Tensor product methods and entanglement optimization for ab initio quantum chemistry

Szilárd Szalay; Max Pfeffer; Valentin Murg; Gergely Barcza; Frank Verstraete; Reinhold Schneider; Örs Legeza

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Tensor product methods and entanglement optimization for ab initio quantum chemistry

Szilárd Szalay, Max Pfeffer, Valentin Murg, Gergely Barcza, Frank Verstraete, Reinhold Schneider, Örs Legeza

arXiv:1412.5829v1Dec 18, 2014physics.chem-phcond-mat.str-elmath-phquant-ph

TL;DR

Accurate, data-sparse wave-function representations remain difficult for strongly correlated systems because high-dimensional quantum problems incur exponential complexity. The paper presents tensor-network and entanglement-based techniques, illustrating their benefits numerically for LiF and achieving energy agreement with the exact value.

Problem
Strongly correlated many-electron systems lack a method-of-choice for sufficiently accurate, data-sparse exact wave-function representations, while high-dimensional Schrödinger problems face exponential complexity.
Method
The paper combines tensor product approximation with entanglement-based tensor-network methods, including QC-DMRG and QC-TTNS, and demonstrates them on LiF.
Results
E = −107.115216925(2) for δεTR ≥10−9, compared with Eexact = −107.1152169273.
Takeaways & Limitations
Entanglement concepts support correlated-orbital identification, active-space construction, and characterization of correlation effects relevant to chemical bonding.
Takeaways & Limitations
QC-TTNS optimization remains more complicated than MPS optimization, with additional topology-related tasks and substantial work still required.

Abstract

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The treatment of high-dimensional problems such as the Schrödinger equation can be approached by concepts of tensor product approximation. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods, -- developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools -- can be combined to attack highly challenging problems in quantum chemistry. The aim of the present paper is to give a pedagogical introduction to the theoretical background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic "New wavefunction methods and entanglement optimizations in quantum chemistry" at the Workshop on Theoretical Chemistry, 18 - 21 February 2014, Mariapfarr, Austria.

1 Introduction

Quantum chemistry needs accurate, data-sparse wavefunction representations for strongly correlated systems, where standard methods lack a method-of-choice solution. The paper introduces tensor-product and entanglement-based methods, emphasizing MPS, TTNS, and controlled entanglement optimization.

Motivation: Strongly correlated systems cannot be sufficiently described by small perturbations of a single Slater determinant, and accurate sparse representations remain unavailable as a standard solution.This is especially relevant to open-shell systems such as transition metal complexes.
Tensor representations: MPS represents a wavefunction over d sites by products of site-associated matrices, while tensor-network representations generalize matrix factorization to higher-order tensors.In quantum chemistry, MPS and TNS can approximate full-CI wavefunctions.
Entanglement-based methods: Entanglement-based methods from distinct communities can be combined for challenging quantum-chemistry problems, including methods based on Tree Tensor Network States.QC-TTNS is presented as a promising direction for problems described as intractable by standard DFT or CC techniques.
High-dimensional problems: Tensor-product approximation addresses high-dimensional Schrödinger-type equations whose computational complexity otherwise scales exponentially with dimension.Related concepts also support optimization and time-dependent problems.
Existing methods: DMRG and its MPS formulation enabled accurate treatment of large active spaces, reaching the full-CI limit in some quantum-chemical applications.Subsequent developments include orbital optimization, dynamic-correlation treatments, and relativistic 2c- and 4c-DMRG.
Optimization challenges: Method performance depends on truncation criteria, orbital ordering, basis optimization, and network initialization, while the minimum effort for a target accuracy remains open.The matrix sizes can be controlled to achieve an a priori error margin, but quantum-chemical ranks depend strongly on ordering.

2 Quantum chemistry

The quantum-chemistry formulation describes antisymmetric N-electron wavefunctions, their Hamiltonian eigenproblem, and finite-dimensional approximations using Slater determinants. The Ritz-Galerkin approximation provides an upper-bound ground-state energy, but full CI scales exponentially with electron number.

