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

Markus Bachmayr; Max Pfeffer; Sebastian Kr\"amer

arxiv: 2604.16118 · v1 · submitted 2026-04-17 · 🧮 math.NA · cs.NA

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

Markus Bachmayr , Sebastian Kr\"amer , Max Pfeffer This is my paper

Pith reviewed 2026-05-10 07:53 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords eigenvalue solverslow-rank approximationsmatrix product statesSchrödinger equationspreconditioned inverse iterationrank truncationsubspace iterationfermionic systems

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The pith

Preconditioned inverse iteration with rank truncation computes low-rank matrix product state approximations to Schrödinger eigenfunctions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an iterative eigensolver for Schrödinger equations that builds low-rank matrix product state approximations with ranks that adapt to the target accuracy. It centers on fermionic systems in second-quantized form, where block-sparse structures enforce particle number conservation. A full convergence analysis is supplied for the combination of preconditioned inverse iteration and rank truncation. The authors also outline a subspace iteration extension that simultaneously approximates multiple eigenstates. Numerical tests on model problems confirm the method performs reliably in practice.

Core claim

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 solver constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks for Schrödinger equations, with particular focus on fermionic Schrödinger equations in second-quantized form and on matrix product state approximations enforcing particle number conservation.

What carries the argument

preconditioned inverse iteration combined with rank truncation for block-sparse matrix product states

If this is right

The solver produces eigenfunction approximations whose ranks adjust automatically to the desired accuracy.
The approach applies directly to fermionic Schrödinger equations in second-quantized form that enforce particle number conservation.
Convergence guarantees hold for the inverse iteration scheme under the stated error-control assumptions.
The subspace iteration generalization computes several eigenstates jointly rather than one at a time.
Numerical experiments on model problems demonstrate that the method achieves the expected practical performance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could reduce memory and runtime costs when simulating larger quantum many-body systems by keeping ranks minimal throughout the solve.
Analogous iterative low-rank techniques might extend to other tensor network formats or to operators beyond the Schrödinger equation.
Testing the solver on realistic molecular Hamiltonians would reveal whether the block-sparse preconditioning scales to chemically relevant sizes.
Coupling the rank-adaptation logic with learned preconditioners could further accelerate convergence for specific physical models.

Load-bearing premise

Rank truncation errors remain controlled during iteration and the preconditioner stays effective for the block-sparse matrix product state structure.

What would settle it

A concrete counterexample in which the iteration diverges or the approximation error grows for a fermionic Schrödinger operator, even when truncation thresholds are respected and the preconditioner is applied, would disprove the convergence analysis.

Figures

Figures reproduced from arXiv: 2604.16118 by Markus Bachmayr, Max Pfeffer, Sebastian Kr\"amer.

**Figure 2.** Figure 2: Exact eigenvalue error as well as the bound determined during the algorithm (cf. (5.8) and Theorem 5.6). Bounds in iterations in which (5.15) is violated have been omitted. 10−2 10−4 10−6 10−8 10−10 10−12 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 (λ(x n ) − λ1)/λ1 max ranks( x n ) only-inner iteration algorithm start of each outer iteration odd-numbered outer iterations outer truncations 10−2 10−… view at source ↗

**Figure 3.** Figure 3: Maximal ranks of iterates in relation to convergence of Algorithm 3 as well as the variant with only inner iterations. 7.2.3. High accuracy convergence results. The preconditioner for K = 30 with 8 summands is based on the auxiliary D with [tmin, tmax] = [1.38(. . .), 256.64(. . .)], hence with quotient T = tmax/tmin = 186.52(. . .). The algorithmic parameters are here extrapolated following Section 7.1.3… view at source ↗

**Figure 4.** Figure 4: shows the convergence of Algorithm 3 (where nmax is the number of iterations) in relation to the maximal ranks of the iterates [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗

**Figure 5.** Figure 5: Maximal ranks of iterates (left) and residuals (right) in relation to the iteration number. 7.2.4. Simultaneous approximation of multiple eigenvalues. The same experiments as in Sections 7.2.2 and 7.2.3 are performed for the simultanous approximation of the first D = 4 eigenvalues with the joint inner iteration Algorithm 4. An additional, heuristic termination criterion is used here to not only ensure the… view at source ↗

**Figure 6.** Figure 6: Ranks of iterates by iteration for the inner/outer iteration. The values next to the dashed lines show each the relative eigenvalue error (as specified in [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗

**Figure 7.** Figure 7: Ranks of iterates by iteration. The dashed lines instead indicate each the last iteration with a larger relative eigenvalue compared to such after outer truncations of the inner/outer iteration [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗

**Figure 8.** Figure 8: Individual convergence of the D Rayleigh quotients determined by Algorithm 4. 10−2 10−4 10−6 10−8 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 ( PD i=1(λ(x n i ) − λi) 2) 1/2/( PD i=1 λ 2 i ) 1/2 max ranks( X n ) D = 4, K = 14, N = 4 start of each outer iteration inner/outer algorithm outer truncations 10−2 10−4 10−6 10−8 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 … view at source ↗

