An Accurate Lanczos Method for the Matrix Product State Representation
cond-mat.str-el, physics.comp-ph, quant-ph
Submitted: 2025-04-30
Updated: 2026-08-20
DOI: 10.1038/s42005-026-02896-x
License: http://creativecommons.org/licenses/by/4.0/
The gist: We improve the convergence of the Lanczos algorithm using the matrix product state representation.
Terminology
Abstract
We improve the convergence of the Lanczos algorithm using the matrix product state representation. As an alternative to the density matrix renormalization group (DMRG), the Lanczos algorithm avoids local minima and can directly find multiple low-lying eigenstates. However, its performance and accuracy are affected by the truncation required to maintain the efficiency of the tensor network representation. In this work, we propose the modified thick-block Lanczos method to enhance the convergence of the Lanczos algorithm with MPS representation. We benchmark our method on one-dimensional instances of the Fermi-Hubbard model and the Heisenberg model in an external field, using numerical experiments targeting the first five lowest eigenstates. Across these tests, our approach attains the best possible accuracy permitted by the given bond dimension. This work establishes the Lanczos method as a reliable and accurate framework for finding multiple low-lying states within a tensor-network representation
Sources
- Inexact subspace projection methods for low-rank tensor eigenvalue problems
- Bound states in doped charge transfer insulators
- Matrix-product-state-based band-Lanczos solver for quantum cluster approaches
- Optimal Symbolic Construction of Matrix Product Operators and Tree Tensor Network Operators
- Accurate, full-dimensional computations of thousands of complex vibrational eigenstates with tree tensor network states
- Successive randomized compression: A randomized algorithm for the compressed MPO-MPS product
Related papers
- Microscopic Constructions of the BF+AAB Topological Field Theory and Borromean-Rings Braiding in (3+1) Dimensions
- Transport in the emergent Bose liquid: Bad metal, strange metal, and weak insulator, all in one system
- Gibbs-state phases beyond microscopic symmetry breaking: A phase invariant for the 3D toric code
- Instabilities in self-consistent diagrammatic approaches and how to cure them
- True vs false Fermi surfaces in the Pseudogap regime and their transformation with doping and temperature in the Hubbard Model
- Orbital magnetic susceptibility and de Haas-van Alphen effect of a flat band from quantum geometry