Electronic Schrödinger equation: An N-electron quantum system is described by a wavefunction depending on 3N spatial variables and N discrete spin variables.The wavefunction is treated within a non-relativistic Born-Oppenheimer framework.
Fermionic structure: The electronic wavefunction is antisymmetric under exchange of electron variables, implying the Pauli exclusion principle.The wavefunction vanishes when two fermions have identical spatial and spin coordinates.
Slater determinants and FCI: A finite orthonormal spin-orbital basis yields Slater determinants that span the full configuration-interaction space.Each determinant selects N distinct spin orbitals from a finite set of d orbitals.
Ritz-Galerkin approximation: The Ritz-Galerkin method projects the electronic eigenvalue problem onto a finite-dimensional subspace and approximates the ground state there.The construction uses an L2-orthogonal projection and a finite-dimensional eigenvalue problem.
Ritz-Galerkin approximation: E0 ≤ E0,d: the Ritz-Galerkin approximate ground-state energy is an upper bound on the exact ground-state energy.The Galerkin eigenfunction is quasi-optimal, while the eigenvalue converges quadratically relative to the eigenfunction.
Computational scaling: Full CI is practically infeasible for molecules with more than a very small number of electrons because its dimension scales exponentially with N.The supplied passage gives dim Vd_N ∼ O(d^N) ≥ O(2^N).
Basis transformation: Changing the one-particle basis can turn a single Slater determinant into a linear combination of many determinants and increase tensor rank.Each factor has rank at most two, so repeated matrix multiplication can at worst double the rank at each step.

3 Tensor product approximation

Tensor product approximation addresses exponential storage and computational costs by representing high-order tensors in data-sparse forms, while different decompositions provide distinct rank concepts and computational properties.

Tensor storage grows exponentially with tensor order, creating the curse of dimensionality.
Data-sparse tensor representations generalize matrix low-rank approximation by expressing multivariate functions as sums of products of univariate functions.
Higher-order tensors admit multiple decompositions, each inducing a different definition of tensor rank.
The canonical decomposition has the smallest exact representation rank, but finding that rank and decomposition is NP-hard.
Canonical representations can exhibit non-closed rank sets, border-rank behavior, slow convergence, and low accuracy during optimization.
General tensor representations use component tensors connected through physical and virtual indices, with contractions over shared virtual indices.

3.2 Tensor networks

Tensor networks encode multilinear tensor representations as graphs, while restricting virtual-index connectivity and graph topology yields tensor trees with more controlled structure.

Each physical index belongs to one component, and each virtual index connects exactly two component tensors.
A tensor network represents component tensors as graph vertices, contractions as edges, and physical indices as half-edges.
Network edge and half-edge weights are given by virtual and physical index dimensions, while the minimal virtual dimensions define network rank.
Connected tensor networks with cycle-free graphs are tensor trees, also called Tree Tensor Network States.
Closed-loop tensor networks can lack algebraic closure, so restricting the network to cycle-free graphs avoids these difficulties.

3.3 Subspace optimization and the Tucker format

Subspace optimization compresses tensors by selecting lower-dimensional mode subspaces and a reduced core; Tucker decomposition provides a constructive polynomial-time route, but its storage can remain exponential.

Tucker decomposition represents a tensor using optimal basis sets for each mode and a reduced core tensor.
The Higher Order SVD computes the Tucker decomposition constructively by applying an SVD to every tensor mode.
For d > 2, finding the best rank-one approximation can be NP-hard, while HOSVD truncation provides only a quasi-optimal l2-norm approximation.
The Tucker format is a tensor tree with one virtual core component and d physical components.
The HOSVD computes a minimal-rank Tucker decomposition in polynomial time and avoids the border rank problem because bounded Tucker-rank sets are closed.
Tucker storage scales as O(dqr + rd) for r := max{ri}, which remains exponential in r and gives no nontrivial reduction when qi = 2.

3.4 Matricization and tensor multiplication

Matricization reorganizes tensor indices into row and column groups, enabling matrix-like contractions and products while preserving the distinctions of complex tensor notation.

