Fast convergence of imaginary time evolution tensor network algorithms by recycling the environment

Abstract

We propose an environment recycling scheme to speed up a class of tensor network algorithms that produce an approximation to the ground state of a local Hamiltonian by simulating an evolution in imaginary time. Specifically, we consider the time-evolving block decimation (TEBD) algorithm applied to infinite systems in 1D and 2D, where the ground state is encoded, respectively, in a matrix product state (MPS) and in a projected entangled-pair state (PEPS). An important ingredient of the TEBD algorithm (and a main computational bottleneck, especially with PEPS in 2D) is the computation of the so-called environment, which is used to determine how to optimally truncate the bond indices of the tensor network so that their dimension is kept constant. In current algorithms, the environment is computed at each step of the imaginary time evolution, to account for the changes that the time evolution introduces in the many-body state represented by the tensor network. Our key insight is that close to convergence, most of the changes in the environment are due to a change in the choice of gauge in the bond indices of the tensor network, and not in the many-body state. Indeed, a consistent choice of gauge in the bond indices confirms that the environment is essentially the same over many time steps and can thus be re-used, leading to very substantial computational savings. We demonstrate the resulting approach in 1D and 2D by computing the ground state of the quantum Ising model in a transverse magnetic field.

17 pages, 28 figures

References in corpus (48)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Area laws for the entanglement entropy - a review
• Efficient classical simulation of slightly entangled quantum computations
• Entropy and Area
• Efficient simulation of one-dimensional quantum many-body systems
• Real time evolution using the density matrix renormalization group
• Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces
• 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
• Time-dependent variational principle for quantum lattices
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• 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
• Competing states in the t-J model: uniform d-wave state versus stripe state
• The iTEBD algorithm beyond unitary evolution
• Corner Transfer Matrix Renormalization Group Method
• From density-matrix renormalization group to matrix product states
• Coarse-graining renormalization by higher-order singular value decomposition
• Simulation of two dimensional quantum systems on an infinite lattice revisited: corner transfer matrix for tensor contraction
• Matrix product operator representations
• Valence Bond Solids for Quantum Computation
• Variational study of hard-core bosons in a 2-D optical lattice using Projected Entangled Pair States (PEPS)
• Second Renormalization of Tensor-Network States
• Stripes in the two-dimensional t-J model with infinite projected entangled-pair states
• Algorithms for finite Projected Entangled Pair States
• Renormalization of tensor-network states
• Applying matrix product operators to model systems with long-range interactions
• Variational quantum Monte Carlo simulations with tensor-network states
• Two-Dimensional Tensor Product Variational Formulation
• Unifying Projected Entangled Pair States contractions
• Exploring corner transfer matrices and corner tensors for the classical simulation of quantum lattice systems
• A Density Matrix Algorithm for 3D Classical Models
• Simulation of fermionic lattice models in two dimensions with Projected Entangled-Pair States: Next-nearest neighbor Hamiltonians
• Higher Order Methods for Simulations on Quantum Computers
• Infinite boundary conditions for matrix product state calculations
• Efficient simulation of infinite tree tensor network states on the Bethe lattice
• Efficient Matrix Product State Method for periodic boundary conditions
• Self-Consistent Tensor Product Variational Approximation for 3D Classical Models
• A canonical form for Projected Entangled Pair States and applications
• Time-reversal symmetry breaking superconducting ground state in the doped Mott insulator on the honeycomb lattice
• Fixed Point of the Finite System DMRG
• Stable Optimization of a Tensor Product Variational State
• Fate of the cluster state on the square lattice in a magnetic field
• Dynamical windows for real-time evolution with matrix product states
• Bounds on universal quantum computation with perturbed 2d cluster states

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• Variational optimization algorithms for uniform matrix product states
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• Lecture Notes of Tensor Network Contractions
• Loop optimization for tensor network renormalization
• Efficient tensor network simulation of IBM's Eagle kicked Ising experiment
• Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions
• Variational Approach to Projected Entangled Pair States at Finite Temperature
• Variational tensor network renormalization in imaginary time: benchmark results in the Hubbard model at finite temperature
• Time Dependent Variational Principle for Tree Tensor Networks
• Time-dependent study of disordered models with infinite projected entangled pair states
• Towards a mathematical formalism for classifying phases of matter
• Simulation of three-dimensional quantum systems with projected entangled-pair states
• Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices
• Evaluation of time-dependent correlators after a local quench in iPEPS: hole motion in the t-J model
• Conversion of projected entangled pair states into a canonical form
• Projected Entangled Pair States at Finite Temperature: Iterative Self-Consistent Bond Renormalization for Exact Imaginary Time Evolution
• Quantum quenches in two spatial dimensions using chain array matrix product states
• Variational tensor network renormalization in imaginary time: Two-dimensional quantum compass model at finite temperature
• Gauging tensor networks with belief propagation
• A beginner's guide to non-abelian iPEPS for correlated fermions
• Explicitly correlated electronic structure calculations with transcorrelated matrix product operators
• Abelian and non-abelian symmetries in infinite projected entangled pair states
• The minimal canonical form of a tensor network
• Real-time chiral dynamics at finite temperature from quantum simulation
• Unsupervised mapping of phase diagrams of 2D systems from infinite projected entangled-pair states via deep anomaly detection
• Efficient tensor network algorithm for layered systems
• Tensor Network enhanced Dynamic Multiproduct Formulas
• 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
• Loop update for infinite projected entangled-pair states in two spatial dimensions
• Accuracy and Efficiency of Simplified Tensor Network Codes
• Simulation of the Ring-exchange Models with Projected Entangled Pair States
• Scalable projected entangled-pair state representation of random quantum circuit states
• Carving-width and contraction trees for tensor networks
• Dispersion Relations in Two- and Three-Dimensional Quantum Systems
• Faster randomized partial trace estimation
• Electric Field Propagation Through Singular Value Decomposition