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Journal of Chemical Theory and Computation

The Embedded Density Matrix Renormalization Group: Size-Extensive and Quasi-Exact for Nonlinear Quantum Chemistry

Shaun Weatherly, Troy Van Voorhis

Published Aug 7, 2025• 3 citations
Publisher

Abstract

Tensor networks (TNs) and the breadth of algorithms acting on them have seen astounding success in simulating quantum many-body systems in the strongly interacting regime with both accuracy and efficiency. In the context of quantum chemistry, Steven White’s density matrix renormalization group (DMRG) continues to take center stage as the TN method of choice, seeing countless theoretical and computational breakthroughs in recent decades yet remaining fettered by a few persistent shortcomings, notably, a lack of size-extensivity and quasi-exactness . Here, we present a simple yet versatile framework for circumventing these issues: the bootstrap embedded density matrix renormalization group (BE-DMRG), and numerically validate its size-extensive and quasi-exact ground-state properties for a test bed of strongly correlated molecular systems (linear H - chains; quasi-linear E - polyacetylene; 2D and 3D H -lattices; 2D arene flakes ). Spanning a breadth of system sizes (10 to 200 orbitals) and entanglement topologies (linear to highly nonlinear), we demonstrate the robustness of the BE-DMRG for problems far beyond the reach of conventional DMRG implementations. Furthermore, by detailing BE-DMRG convergence behavior with respect to exact diagonalization, we find the rate of convergence with bond dimension to be significantly faster than, yet just as reliable as, that of conventional DMRG. Ultimately, we find that the embedded DMRG might serve as a natural extension of White’s original formulation to higher dimensions without the need for higher-order tensor networks. The coupling of tensor network theories to the framework of quantum embedding, more broadly, may become an incomparably powerful tool for the study of strongly correlated molecules and materials.

