🔬 condensed matter

An Accurate Lanczos Method for the Matrix Product State Representation

This paper proposes a modified thick-block Lanczos method that significantly improves the convergence and accuracy of finding multiple low-lying eigenstates within the matrix product state representation, establishing it as a reliable alternative to DMRG that avoids local minima while achieving optimal precision for a given bond dimension.

Original authors: Yu Wang, Zhangyu Yang, Xingyao Wu, Christian B. Mendl

Published 2026-08-21

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CC BY 4.0

Original authors: Yu Wang, Zhangyu Yang, Xingyao Wu, Christian B. Mendl

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of atoms and electrons, scientists often face a problem of overwhelming scale. To predict how a material behaves, they must solve a massive mathematical puzzle involving the interactions of countless particles. The difficulty lies in the sheer number of possibilities; as the system grows, the number of potential states explodes so rapidly that even the most powerful supercomputers cannot track them all. To navigate this, researchers use a clever shortcut called the matrix product state. Imagine trying to describe a long, complex sentence by breaking it down into a chain of smaller, manageable phrases that fit together. This method compresses the vast information of a quantum system into a more compact form, allowing computers to handle calculations that would otherwise be impossible. However, this compression comes with a cost: it inevitably discards some details, introducing small errors that can accumulate and distort the final answer.

One of the most trusted tools for finding the specific energy levels of these quantum systems is an algorithm known as the Lanczos method. It works like a skilled explorer, gradually mapping out the most important parts of the energy landscape to find the lowest valleys, which represent the most stable states of a material. While this method is excellent at finding these states simultaneously and avoiding false leads, it struggles when paired with the compressed matrix product state representation. The small errors introduced by the compression cause the algorithm to lose its way, stalling its progress and leaving it unable to find the precise energy levels needed for accurate predictions. This limitation has hindered the study of complex materials where multiple energy states are equally important.

In a recent study, researchers Yu Wang, Zhangyu Yang, Xingyao Wu, and Christian B. Mendl addressed this stumbling block by developing a refined version of the algorithm they call the modified thick-block Lanczos method. Their work focuses on a specific failure mode: when the algorithm tries to find several energy states at once, the errors from the compression cause the correction steps for each state to point in different, conflicting directions. In the standard approach, the algorithm attempts to use a single correction step to fix all states simultaneously, but because the errors have scrambled the directions, this single step fails to guide the system accurately. The researchers realized that instead of forcing a single path, they needed to treat each state's correction individually.

To solve this, the team introduced a strategy that keeps a block of vectors, including both the current best guesses for the energy states and the specific error corrections for each one. By restarting the calculation with this entire block of information, the algorithm can simultaneously refine multiple states without them interfering with one another. They tested this new method on two classic models of quantum matter: a chain of fermions known as the Fermi-Hubbard model and a chain of magnetic spins called the Heisenberg model. In simulations involving chains of up to 120 sites, the new method proved to be vastly superior. While older methods would stall with errors as large as one part in a thousand, the modified approach drove the errors down to one part in a million or better, reaching the theoretical limit of accuracy allowed by the compression itself.

The results demonstrate that the new method can find not just the lowest energy state, but also the excited states that sit just above it, all with equal precision. In one test case involving a chain of 16 spin sites, the researchers found that the new method improved the accuracy of the results by three to seven orders of magnitude compared to previous techniques. It successfully identified degenerate states—situations where multiple different configurations share the exact same energy—which are notoriously difficult to distinguish. Furthermore, the team showed that by combining their method with a technique that inverts the energy spectrum, they could target specific excited states directly, bypassing the need to calculate all the lower energy states first. This capability means that researchers can now study complex quantum phenomena with a level of precision that was previously out of reach, opening the door to more reliable simulations of materials and chemical reactions.

The study confirms that the primary obstacle to accuracy was not the compression itself, but how the algorithm handled the errors that compression produced. By acknowledging that each state requires its own unique correction path and providing the algorithm with the tools to follow them all at once, the researchers have restored the power of the Lanczos method for modern quantum simulations. Their work suggests that the limitations of current quantum simulations are not inherent to the physics of the problem, but rather to the mathematical tools used to solve it. With this new approach, the field can now push toward larger and more complex systems, confident that the results will reflect the true behavior of the quantum world rather than the artifacts of the calculation.

Problem Statement
The Lanczos algorithm is a standard iterative method for computing low-lying eigenstates of large linear systems, offering a global treatment of the spectrum that avoids local minima and root flipping issues common in variational methods like DMRG. However, when applied to quantum many-body systems using the Matrix Product State (MPS) formalism, the algorithm faces a critical bottleneck: MPS truncation. To maintain computational feasibility, the bond dimension of the MPS must be capped after operations such as MPO-MPS multiplication or vector addition. This truncation introduces non-negligible errors that manifest in two ways:

Loss of Orthogonality: The iterative generation of Krylov vectors leads to a loss of global orthogonality, which is typically addressed by re-orthogonalization techniques.
Divergence from the Ideal Krylov Subspace: More critically, truncation errors accumulate, causing the generated Krylov subspace to deviate from the ideal one. This prevents the subspace from introducing new independent information, leading to a pronounced "high-error plateau" where convergence stalls far above the accuracy limits imposed by the bond dimension.