General matricization partitions physical indices into a selected set of row dimensions and its complement as column dimensions.
The i-th matricization places the first i variables in the row index and the remaining d − i variables in the column index.
Tensor multiplication is defined through matrix multiplication of corresponding matricizations and contraction over shared indices.
When no contraction is performed, the same notation yields the tensor product.
For complex tensors, matricization only reorders and groups indices; changing index order produces a transpose rather than a Hermitian matrix.
Multiplication over all common indices is denoted by a circle and interpreted as composition of two linear operators.

3.5 Matrix product states or the tensor train format

The Tensor Train format represents high-order tensors through a chain of low-order component tensors, with ranks controlling complexity and singular-value-based procedures supporting orthogonalization and approximation.

Tensor Train representation: Tensor Train decomposition represents a tensor using d component tensors of order 2 or 3 arranged in a chain.This chain structure motivates the name Tensor Train and corresponds to Matrix Product States with open boundary conditions.
Tensor Train representation: The TT format has complexity O(qdr^2), where r is the maximum TT rank.This quadratic rank scaling overcomes most disadvantages of the canonical format while retaining favorable tensor-approximation properties.
Hierarchical structure: TT tensors use hierarchical subspaces whose minimal dimensions are the ranks of the corresponding matricizations.The separation theorem establishes minimal TT representations through these ranks.
Orthogonalization: Orthogonality can be shifted between components by applying SVD or moving the diagonal singular-value matrices between adjacent tensors.This produces the standard, HSVD, or Vidal representation and allows the root to be changed.
Approximation: HSVD recovers tensors exactly, while thresholding its singular values provides an approximation with quasi-optimality rather than guaranteed best approximation.The truncated representation is estimated relative to tensors with bounded TT rank.

3.6 Dimension trees and the hierarchical tensor decomposition

Hierarchical Tucker decompositions organize tensor components on dimension trees, generalizing Tensor Trains and enabling complexity bounds for tree-structured representations.

Scope: The paper restricts its detailed treatment to Tensor Trains because general tree notation becomes more complex despite shared theoretical results.The HT format is discussed as a closely related generalization.
Dimension trees: The HT format is defined by a usually binary dimension tree with physical leaf components and virtual inner vertices.Each inner vertex represents a subspace built from its sons, and the root recursively reconstructs the tensor.
Complexity: For a tree with at most O(d) vertices, HT complexity is O(qdr + dr^(p+1)); for binary trees it becomes O(qdr + dr^3).Here p denotes the number of sons of an inner vertex.
Construction and approximation: HT representations can be constructed by successive hierarchical SVDs and retain separation theorems and quasi-optimality for truncated approximations.These properties parallel those established for Tucker and Tensor Train formats.
Relation to Tensor Train: Tensor Train is a special HT case using an unbalanced tree and omitting optimal subspaces at the leaves.Binary tree structures can nevertheless be advantageous in some cases.

3.7 Fixed rank manifolds and varieties

Fixed-rank tensor sets provide low-dimensional structures for computation, but their parametrizations are redundant and their closures include singular lower-rank tensors.

Parametrization: The component parametrization is nonunique because invertible matrices can be inserted between adjacent component tensors without changing the represented tensor.This redundancy motivates quotient-space and gauge-based formulations.
Quotient manifold: The group action partitions component representations into equivalence classes, yielding a smooth quotient manifold M_r.The quotient removes the representation redundancy while preserving the represented tensors.
Tangent spaces: The fixed-rank tensor manifold can be embedded in the ambient tensor space, allowing tangent-space constructions for optimization.Horizontal and vertical spaces separate physical tangent directions from directions along representation orbits.
Varieties and singularities: The bounded-rank set is algebraically closed, and its singular points are exactly tensors whose actual rank is not maximal.Thus the closure contains lower-rank boundary points and is an algebraic variety.

3.8 Dirac-Frenkel variational principle or dynamical low rank approximation

The Dirac-Frenkel and dynamical low-rank framework projects optimization gradients and time-dependent dynamics onto the tangent space of a fixed-rank tensor manifold.

