Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

SIAM Journal on Scientific Computing

Computing Eigenspaces With Low Rank Constraints

Christian Krumnow, Max J. Pfeffer, André Uschmajew

Published Jan 1, 2021• 1 citation
Publisher

Abstract

In this work, the task to find a simultaneous low rank approximation of several lowest eigenpairs of a matrix-valued symmetric operator is considered. This problem arises, for instance, in the density matrix renormalization group algorithm (DMRG) for the accurate simulation of quantum chains. The usual approach is to compute the desired set of eigenvectors and then to identify a low rank approximation by truncating the singular value decomposition (SVD). Since SVD truncation is a norm projection, this yields only suboptimal results for the eigenvalues. As an alternative, this article explores a direct trace minimization on the intersection of the Stiefel and a low rank manifold using a Riemannian optimization method, which gives better approximations to the eigenspace. A second algorithm based on alternating optimization is also considered but is less stable. Compared to SVD truncation, the proposed Riemannian method can be seen as a more natural choice for a subsolver in the DMRG algorithm and appears to yield better results when applied to spin chains, for which the singular values of the exact eigenvectors decay only moderately.

DOI
10.1137/19m1308384

Related papers

Inexact subspace projection methods for low-rank tensor eigenvalue problems

Alec Dektor, P Delmastro, Erika Ye +2 more

We propose inexact subspace iteration for solving high-dimensional eigenvalue problems with low-rank structure. Inexactness stems from low-rank compression, enabling efficient representation of high-dimensional vectors in a low-rank tensor format. A primary challenge in these methods is that standard operations, such as matrix-vector products and linear combinations, increase tensor rank, necessitating rank truncation and hence approximation. We compare the proposed methods with an existing inexact Lanczos method with low-rank compression. This method constructs an approximate orthonormal Krylov basis, which is often difficult to represent accurately in low-rank tensor formats, even when the eigenvectors themselves exhibit low-rank structure. In contrast, inexact subspace iteration uses approximate eigenvectors (Ritz vectors) directly as a subspace basis, bypassing the need for an orthonormal Krylov basis. Our analysis and numerical experiments demonstrate that inexact subspace iteration is much more robust to rank-truncation errors compared to the inexact Lanczos method. We also demonstrate that rank-truncated subspace iteration can converge for problems where the DMRG method stagnates. Furthermore, the proposed subspace iteration methods do not require a Hermitian matrix, in contrast to DMRG, which is designed specifically for Hermitian matrices.

arXiv:2502.19578•Linear Algebra and its Applications•Apr 23, 2026•0 citations

Zero-site density matrix renormalization group and the optimal low-rank correction

Y. Núñez-Fernández, Gonzalo Torroba

A zero-site density matrix renormalization algorithm (DMRG0) is proposed to minimize the energy of matrix product states (MPSs). Instead of the site tensors themselves, we propose to optimize sequentially the ``message'' tensors between neighbor sites, which contain the singular values of the bipartition. This leads to a local minimization step that is independent of the physical dimension of the site. Conceptually, it separates the optimization and decimation steps in DMRG. Furthermore, we introduce two global perturbations based on the optimal low-rank correction to the current state, which are used to avoid local minima. They are determined variationally as the MPSs closest to the one-step correction of the Lanczos or Jacobi-Davidson eigensolver. These perturbations mainly decrease the energy and are free of hand-tuned parameters. Compared to existing single-site enrichment proposals, our approach gives similar convergence ratios per sweep, while the computations are cheaper by construction. Our methods may be useful in systems with many physical degrees of freedom per lattice site. We test our approach on the periodic Heisenberg spin chain for various spins and on free electrons on the lattice.

arXiv:1908.10880•Physical review. B./Physical review. B•Feb 24, 2020•2 citations

Hybrid stochastic method for the tensor renormalization group

Hiroshi Ohki, Masaaki Tomii, Erika Arai

We propose a hybrid stochastic method for the tensor renormalization group (TRG) approach. TRG is known as a powerful tool to study the many-body systems and quantum field theory on the lattice. It is based on a low-rank approximation of the tensor using the truncated singular value decomposition (SVD), whose computational cost increases as the bond dimension increases, so that efficient cost reduction techniques are highly demanded. We use noise vectors for the low-rank approximation with the truncated SVD, by which the truncation error is replaced with a statistical error due to noise, and an error estimation could be improved. We test this method in the classical Ising model and observe a better accuracy than TRG. We also discuss a cross contamination issue in a multiple use of the same noise vectors, and to remove this systematic error we consider position-dependent noise vectors.

arXiv:2110.11921•International Symposium on Lattice Field Theory (LATTICE)•May 16, 2022•0 citations

Density matrix renormalization group (DMRG) for cyclic and centrosymmetric linear chains

Manoranjan Kumar, Dayasindhu Dey, Aslam Parvej +2 more

The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of $N$ sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin $S \ge 1$ and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.

arXiv:1610.00998•Oct 4, 2016•0 citations

Low-rank eigenvalue solvers for block-sparse matrix product states

Markus Bachmayr, Sebastian Krämer, Max J. Pfeffer

We consider an iterative eigensolver for Schrödinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermionic Schrödinger equations in second-quantized form and on matrix product state approximations enforcing particle number conservation. We provide a complete analysis of a solver based on preconditioned inverse iteration combined with rank truncation and propose a generalization to subspace iteration for the joint approximation of several eigenspaces. The practical performance of the method is illustrated by numerical tests for several model problems.

