An Optimally Accurate Lanczos Algorithm in the Matrix Product State Representation

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

💡 Research Summary

This paper addresses a long‑standing difficulty in applying the Lanczos algorithm within the matrix‑product‑state (MPS) framework: the truncation required to keep the bond dimension bounded introduces errors that destroy orthogonality of the Krylov vectors and cause the generated subspace to diverge from the ideal Krylov space. In exact arithmetic the Lanczos recurrence is a three‑term relation, and thick‑restart Lanczos (TRL) can reuse a single residual vector to correct all retained Ritz pairs. However, when MPS truncation is present the recurrence no longer holds, each Ritz pair acquires its own residual, and the single‑vector restart fails, leading to a pronounced high‑error plateau.

To overcome this, the authors propose the Modified Thick‑Block Lanczos (MTBL) method. After a standard Lanczos sweep they compute both the Ritz vectors { |ϕ_i⟩ } and their individual residuals |r_i⟩ = H|ϕ_i⟩ − θ_i|ϕ_i⟩. These vectors are assembled into a block, compressed back to the prescribed bond dimension using SVD‑based truncation, and used as the new starting basis for the next Lanczos cycle. Because each Ritz pair now has its own correction direction, the block‑restart restores independent information that would otherwise be lost to truncation. Orthogonalization is performed on the whole block rather than on each vector sequentially, reducing the cost of re‑orthogonalization.

The paper also integrates the shift‑and‑invert (SI) spectral transformation. By applying (H − σI)⁻¹ (implemented as an MPO approximation) the spectrum is inverted around a shift σ, allowing the Lanczos process to target excited states directly without first computing lower‑energy states. This eliminates cumulative errors from previously obtained eigenvectors and mitigates root‑flipping in near‑degenerate sectors.

Numerical benchmarks are carried out on two one‑dimensional models: a half‑filled Fermi‑Hubbard chain (L = 30) and a spin‑½ Heisenberg XXZ chain in a longitudinal field (L = 40). For each model the first five eigenstates are targeted. Across a range of bond dimensions (M = 100–500) MTBL consistently reaches the accuracy limit imposed by the chosen M, often improving the energy error by three to seven orders of magnitude compared with the earlier MPS‑Lanczos implementation. In particular, MTBL matches or exceeds the precision of density‑matrix renormalization group (DMRG) calculations at the same bond dimension, while avoiding DMRG’s local‑minimum traps and root‑flipping problems.

A scalability test on a larger Heisenberg chain (L = 120) demonstrates that MTBL maintains O(L M²) memory usage and still achieves energy errors on the order of 10⁻⁸, confirming that the algorithm can be applied to substantially larger systems without loss of accuracy.

The authors conclude that MTBL resolves the two principal sources of error in MPS‑based Lanczos methods: (1) loss of orthogonality due to truncation, and (2) divergence of the Krylov subspace from the true subspace. By employing a block of Ritz vectors and their residuals, and by optionally using shift‑and‑invert, the method provides a globally robust, multi‑state eigensolver that is competitive with, and in some aspects superior to, state‑of‑the‑art DMRG. The paper suggests that the same block‑residual strategy could be extended to higher‑dimensional tensor networks (e.g., PEPS), multi‑orbital quantum chemistry Hamiltonians, and real‑time dynamics, opening a pathway for accurate, scalable spectral calculations in a broad class of many‑body quantum problems.