Variational principle: For fixed-rank approximation, minimizing an energy functional on M_r requires the ambient gradient to be orthogonal to the manifold’s tangent space.Equivalently, the projected gradient vanishes at a constrained minimizer.
Dynamical low rank: Time-dependent problems are approximated by projecting the governing vector field onto the tangent space T_U M_r.The resulting projected differential equation evolves within the fixed-rank manifold.
Gradient flow: Replacing f(U) with −∇J(U) produces the gradient flow associated with the objective J.The same tangent-space projection machinery applies to this optimization flow.
Gradient flow: The gradient flow on M_r can be solved with the methods described in the paper.The section links the projected dynamics to an illustration of the flow on the tensor manifold.

3.9 The alternating least squares algorithm

The ALS algorithm alternates optimization over small tensor-component subproblems represented in reduced bases. It is closely related to one-site DMRG, while two-site DMRG enables dynamic rank increases through SVD and truncation.

Alternating least squares: ALS fixes all tensor components except one and solves the resulting small subproblem successively in alternating directions.For least-squares fitting, this gives the Alternating Least Squares interpretation.
Reduced-basis formulation: The TT format provides closed-form ALS subproblems solvable with standard linear-algebra and numerical-optimization tools.The reduced-basis formulation turns large linear systems or eigenvalue problems into smaller problems of the same type.
Renormalization and conditioning: The small subproblems support a renormalization picture by reducing a large system to a smaller one while retaining its ground-state energy and related quantities.The reduced-space conditioning is bounded by the conditioning of the full operator when the relevant invertibility condition holds.
ALS and one-site DMRG: ALS optimizes over very small subspaces, but its ranks remain fixed and must be guessed initially.Higher ranks require a greedy procedure such as adding rank-one approximations.
Two-site DMRG: Two-site DMRG optimizes two neighboring components simultaneously in a larger reduced subspace.A subsequent SVD separates the components and can produce a new, possibly higher, rank; truncation controls rank growth.
Limitations: Neither ALS nor DMRG has guaranteed global convergence; general convergence theory remains an open research topic.The methods converge only to stationary points or, at most, local minima, although modified schemes have published convergence results.

4 Numerical techniques

This section reviews optimization and entanglement techniques for numerical quantum-chemistry calculations, using LiF as a pedagogical test case. The numerical focus leaves molecular entropic measures only briefly discussed.

Numerical techniques: The section surveys iterative blocking-based optimization methods and introduces entanglement localization, tensor-network topology optimization, and related choices.Its emphasis is numerical methods rather than a comprehensive treatment of molecular entropic properties.
LiF example: LiF is used as a textbook example because an ionic-neutral curve crossing between its two lowest 1Σ+ states tests QC-DMRG and tree tensor-network methods.Near equilibrium, the two states are qualitatively described by distinct configurations.
LiF example: The molecular-orbital basis is obtained through CASSCF optimizations with two active electrons in two active orbitals, optimized simultaneously for both 1Σ+ states.The resulting integrals are expressed in this molecular-orbital basis, with lower orbitals frozen in the presented configuration-interaction calculations.

4.1 Basic terms and brief overview

The paper represents molecular-orbital wavefunctions as tensor products and reduces their exponential dimensionality through truncated bases, symmetry sectors, and tensor-network approximations. These constructions establish the basic local spaces and many-body representations used later.

