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Physical review. B, Condensed matter

Optimizing the density-matrix renormalization group method using quantum information entropy

Örs Legeza, Jenő Sólyom

Published Nov 21, 2003• 319 citations
Publisher

Abstract

In order to optimize the ordering of the lattice sites in the momentum space and quantum chemistry versions of the density-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various molecules. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chemistry DMRG to a great extent. The effect of lattice site ordering on the number of block states to be kept during the RG procedure is also investigated.

DOI
10.1103/physrevb.68.195116

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In order to optimize the ordering of the lattice sites in the momentum space and quantum chemistry versions of the density-matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the one-dimensional Hubbard model and various molecules. By analyzing the behavior of von Neumann entropy we have found criteria that help to fasten convergence. An initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space in a dynamical fashion. The dynamically extended active space procedure reduces significantly the effective system size during the first half-sweep and accelerates the speed of convergence of momentum space DMRG and quantum chemistry DMRG to a great extent. The effect of lattice site ordering on the number of block states to be kept during the RG procedure is also investigated.
In order to optimize the ordering of the lattice sites in the momentum space and quantum chemistry versions of the density matrix renormalization group (DMRG) method we have studied the separability and entanglement of the target state for the 1-D Hubbard model and various molecules. By analyzing the behavior of von Neumann and Neumann-Renyi entropies we have found criteria that help to fasten convergence. A new initialization procedure has been developed which maximizes the Kullback-Leibler entropy and extends the active space (AS) in a dynamical fashion. The dynamically extended active space (DEAS) procedure reduces significantly the effective system size during the first half sweep and accelerates the speed of convergence of momentum space DMRG and quantum chemistry DMRG to a great extent. The effect of lattice site ordering on the number of block states to be kept during the RG procedure is also investigated.
We have studied quantum data compression for finite quantum systems where the site density matrices are not independent, i.e., the density matrix cannot be given as direct product of site density matrices and the von Neumann entropy is not equal to the sum of site entropies. Using the density-matrix renormalization-group (DMRG) method for the one-dimensional Hubbard model, we have shown that a simple relationship exists between the entropy of the left or right block and dimension of the Hilbert space of that block as well as of the superblock for any fixed accuracy. The information loss during the RG procedure has been investigated and a more rigorous control of the relative error has been proposed based on Kholevo's theory. Our results are also supported by the quantum chemistry version of DMRG applied to various molecules with system lengths up to 60 lattice sites. A sum rule that relates site entropies and the total information generated by the renormalization procedure has also been given, which serves as an alternative test of convergence of the DMRG method.
Despite the success of modern quantum chemistry in predicting properties of organic molecules, the treatment of inorganic systems, which have many close lying states, remains out of quantitative reach for current methods. To treat non-dynamic correlation, we take advantage of the density matrix renormalization group (DMRG) method that has become very successful in the field of solid state physics. We present a detailed study of the DMRG method, and we pay special attention to the evolution of the understanding behind the mathematical structure of the DMRG wave function. Our primary goal is to develop a density matrix renormalization group self--consistent--field (DMRG-SCF) approach, analogous to the complete active space self--consistent field (CASSCF) method, but dealing with large active spaces that are too demanding for the full configuration interaction (FCI) method. \nAs a first step towards such a DMRG-SCF procedure, we present a spin-adapted DMRG algorithm designed to target spin- and spatial-symmetry states that are hard to obtain while using an unrestricted algorithm. \nOur next step is a modification of the DMRG algorithm to obtain decreasing energy at every step during the sweep. This monotonically convergent DMRG scheme lets us obtain the two-body density matrix as a by--product of the existing procedure without any additional cost in storage. Additionally, the two-body density matrix produced at convergence using this scheme is free from the N-representability problem that is present when the two--body density matrix is produced with the two-site DMRG scheme without additional storage cost. Finally, taking advantage of the modifications developed herein, we present results obtained from our DMRG-SCF method. Lastly, we discuss possible ways of incorporating dynamical correlation into the DMRG scheme, in order to obtain a modern multireference approach.
The density-matrix renormalization group algorithm has emerged as a promising new method in ab initio quantum chemistry. However, many problems still need to be solved before this method can be applied routinely. At the start of such a calculation, the orbitals originating from a preceding quantum chemical calculation must be placed in a specific order on a one-dimensional lattice. This ordering affects the convergence of the density-matrix renormalization group iterations significantly. In this paper, we present two approaches to obtain optimized orderings of the orbitals. First, we use a genetic algorithm to optimize the ordering with respect to a low total electronic energy obtained at a predefined stage of the density-matrix renormalization group algorithm with a given number of total states kept. In addition to that, we derive orderings from the one- and two-electron integrals of our test system. This test molecule is the chromium dimer, which is known to possess a complicated electronic structure. For this molecule, we have carried out calculations for the various orbital orderings obtained. The convergence behavior of the density-matrix renormalization group iterations is discussed in detail.
Density matrix renormalization group is a powerful numerical approach originating in solid state physics. Since its introduction to quantum chemis- try it has been successfully applied to many challenging problems and proved especially efficient for systems with strong multi-reference character. As the method is not invariant to the change of orbital ordering, we implemented the automatic orbital ordering optimization procedure built around the entan- glement analysis and employing the Fiedler method. A technique for active space selection based on one-orbital entropy is introduced. We followed the study by Berardo et al. in which EOM-CC methods were applied to excited states of TiO2 clusters. Four lowest-lying singlet excited states of monomer, dimer and trimer were calculated by means of the DMRG method. In case of the monomer there was good agreement for two states, but the calculations suffered from insufficient treatment of dynamic correlation. For the dimer our results were in very good agreement with EOM-CCSDT results and for the trimer further study would be appropriate. 1
We have applied the momentum space version of the density-matrix renormalization-group method (k-DMRG) in quantum chemistry in order to study the accuracy of the algorithm in this new context. We have shown numerically that it is possible to determine the desired accuracy of the method in advance of the calculations by dynamically controlling the truncation error and the number of block states using a novel protocol that we dubbed dynamical block state selection protocol. The relationship between the real error and truncation error has been studied as a function of the number of orbitals and the fraction of filled orbitals. We have calculated the ground state of the molecules CH2,{\mathrm{CH}}_{2},CH2​, H2O,{\mathrm{H}}_{2}\mathrm{O},H2​O, and F2{\mathrm{F}}_{2}F2​ as well as the first excited state of CH2.{\mathrm{CH}}_{2}.CH2​. Our largest calculations were carried out with 57 orbitals, the largest number of block states was 1500--2000, and the largest dimensions of the Hilbert space of the superblock configuration was 800 000--1 200 000.