Existing attempts to improve convergence, such as the Thick-Restart Lanczos (TRL) method, fail in the MPS context. In exact arithmetic, a single Krylov vector serves as the residual direction for all Ritz vectors (approximate eigenpairs). However, under MPS truncation, the residual vectors associated with different Ritz pairs are no longer parallel. Consequently, a single restart vector cannot provide the necessary correction information for all target states simultaneously.

Methodology
The authors propose the Modified Thick-Block Lanczos (MTBL) method to address these convergence issues. The core innovation lies in adapting the restart strategy to account for the non-parallelism of residual vectors caused by MPS truncation.

Block-Restart Strategy: Instead of retaining a single correction vector, MTBL retains a block of vectors upon restarting. This block consists of:
1. The $t$ Ritz vectors (approximate eigenstates) of interest.
2. The $t$ corresponding residual vectors, calculated explicitly as $|r_i\rangle = H|\phi_i\rangle - \theta_i|\phi_i\rangle$ .
Algorithm Flow: The method initializes with a block of vectors, generates new Krylov vectors by applying the Hamiltonian, and orthogonalizes them against previous vectors. Crucially, at each restart, the algorithm explicitly computes the residual vectors for the current Ritz pairs and includes them in the restart block. This ensures that the correction directions for all target eigenstates are preserved and independent, even under truncation.
Shift-and-Invert: The framework incorporates the shift-and-invert spectral transformation ( $B = (H - \lambda I)^{-1}$ ) to target excited states directly. Since inverting the Hamiltonian as an MPO is prohibitively expensive, the method uses a variational approach to find the MPS $|w_j\rangle$ that minimizes $\|(H - \lambda I)|w_j\rangle - |q_{j-1}\rangle\|^2$ , allowing for the direct targeting of excited states without sequentially converging lower-energy states.
Accuracy Metric: The paper utilizes energy variance ( $\sigma^2 = \langle \phi_i | (H - \theta_i I)^2 | \phi_i \rangle$ ) as a diagnostic tool to assess the quality of approximate eigenpairs, particularly when exact diagonalization is unavailable for larger systems.

Key Results
The authors benchmarked MTBL against the Plain Lanczos Method (PLM) and the standard Thick-Restart Lanczos (TRL) on the Fermi-Hubbard and Heisenberg XXZ models.

Convergence and Accuracy: On a half-filled 8-site Fermi-Hubbard chain (16 spin sites) with a bond dimension capped at $M=32$ , PLM and TRL failed to converge excited states accurately, with errors stalling at high levels. In contrast, MTBL allowed all five target eigenstates (ground state and four excited states) to converge to the optimal accuracy allowed by the bond dimension.
Error Reduction: MTBL improved the relative energy error by three to seven orders of magnitude compared to previous MPS-based Lanczos methods. The method successfully captured degenerate excited states, which are often problematic for other techniques.
Scalability: The method was demonstrated to scale to larger systems, specifically a 50-site Fermi-Hubbard model (100 spin sites) and a 120-site Heisenberg XXZ chain.
Computational Cost: While MTBL requires additional operations to compute residual vectors at each restart, it converges faster than PLM and TRL when measured by the total number of MPO-MPS multiplication-compressions. The geometric mean of relative energy errors for MTBL decreased significantly faster than its counterparts under the same computational cost.

Significance and Claims
The paper claims that MTBL resolves the primary limitation hindering the broader application of Lanczos-type methods within the MPS formalism: the inability to accurately target multiple eigenstates simultaneously due to truncation-induced divergence.

Optimal Accuracy: The authors assert that MTBL achieves the "optimal accuracy allowed by the bond dimension," effectively raising the accuracy of Lanczos-type methods to the level of mainstream high-accuracy MPS approaches like DMRG.
Robustness: By explicitly handling the non-parallelism of residual vectors, the method provides a robust treatment of the spectrum without the risk of root flipping or getting trapped in local minima.
Direct Excited State Targeting: The integration with shift-and-invert allows for the direct targeting of excited states without the cumulative errors associated with sequentially solving lower-energy states, a significant advantage over standard DMRG-based excited state methods.

In summary, the work presents a modified algorithmic framework that restores the theoretical advantages of the Lanczos algorithm (global spectrum treatment, simultaneous multi-state targeting) within the practical constraints of tensor network representations.

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