Efficient Tree Tensor Network States (TTNS) for Quantum Chemistry: Generalizations of the Density Matrix Renormalization Group Algorithm

Abstract

We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an ilustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.

15 pages, 19 figures

References in corpus (25)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Entanglement renormalization
• Classical simulation of quantum many-body systems with a tree tensor network
• Ab Initio Quantum Chemistry using the Density Matrix Renormalization Group
• From density-matrix renormalization group to matrix product states
• Matrix product operator representations
• Orbital Optimization in the Density Matrix Renormalization Group, with applications to polyenes and β-carotene
• A spin-adapted Density Matrix Renormalization Group algorithm for quantum chemistry
• Measuring orbital interaction using quantum information theory
• Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
• Controlling the accuracy of the density matrix renormalization group method: The Dynamical Block State Selection approach
• Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks
• Density matrix renormalization group algorithms with a single center site
• Quantum information analysis of electronic states at different molecular structures
• Multireference Correlation in Long Molecules with the Quadratic Scaling Density Matrix Renormalization Group
• Density-matrix renormalization-group method in momentum space
• Finite automata for caching in matrix product algorithms
• Efficient simulation of infinite tree tensor network states on the Bethe lattice
• Tensor operators: constructions and applications for long-range interaction systems
• A density matrix renormalisation group algorithm for quantum lattice systems with a large number of states per site
• Heisenberg antiferromagnet on Cayley trees: low-energy spectrum and even/odd site imbalance
• Density matrix renormalization group algorithm for Bethe lattices of spin 1/2 or 1 sites with Heisenberg antiferromagnetic exchange

Cited by in corpus (94)

• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Quantum Chemistry in the Age of Quantum Computing
• Heat-bath Configuration Interaction: An efficient selected CI algorithm inspired by heat-bath sampling
• The density matrix renormalization group for ab initio quantum chemistry
• Tensor product methods and entanglement optimization for ab initio quantum chemistry
• Is there evidence for exponential quantum advantage in quantum chemistry?
• Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory
• CheMPS2: a free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry
• Quantum Machine Learning for Chemistry and Physics
• Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement, and Holography
• The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems
• Time-Dependent Density Matrix Renormalization Group Algorithms for Nearly Exact Absorption and Fluorescence Spectra of Molecular Aggregates at Both Zero and Finite Temperature
• Tensor Networks for Lattice Gauge Theories with continuous groups
• Density Matrix Renormalization Group with Efficient Dynamical Electron Correlation Through Range Separation
• Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement
• Time-dependent Density Matrix Renormalization Group Quantum Dynamics for Realistic Chemical Systems
• Fermionic orbital optimisation in tensor network states
• Second-Order Self-Consistent-Field Density-Matrix Renormalization Group
• Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks
• Can Density Matrix Embedding Theory with the Complete Activate Space Self-Consistent Field Solver Describe Single and Double Bond Breaking in Molecular Systems?
• Computing vibrational eigenstates with tree tensor network states (TTNS)
• Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops
• Selected Configuration Interaction in a Basis of Cluster State Tensor Products
• A General Automatic Method for Optimal Construction of Matrix Product Operators Using Bipartite Graph Theory
• T3NS: three-legged tree tensor network states
• Advanced density matrix renormalization group method for nuclear structure calculations
• X-Symbols for Non-Abelian Symmetries in Tensor Networks
• Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer
• Tree tensor network state approach for solving hierarchical equations of motion
• Matrix product states with large sites
• Validating Quantum-Classical Programming Models with Tensor Network Simulations
• Accurate simulation for finite projected entangled pair states in two dimensions
• A tensor network view of multilayer multiconfiguration time-dependent Hartree methods
• Entanglement patterns and generalized correlation functions in quantum many body systems
• Measurement-induced phase transitions on dynamical quantum trees
• Automatic structural optimization of tree tensor networks
• Gauging tensor networks with belief propagation
• Efficient reconstruction of CASCI-type wave functions for a DMRG state using quantum information theory and genetic algorithm
• Hybrid grid/basis set discretizations of the Schrödinger equation
• Studying dynamics in two-dimensional quantum lattices using tree tensor network states
• Block product density matrix embedding theory for strongly correlated spin systems
• Few-body systems capture many-body physics: tensor network approach
• Analysis of Electron Correlation Effects in Strongly Correlated Systems (${\mathrm{N}}_2$ and ${\mathrm{N}}_2^{+}$) by applying DMRG and Quantum Information Theory
• Adaptive-weighted tree tensor networks for disordered quantum many-body systems
• Density-Matrix Renormalization Group Algorithm with Multi-Level Active Space
• Tree tensor networks and entanglement spectra
• Entanglement bipartitioning and tree tensor networks
• Optimal Tree Tensor Network Operators for Tensor Network Simulations: Applications to Open Quantum Systems
• Expressibility of comb tensor network states (CTNS) for the P-cluster and the FeMo-cofactor of nitrogenase
• Cluster many-body expansion: a many-body expansion of the electron correlation energy about a cluster mean-field reference
• Quantum Quench and Charge Oscillations in the SU(3) Hubbard Model: a Test of Time Evolving Block Decimation with general non-Abelian Symmetries
• The Three-Legged Tree Tensor Networks with SU(2)- and molecular point group symmetry
• Probabilistic low-rank factorization accelerates tensor network simulations of critical quantum many-body ground states
• Quantum information-based analysis of electron-deficient bonds
• Efficient Quantum Simulation for Thermodynamics of Infinite-size Many-body Systems in Arbitrary Dimensions
• Tensor network representations from the geometry of entangled states
• Self-Adaptive Tensor Network States with Multi-Site Correlators
• Quantum State Concentration and Classification of Multipartite Entanglement
• Machine learning with tree tensor networks, CP rank constraints, and tensor dropout
• Two dimensional quantum lattice models via mode optimized hybrid CPU-GPU density matrix renormalization group method
• The seniority quantum number in Tensor Network States
• Automatic Contraction of Unstructured Tensor Networks
• Algorithms for Tensor Network Contraction Ordering
• Efficient Density Matrix Renormalization Group algorithm to study Y-Junctions with integer and half-integer spin
• Efficient Decomposition of High-Rank Tensors
• Tensor-based quantum phase difference estimation for large-scale demonstration
• Optimal Symbolic Construction of Matrix Product Operators and Tree Tensor Network Operators
• Tensor Network Computations That Capture Strict Variationality, Volume Law Behavior, and the Efficient Representation of Neural Network States
• On Fermionic Shadow Wave Functions for strongly-correlated multi-reference systems based on a single Slater determinant
• Hilbert space renormalization for the many-electron problem
• The Hubbard model on the Bethe lattice via variational uniform tree states: metal-insulator transition and a Fermi liquid
• State Diagrams to determine Tree Tensor Network Operators
• Tensor Network States with Three-Site Correlators
• Efficient simulation of inhomogeneously correlated systems using block interaction product states
• The resource theory of tensor networks
• QCMaquis 4.0: Multi-Purpose Electronic, Vibrational, and Vibronic Structure and Dynamics Calculations with the Density Matrix Renormalization Group
• Time-reversal symmetry adaptation in relativistic density matrix renormalization group algorithm
• Image Classification by Throwing Quantum Kitchen Sinks at Tensor Networks
• Comparative study on compact quantum circuits of hybrid quantum-classical algorithms for quantum impurity models
• Constructing Local Bases for a Deep Variational Quantum Eigensolver for Molecular Systems
• Automatic Structural Search of Tensor Network States including Entanglement Renormalization
• Survey on Computational Applications of Tensor Network Simulations
• Strictly local tensor networks for short-range topological insulators
• Variational Quantum Imaginary Time Evolution for Matrix Product State Ansatz with Tests on Transcorrelated Hamiltonians
• Coupled-cluster pairing models for radicals with strong correlations
• Chemical master equation parameter exploration using DMRG
• Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm
• Tensor network method for real-space topology in quasicrystal Chern mosaics
• Seeding neural network quantum states with tensor network states
• The Software Landscape for the Density Matrix Renormalization Group
• Simulating Quantum Circuits with Tree Tensor Networks using Density-Matrix Renormalization Group Algorithm
• TTNOpt: Tree tensor network package for high-rank tensor compression
• Fractal Path Strategies for Efficient 2D DMRG Simulations
• Hierarchical Search of Tree Tensor Networks for High-Dimensional Data