Local tensor spaces: A spin-orbital is treated as a local tensor space Λ ≅ C^q; the fermionic occupation basis has q = 2, while a molecular-orbital representation has q = 4.The q = 4 basis distinguishes empty, singly occupied spin-down or spin-up, and doubly occupied states.
Local tensor spaces: Combining two q = 4 orbital spaces produces a two-orbital tensor space of dimension q^2 = 16.The composite basis is indexed from the two local basis indices.
Basis transformation and truncation: A unitary basis transformation can diagonalize the two-orbital Hamiltonian without changing its eigenvalue spectrum.Selecting only M < q^2 eigenstates yields a smaller rectangular transformation and truncates the original tensor space, losing information while potentially retaining a good description.
Eigenvalue problems: Power, Lanczos, and Davidson methods target low-lying eigenstates without requiring full Hamiltonian diagonalization.Lanczos- and Davidson-type extensions provide faster convergence and can calculate excited states.
Symmetries: Symmetry operators decompose the Hilbert space into quantum-number sectors, replacing one large eigenproblem with several smaller problems.For U(1) symmetries, composite orbital quantum numbers add; non-Abelian symmetries require more complex algebra.
Many-orbital wavefunctions: For d orbitals, the wavefunction tensor dimension grows exponentially with d, motivating approximations by products of lower-order tensors with smaller ranks.Successive basis transformations and truncations extend the two-orbital construction to many orbitals.

4.2 Entanglement and correlations

Entanglement is characterized by reduced density matrices and their spectra for bipartitions of molecular orbitals. Entropy and mutual-information measures then quantify how correlations are distributed across orbitals and blocks.

Entropy measures: The von Neumann entropy distinguishes the example entangled and separable states, with S(A) = ln 2 and S(A) = 0, respectively.The entangled example is maximally entangled within the two-electron subspace of a two-orbital system.
Reduced density matrices: For a pure state partitioned into blocks A and B, the reduced density matrix ρ(A) is obtained by tracing out subsystem B.Its eigenvalues equal the squared Schmidt coefficients and completely characterize bipartite entanglement.
Orbital correlations: The choice of orbital subset determines the entanglement information extracted, including single-orbital and two-orbital entropies and mutual information.These quantities characterize the distribution and types of correlations across orbitals.
Block entropy: Block entropy uses contiguous orbital blocks in DMRG and diverges logarithmically with block size for critical one-dimensional systems but saturates for gapped systems.The cited profile concerns a critical model with soft modes at k = ±2π/3.
Orbital entropies: Single-orbital entropy measures local orbital mixedness, while summing all single-orbital entropies gives the total correlation encoded in a pure-state wavefunction.This sum can monitor entanglement changes as molecular parameters such as bond length vary.

4.3 Methods based on block transfromation procedures

Block-transformation methods iteratively reduce high-dimensional orbital Hilbert spaces, with DMRG refining truncated block bases through sweeps and singular-value decompositions. For QC-DMRG and QC-TTNS, computational effort scales with orbital count, retained block states, and tree coordination.

Block Renormalization Group method: Block Renormalization Group groups orbitals into blocks and repeatedly transforms the Hamiltonian to reduce the representation.The procedure truncates states during successive block transformations.
Block Renormalization Group method: BRG truncation accumulates information loss and is inefficient for quantum chemistry because of dramatic truncation and non-local interactions.It nevertheless provides a basis for hierarchical tensor representations and tree tensor network states.
Numerical Renormalization Group method: NRG adds orbitals along a Wilson chain with exponentially decreasing hopping, retaining q < M ≪ q^d low-energy states at each iteration.The retained states define transformed block Hamiltonians on progressively different energy scales.
Density Matrix Renormalization Group method: DMRG enlarges left and right blocks by adjacent orbitals, transforms them using singular-value decomposition, and improves the system block through forward and backward sweeps.The finite-lattice procedure reaches length d and alternates optimization between system and environment blocks.
Density Matrix Renormalization Group method: DEAS initializes quantum-chemical DMRG with l = 1 and r = d −3, then applies finite-system sweeps using an approximated environment Hilbert space.The right-block approximation is needed because its exact dimension grows exponentially with the number of orbitals.
Efficient factorization of the interaction terms: The numerical effort of QC-DMRG and QC-TTNS scales as d3M^(z+1), where M is the block-state count and z is tree coordination.The scaling combines tensor-contraction cost, Hamiltonian-term summation, and O(d) iteration steps.

4.4 Optimization of convergence properties

The paper presents entanglement-based optimization procedures for QC-DMRG and QC-TTNS, targeting orbital ordering, topology, basis choice, initialization, convergence, and computational efficiency. Numerical LiF examples show that reducing or localizing entanglement can preserve accuracy while lowering block-state and iteration requirements.