arXiv:2604.16118•Apr 17, 2026•0 citations

Quantum singular-value decomposition of nonsparse low-rank matrices

Patrick Rebentrost, Adrian Steffens, Iman Marvian +1 more

We present a method to exponentiate nonsparse indefinite low-rank matrices on a quantum computer. Given access to the elements of the matrix, our method allows one to determine the singular values and their associated singular vectors in time exponentially faster in the dimension of the matrix than known classical algorithms. The method extends to non-Hermitian and nonsquare matrices via matrix embedding. Moreover, our method preserves the phase relations between the singular spaces allowing for efficient algorithms that require operating on the entire singular-value decomposition of a matrix. As an example of such an algorithm, we discuss the Procrustes problem of finding a closest isometry to a given matrix.

arXiv:1607.05404•Physical Review A•Jan 24, 2018•135 citations
In this work, the task to find a simultaneous low rank approximation of several lowest eigenpairs of a matrix-valued symmetric operator is considered. This problem arises, for instance, in the density matrix renormalization group algorithm (DMRG) for the accurate simulation of quantum chains. The usual approach is to compute the desired set of eigenvectors and then to identify a low rank approximation by truncating the singular value decomposition (SVD). Since SVD truncation is a norm projection, this yields only suboptimal results for the eigenvalues. As an alternative, this article explores a direct trace minimization on the intersection of the Stiefel and a low rank manifold using a Riemannian optimization method, which gives better approximations to the eigenspace. A second algorithm based on alternating optimization is also considered but is less stable. Compared to SVD truncation, the proposed Riemannian method can be seen as a more natural choice for a subsolver in the DMRG algorithm and appears to yield better results when applied to spin chains, for which the singular values of the exact eigenvectors decay only moderately.
We propose inexact subspace iteration for solving high-dimensional eigenvalue problems with low-rank structure. Inexactness stems from low-rank compression, enabling efficient representation of high-dimensional vectors in a low-rank tensor format. A primary challenge in these methods is that standard operations, such as matrix-vector products and linear combinations, increase tensor rank, necessitating rank truncation and hence approximation. We compare the proposed methods with an existing inexact Lanczos method with low-rank compression. This method constructs an approximate orthonormal Krylov basis, which is often difficult to represent accurately in low-rank tensor formats, even when the eigenvectors themselves exhibit low-rank structure. In contrast, inexact subspace iteration uses approximate eigenvectors (Ritz vectors) directly as a subspace basis, bypassing the need for an orthonormal Krylov basis. Our analysis and numerical experiments demonstrate that inexact subspace iteration is much more robust to rank-truncation errors compared to the inexact Lanczos method. We also demonstrate that rank-truncated subspace iteration can converge for problems where the DMRG method stagnates. Furthermore, the proposed subspace iteration methods do not require a Hermitian matrix, in contrast to DMRG, which is designed specifically for Hermitian matrices.
A zero-site density matrix renormalization algorithm (DMRG0) is proposed to minimize the energy of matrix product states (MPSs). Instead of the site tensors themselves, we propose to optimize sequentially the ``message'' tensors between neighbor sites, which contain the singular values of the bipartition. This leads to a local minimization step that is independent of the physical dimension of the site. Conceptually, it separates the optimization and decimation steps in DMRG. Furthermore, we introduce two global perturbations based on the optimal low-rank correction to the current state, which are used to avoid local minima. They are determined variationally as the MPSs closest to the one-step correction of the Lanczos or Jacobi-Davidson eigensolver. These perturbations mainly decrease the energy and are free of hand-tuned parameters. Compared to existing single-site enrichment proposals, our approach gives similar convergence ratios per sweep, while the computations are cheaper by construction. Our methods may be useful in systems with many physical degrees of freedom per lattice site. We test our approach on the periodic Heisenberg spin chain for various spins and on free electrons on the lattice.
We propose a hybrid stochastic method for the tensor renormalization group (TRG) approach. TRG is known as a powerful tool to study the many-body systems and quantum field theory on the lattice. It is based on a low-rank approximation of the tensor using the truncated singular value decomposition (SVD), whose computational cost increases as the bond dimension increases, so that efficient cost reduction techniques are highly demanded. We use noise vectors for the low-rank approximation with the truncated SVD, by which the truncation error is replaced with a statistical error due to noise, and an error estimation could be improved. We test this method in the classical Ising model and observe a better accuracy than TRG. We also discuss a cross contamination issue in a multiple use of the same noise vectors, and to remove this systematic error we consider position-dependent noise vectors.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of NNN sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin S≥1S \ge 1S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
We consider an iterative eigensolver for Schrödinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermionic Schrödinger equations in second-quantized form and on matrix product state approximations enforcing particle number conservation. We provide a complete analysis of a solver based on preconditioned inverse iteration combined with rank truncation and propose a generalization to subspace iteration for the joint approximation of several eigenspaces. The practical performance of the method is illustrated by numerical tests for several model problems.
We present a method to exponentiate nonsparse indefinite low-rank matrices on a quantum computer. Given access to the elements of the matrix, our method allows one to determine the singular values and their associated singular vectors in time exponentially faster in the dimension of the matrix than known classical algorithms. The method extends to non-Hermitian and nonsquare matrices via matrix embedding. Moreover, our method preserves the phase relations between the singular spaces allowing for efficient algorithms that require operating on the entire singular-value decomposition of a matrix. As an example of such an algorithm, we discuss the Procrustes problem of finding a closest isometry to a given matrix.