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arXiv:2506.12629 · Computer Physics Communications

The software landscape for the density matrix renormalization group

Per Sehlstedt, Jan Brandejs, Paolo Bientinesi, Lars Karlsson

Published Mar 20, 2026• 1 citation
arXivHTMLPDF

Abstract

The density matrix renormalization group (DMRG) algorithm is a cornerstone computational method for studying quantum many-body systems, renowned for its accuracy and adaptability. Because DMRG provides a general framework applicable across various fields such as materials science, quantum chemistry, and quantum computing, one might expect a shared, flexible library to serve most users. Nevertheless, numerous independent implementations continue to appear, resulting in significant duplication of effort. To identify collaboration opportunities that can promote a more unified approach, we map the rapidly expanding DMRG software landscape and provide a comprehensive comparison of features across 37 existing packages. When comparing key features, such as parallelism strategies for high-performance computing and symmetry-adapted formulations that enhance efficiency, we found significant overlap among the packages. This overlap suggests opportunities for collaboration to modularize common functionality—e.g., tensor operations, symmetry representations, and eigensolvers—as the packages are mostly independent and share few third-party library dependencies. More collaboration on modularization could reduce duplication of effort, improve interoperability, and enable prioritization and quicker spread of new advances. We believe the current lack of modularity is more socially driven than a technical issue; hence, we see raising awareness about the existing implementations as a first step in the right direction. Ultimately, this work emphasizes the value of greater cohesion through modularity, which would benefit DMRG software and related tensor-network-centered software, enabling the solution of more complex and ambitious problems.

DOI
10.1016/j.cpc.2026.110136

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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

arXiv:2304.13395•The European Physical Journal B•Aug 1, 2023•23 citations
The density matrix renormalization group (DMRG) algorithm is a cornerstone computational method for studying quantum many-body systems, renowned for its accuracy and adaptability. Because DMRG provides a general framework applicable across various fields such as materials science, quantum chemistry, and quantum computing, one might expect a shared, flexible library to serve most users. Nevertheless, numerous independent implementations continue to appear, resulting in significant duplication of effort. To identify collaboration opportunities that can promote a more unified approach, we map the rapidly expanding DMRG software landscape and provide a comprehensive comparison of features across 37 existing packages. When comparing key features, such as parallelism strategies for high-performance computing and symmetry-adapted formulations that enhance efficiency, we found significant overlap among the packages. This overlap suggests opportunities for collaboration to modularize common functionality—e.g., tensor operations, symmetry representations, and eigensolvers—as the packages are mostly independent and share few third-party library dependencies. More collaboration on modularization could reduce duplication of effort, improve interoperability, and enable prioritization and quicker spread of new advances. We believe the current lack of modularity is more socially driven than a technical issue; hence, we see raising awareness about the existing implementations as a first step in the right direction. Ultimately, this work emphasizes the value of greater cohesion through modularity, which would benefit DMRG software and related tensor-network-centered software, enabling the solution of more complex and ambitious problems.
The performance of scientific software often determines the scale of problems that can be solved in practice. As multiple implementations of the same algorithm emerge, systematic evaluation is needed to compare their strengths and limitations. The density matrix renormalization group (DMRG) algorithm, widely used to study quantum systems, has over 50 software implementations. These implementations vary in multiple aspects that can strongly affect performance. However, despite the need, performance evaluations of these implementations are scarce and lack a consistent standard; many existing evaluations are either too incomplete to enable meaningful comparisons or focus on objectives other than direct performance comparisons, thereby limiting understanding of how the implementations compare. Here, we present a performance-oriented benchmarking framework to facilitate meaningful comparisons of DMRG implementations, and we apply it to quantify the performance of eight implementations, highlighting similarities and differences among them. Furthermore, we examine multiple parameter settings, optimization strategies, and implementation-specific features to demonstrate how parameter configuration can affect performance and how systematic evaluation can reveal non-obvious trade-offs. The results show significant performance differences, up to two orders of magnitude in some cases, not only between different implementations when aligning parameters, but also within the same implementation when comparing different parameter configurations. Hence, our results demonstrate the significant value and insight that can be gained from conducting rigorous performance evaluations. Using our results and framework as a starting point, more rigorous benchmarking will ultimately help users and developers make informed decisions and support future development efforts to build better, more efficient software.
Recent advances in high-resolution spectroscopy and cavity quantum technologies have driven growing interest in multicomponent quantum chemistry, where electronic degrees of freedom are treated on an equal footing with other quantum particles. Here, we present a unified symmetry-aware density matrix renormalization group (DMRG) framework for multicomponent quantum chemistry. By decoupling physical model specification from tensor-network representation and numerical optimization, new multicomponent models are introduced through symmetry quantum numbers, local basis, and operators, without modifying the underlying tensor algorithms. Built upon a mature tensor infrastructure, the framework provides a reusable symmetry-aware numerical engine, allowing new models to be implemented through concise Python-level specifications while fully reusing the optimized computational backend. The framework is demonstrated through symmetry-adapted nuclear-electronic orbital (NEO) and cavity quantum electrodynamics (CQED) DMRG calculations, yielding accurate electronic structures, excited-state energies, and polaritonic properties. This work establishes a reusable and extensible computational framework for symmetry-aware multicomponent quantum chemistry.
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.
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.
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