DOI
10.1021/acs.jctc.5c00725

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Tensor networks (TNs) and the breadth of algorithms acting on them have seen astounding success in simulating quantum many-body systems in the strongly interacting regime with both accuracy and efficiency. In the context of quantum chemistry, Steven White’s density matrix renormalization group (DMRG) continues to take center stage as the TN method of choice, seeing countless theoretical and computational breakthroughs in recent decades yet remaining fettered by a few persistent shortcomings, notably, a lack of size-extensivity and quasi-exactness . Here, we present a simple yet versatile framework for circumventing these issues: the bootstrap embedded density matrix renormalization group (BE-DMRG), and numerically validate its size-extensive and quasi-exact ground-state properties for a test bed of strongly correlated molecular systems (linear H - chains; quasi-linear E - polyacetylene; 2D and 3D H -lattices; 2D arene flakes ). Spanning a breadth of system sizes (10 to 200 orbitals) and entanglement topologies (linear to highly nonlinear), we demonstrate the robustness of the BE-DMRG for problems far beyond the reach of conventional DMRG implementations. Furthermore, by detailing BE-DMRG convergence behavior with respect to exact diagonalization, we find the rate of convergence with bond dimension to be significantly faster than, yet just as reliable as, that of conventional DMRG. Ultimately, we find that the embedded DMRG might serve as a natural extension of White’s original formulation to higher dimensions without the need for higher-order tensor networks. The coupling of tensor network theories to the framework of quantum embedding, more broadly, may become an incomparably powerful tool for the study of strongly correlated molecules and materials.
Abstract The physical properties of a quantum many-body system can, in principle, be determined by diagonalizing the respective Hamiltonian, but the dimensions of its matrix representation scale exponentially with the number of degrees of freedom. Hence, only small systems that are described through simple models can be tackled via exact diagonalization. To overcome this limitation, numerical methods based on the renormalization group paradigm that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space have been put forth. A striking example is the density-matrix renormalization group (DMRG), which has become the reference numerical method to obtain the low-energy properties of one-dimensional quantum systems with short-range interactions. Here, we provide a pedagogical introduction to DMRG, presenting both its original formulation and its modern tensor-network-based version. This colloquium sets itself apart from previous contributions in two ways. First, didactic code implementations are provided to bridge the gap between conceptual and practical understanding. Second, a concise and self-contained introduction to the tensor-network methods employed in the modern version of DMRG is given, thus allowing the reader to effortlessly cross the deep chasm between the two formulations of DMRG without having to explore the broad literature on tensor networks. We expect this pedagogical review to find wide readership among students and researchers who are taking their first steps in numerical simulations via DMRG. Graphic abstract
The density-matrix renormalization group (DMRG) method has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. The DMRG is a method that shares features of a renormalization group procedure (which here generates a flow in the space of reduced density operators) and of a variational method that operates on a highly interesting class of quantum states, so-called matrix product states (MPSs). The DMRG method is presented here entirely in the MPS language. While the DMRG generally fails in larger two-dimensional systems, the MPS picture suggests a straightforward generalization to higher dimensions in the framework of tensor network states. The resulting algorithms, however, suffer from difficulties absent in one dimension, apart from a much more unfavourable efficiency, such that their ultimate success remains far from clear at the moment.
The density-matrix renormalization group (DMRG) method has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. The DMRG is a method that shares features of a renormalization group procedure (which here generates a flow in the space of reduced density operators) and of a variational method that operates on a highly interesting class of quantum states, so-called matrix product states (MPSs). The DMRG method is presented here entirely in the MPS language. While the DMRG generally fails in larger two-dimensional systems, the MPS picture suggests a straightforward generalization to higher dimensions in the framework of tensor network states. The resulting algorithms, however, suffer from difficulties absent in one dimension, apart from a much more unfavourable efficiency, such that their ultimate success remains far from clear at the moment.
The recently proposed Clifford augmented density matrix renormalization group (CA-DMRG) method seamlessly integrates Clifford circuits with matrix product states and takes advantage of the expression power from both. CA-DMRG has been shown to be able to achieve higher accuracy than standard DMRG on commonly used lattice models, with only moderate computational overhead compared to the latter. In this work, we propose an efficient scheme in CA-DMRG to deal with ab initio quantum chemistry Hamiltonians and apply it to study several molecular systems. Our numerical results show that CA-DMRG can reach higher accuracy than DMRG using the same bond dimension, pointing out a promising route to push the boundary of solving ab initio quantum chemistry with strong static correlations.
During the past 15 years, the density matrix renormalization group (DMRG) has become increasingly important for ab initio quantum chemistry. The underlying matrix product state (MPS) ansatz is a low-rank decomposition of the full configuration interaction tensor. The virtual dimension of the MPS controls the size of the corner of the many-body Hilbert space that can be reached. Whereas the MPS ansatz will only yield an efficient description for noncritical one-dimensional systems, it can still be used as a variational ansatz for other finite-size systems. Rather large virtual dimensions are then required. The two most important aspects to reduce the corresponding computational cost are a proper choice and ordering of the active space orbitals, and the exploitation of the symmetry group of the Hamiltonian. By taking care of both aspects, DMRG becomes an efficient replacement for exact diagonalization in quantum chemistry. DMRG and Hartree-Fock theory have an analogous structure. The former can be interpreted as a self-consistent mean-field theory in the DMRG lattice sites, and the latter in the particles. It is possible to build upon this analogy to introduce post-DMRG methods. Based on an approximate MPS, these methods provide improved ansätze for the ground state, as well as for excitations. Exponentiation of the single-particle (single-site) excitations for a Slater determinant (an MPS with open boundary conditions) leads to the Thouless theorem for Hartree-Fock theory (DMRG), an explicit nonredundant parameterization of the entire manifold of Slater determinants (MPS wavefunctions). This gives rise to the configuration interaction expansion for DMRG. The Hubbard-Stratonovich transformation lies at the basis of auxiliary field quantum Monte Carlo for Slater determinants. An analogous transformation for spin-lattice Hamiltonians allows to formulate a promising variant for MPSs.
The simulation of strongly correlated quantum systems on classical computers is an essential tool for investigating and understanding quantum materials.Due to the inherent "curse of dimensionality" of quantum systems, such simulations pose a challenging task, and have spurred the development of several computational approaches and approximations.For quasi one-dimensional lattice systems, the density matrix renormalization group (DMRG) framework (Schollwöck, 2005;White, 1992White, , 1993) has emerged as one of the most successful methods.Modern formulations are based on matrix product states (MPS) and operators (MPO) (Schollwöck, 2011): briefly speaking, a matrix product state is an Ansatz for the quantum many-body wavefunction amendable to efficient numerical manipulation.The favorable approximation properties of matrix product states have been theoretically proven for ground states of gapped Hamiltonians with local interactions (Hastings, 2007;Verstraete & Cirac, 2006), and recent work has extended this to quantum systems at finite temperature (Barthel, 2017).Over the years many variations and generalizations of the DMRG framework have been devised (Barthel, 2013;Stoudenmire & White, 2010;Verstraete, Murg, & Cirac, 2008;Vidal, 2003Vidal, , 2008)), and further developments and improvements of associated algorithms remain an active field of research.