant Matrix Product States
• Topological transitions from multipartite entanglement with tensor networks: a procedure for sharper and faster characterization
• Tensor Network Simulation of compact one-dimensional lattice Quantum Chromodynamics at finite density
• Variational Monte Carlo simulation with tensor networks of a pure ${\mathbb{Z}}_3$ gauge theory in (2+1)d
• Simulating 2+1d ${\mathbb{Z}}_3$ lattice gauge theory with iPEPS
• Global symmetries in tensor network states: symmetric tensors versus minimal bond dimension
• Tensor network simulation of QED on infinite lattices: learning from (1+1)d, and prospects for (2+1)d
• Robust topological order in fermionic ${\mathbb{Z}}_2$ gauge theories: from Aharonov-Bohm instability to soliton-induced deconfinement
• Algebraically contractible topological tensor network states
• Simulating (2+1)D SU(2) Yang-Mills Lattice Gauge Theory at finite density with tensor networks
• Finding the ground state of a lattice gauge theory with fermionic tensor networks: a $2+1d$ ${\mathbb{Z}}_2$ demonstration
• Matrix product states and the nonabelian rotor model
• Quantum gravity states, entanglement graphs and second-quantized tensor networks
• Mean-field ansatz for topological phases with string tension
• Robustness of Topological Order in the Toric Code with Open Boundaries
• Different models of gravitating Dirac fermions in optical lattices
• Quantum Yang-Mills theory: an overview of a programme
• Gauss Law, Minimal Coupling and Fermionic PEPS for Lattice Gauge Theories
• Holographic maps from quantum gravity states as tensor networks
• Circuit-based digital adiabatic quantum simulation and pseudoquantum simulation as new approaches to lattice gauge theory
• An Operational Definition of Topological Order
• Matrix product states for Hamiltonian lattice gauge theories
• Entanglement and confinement in lattice gauge theory tensor networks
• Duality transformations and the entanglement entropy of gauge theories
• Gauged Gaussian PEPS -- A High Dimensional Tensor Network Formulation for Lattice Gauge Theories
• Topological Minimally Entangled States via Geometric Measure
• Matrix Product States for Lattice Field Theories
• Wilson Loops and Area Laws in Lattice Gauge Theory Tensor Networks
• From entanglement renormalisation to the disentanglement of quantum double models
• Entanglement renormalization for gauge invariant quantum fields
• Error-correction properties of an interacting topological insulator
• Discrete Abelian lattice gauge theories on a ladder and their dualities with quantum clock models
• Internal structure of gauge-invariant Projected Entangled Pair States
• Quantum Entanglement of Tensor Networks with Symmetry Projections
• Projected Entangled Pair States for Lattice Gauge Theories with Dynamical Fermions
• Topological order on the Bloch sphere
• Algorithmic Aspects of Gauged Gaussian Fermionic PEPS: Gauge Fixing and Translation Invariance
• Free fermion representation of the topological surface code
• Superselection-Resolved Entanglement in Lattice Gauge Theories: A Tensor Network Approach