Optimization targets: Entanglement-based optimization treats ordering, topology, orbital basis, initialization, and convergence control as determinants of tensor-network efficiency.The overall entanglement depends on network topology and can be manipulated through distances, topology, mutual information, or orbital basis.
Convergence control: DBSS dynamically adjusts block states according to entanglement, while an entropy sum rule provides an alternative convergence test.The approach is designed to reach a predefined accuracy by adapting the number of retained block states.
Ordering: Optimized one-dimensional orbital ordering reduces overall entanglement Ioverall from 126.47 to 19.63 while preserving total quantum information Itot = 1.32.The reduced block-entropy height and spread allow the same accuracy with fewer block states.
Topology: Optimized tree topology reaches Ioverall = 5.53 and can attain the MPS numerical accuracy with fewer block states and fewer iteration steps.Tree construction groups orbitals with large mutual-information bonds and places high-entropy orbitals near the network center.
Computational trade-offs: QC-TTNS efficiency trades lower tensor ranks against higher tensor orders, with an expected crossover in CPU time relative to QC-MPS as systems grow.Topology optimization remains less established, and further developments are described as necessary to fully exploit TTNS methods.

4.5 Miscellaneous

The paper illustrates tensor-network methods through simulations of polydiacetylene materials, relativistic quantum chemistry, and hybrid CPU-GPU acceleration. These applications address strongly correlated excitations, relativistic effects, and computational performance.

Simulation of real materials, geometrical optimization and excited states: MPS-based methods efficiently simulate strongly anisotropic polydiacetylene chains using effective Hamiltonians.Polydiacetylene chains provide quasi one-dimensional systems for modeling interacting electrons on ordered chains.
Simulation of real materials, geometrical optimization and excited states: 20% exciton binding energy relative to the single-particle gap makes Coulomb interactions substantial in polydiacetylene materials.Earlier LDA calculations of bare band structures failed to reproduce the experimentally measured excitation spectrum.
Simulation of real materials, geometrical optimization and excited states: Correlating some 100 electrons on 100 orbitals with DMRG and Hellmann–Feynman geometrical optimization produces a very accurate energy spectrum.The calculated bond lengths are rt = 1.22, rd = 1.37, and rs = 1.43.
Relativistic quantum chemistry: Relativistic 2c- and 4c-DMRG provides variational descriptions of scalar-relativistic effects and spin–orbit coupling.The method was applied by correlating 14 electrons on 94 spinors with the Dirac–Coulomb Hamiltonian.
Possible technical developments: hybrid CPU/GPU parallelization: Hybrid CPU-GPU acceleration of DMRG tolerates problems exceeding GPU memory and supports a wide range of problem sizes and GPU configurations.On the one-dimensional Hubbard model, reported speedups reached approximately 2.3-2.4 times for a mid-range configuration and 3.4-3.5 times for a high-end configuration.
Possible technical developments: hybrid CPU/GPU parallelization: The projection operation is well suited to GPUs because it can be formulated as independent dense matrix multiplications.The strong projection speedup motivates accelerating additional parts of the algorithm for better overall performance.

5 Summary and outlook

Entanglement concepts have advanced electronic-structure calculations by improving active-space construction and motivating tensor-network formulations beyond DMRG/MPS. TTNS offers favorable orbital-distance scaling for multireference problems, but its theory and optimization remain incomplete and more complicated.

Entanglement concepts enable identification of highly correlated molecular orbitals, supporting efficient active-space construction and characterization of chemical-bonding correlation effects.
Reformulating DMRG as Matrix Product States places it within the broader Tensor Network States framework, which may outperform DMRG/MPS.
TTNS analysis remains partially incomplete, while QC-TTNS benefits are not fully exploited and its optimization is more complicated than MPS optimization.
QC-TTNS represents wave functions with variable tensor order in a multiparticle basis spanning a truncated CAS-CI Hilbert space.
TTNS orbital distances scale logarithmically with orbital number rather than linearly as in MPS, making the ansatz better suited to multireference problems with many correlated orbitals.