An Optimally Accurate Lanczos Algorithm in the Matrix Product State Representation

TL;DR

This work addresses the convergence and accuracy challenges of the Lanczos method when implemented with matrix product state representations, where MPS truncation can destroy orthogonality and hinder exploration of the Krylov space. It introduces the Modified Thick-Block Lanczos (MTBL) method, combining block-Lanczos with thick-restart and shift-and-invert to preserve independent correction directions and directly target excited states. Through numerical experiments on the 1D Fermi-Hubbard and XXZ Heisenberg models, MTBL achieves accuracy at the limit imposed by the chosen bond dimension, dramatically improving over plain Lanczos and demonstrating scalability to large systems (up to L=120). The method provides a practical, robust tool for obtaining multiple low-lying eigenstates in tensor-network frameworks and may generalize to other TN ansätze and compression strategies.

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

Figures (12)

• Figure 1: Diagrammatic representation of matrix product states and operators.
• Figure 2: The classical SVD method to compress the MPS. First, the MPS is brought into right-canonical form via successive QR decompositions. Then, sweeping from left to right, we carry out SVDs at each site and truncate the smallest singular values. Finally, the $\Sigma$ and $V^{\dagger}$ matrices are merged into the next site.
• Figure 3: Convergence of the first five low-lying eigenstates of the Fermi–Hubbard model obtained with PLM in MPS representation. The excited states cannot converge to high accuracy.
• Figure 4: Convergence behavior of the first five low-lying eigenstates of the Fermi–Hubbard model using the TRL method with the MPS representation. We restart the procedure every 30 iterations. Only the ground state converges well since TRL using MPS can capture the residual vector of the ground state correctly.
• Figure 5: Parallelism of the residual vectors associated with the five Ritz vectors for bond dimensions $[8,16,32,64,128,256]$, measured by $\rho$. The circled point corresponds to the case $M=32$. Only when the bond dimension is set to $M=256$, which corresponds to the no-truncation case, are the residual vectors parallel.