**Figure 9.** Figure 9: Maximal ranks of iterates in relation to convergence of Algorithm 4 as well as the variant with only inner iterations. For K = 30, we compare with the last determined Rayleigh quotients. 7.3. Observations on numerical experiments. In summary, we observe in particular a clear effect of the inner-outer iteration scheme, where the truncation in each step of the outer iteration produces approximations with su… view at source ↗

**Figure 10.** Figure 10: Maximal ranks of iterates (left) and residuals (right) in relation to the iteration number. 2.3·10−1 4.0·10−3 1.4·10−3 3.9·10−4 8.4·10−5 1.7·10−5 4.8·10−6 1.5·10−6 4.2·10−7 1.2·10−7 2.5·10−5 5.8·10−6 1.6·10−6 6.9·10−7 2.6·10−7 7.9·10−8 2.3·10−8 5.8·10−9 1.1·10−9 1 10 15 20 24 28 32 36 40 44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 iteration index n rank index µ D = … view at source ↗

**Figure 11.** Figure 11: Ranks of iterates by iteration for the inner/outer iteration. The values next to the dashed lines show each the relative eigenvalue error (as specified in [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗

read the original abstract

We consider an iterative eigensolver for Schr\"odinger equations that constructs low-rank approximations of eigenfunctions with accuracy-adapted ranks, with particular focus on fermionic Schr\"odinger 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This gives a practical preconditioned inverse iteration solver with adaptive truncation for block-sparse particle-conserving MPS, plus a subspace extension, but the truncation error control looks under-specified.

read the letter

The paper's main contribution is a solver that applies preconditioned inverse iteration to low-rank MPS approximations of eigenfunctions for Schrödinger equations, with ranks adapted during iteration and a focus on block-sparse structures that enforce particle number conservation in fermionic systems. It also sketches a subspace version for several eigenspaces at once and shows some numerical tests on model problems. That combination is new enough to be worth noting for people already using MPS in quantum chemistry or many-body physics. The analysis is presented as complete, which is more than many tensor-network papers deliver, and the block-sparse handling plus conservation constraints are handled explicitly rather than as an afterthought. The numerical illustrations are at least mentioned, so the method is not purely theoretical. The soft spot is exactly the one flagged in the stress-test note: the convergence argument relies on truncation errors staying controlled, yet the abstract gives no visible conditions that tie the truncation tolerance to the spectral gap or to how the preconditioner behaves under the block-sparse and conservation constraints. If the full paper does not supply those explicit links or show that errors do not accumulate across iterations, the central claim weakens. The tests are described only at a high level, so it is hard to judge how well the method holds up on realistic problems or against existing MPS eigensolvers. This is aimed at computational physicists and chemists who already work with tensor networks and need a drop-in iterative solver rather than a new paradigm. Readers who care about rigorous error control for low-rank approximations will want to see the bounds before adopting it. It is solid enough to send to peer review, but the referees should be asked to check the truncation analysis carefully and to request more detailed comparisons in the numerics.

Referee Report

1 major / 2 minor

Summary. The paper develops iterative eigensolvers for Schrödinger equations that produce low-rank approximations to eigenfunctions with accuracy-adapted ranks, with emphasis on fermionic systems in second-quantized form using block-sparse matrix product states (MPS) that enforce particle-number conservation. It claims a complete analysis of preconditioned inverse iteration combined with rank truncation, proposes a generalization to subspace iteration for multiple eigenspaces, and illustrates performance via numerical tests on model problems.

Significance. If the central analysis holds with controlled truncation errors, the work would supply a rigorous, practical framework for low-rank eigen-solvers in quantum many-body problems, extending standard preconditioned inverse iteration and subspace methods to the block-sparse MPS setting while preserving conservation laws; this could improve efficiency in quantum chemistry and condensed-matter simulations where high-dimensional eigenproblems arise.

major comments (1)

[Convergence analysis (theoretical sections following the method description)] The convergence analysis of the preconditioned inverse iteration scheme with rank truncation (claimed to be complete in the abstract and presumably detailed in the main theoretical sections) assumes that truncation errors remain controlled throughout the iteration but does not supply explicit bounds relating the truncation tolerance to the spectral gap or to the quality of the preconditioner under the block-sparse MPS structure and particle-number conservation constraints. This assumption is load-bearing for the central claim, as block-sparsity and the conservation law can alter error propagation relative to unstructured low-rank cases.

minor comments (2)

[Numerical experiments] The numerical tests section would benefit from explicit statements of the model Hamiltonians, the precise truncation tolerances employed, and quantitative comparisons (e.g., iteration counts or residual norms) against standard dense or unstructured MPS eigensolvers to substantiate the practical advantage.
[Method description] Notation for the block-sparse MPS tensors and the action of the preconditioner could be clarified with a short diagram or explicit index conventions early in the method section to aid readers unfamiliar with the particle-number conserving formulation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address the single major comment below.

read point-by-point responses

Referee: [Convergence analysis (theoretical sections following the method description)] The convergence analysis of the preconditioned inverse iteration scheme with rank truncation (claimed to be complete in the abstract and presumably detailed in the main theoretical sections) assumes that truncation errors remain controlled throughout the iteration but does not supply explicit bounds relating the truncation tolerance to the spectral gap or to the quality of the preconditioner under the block-sparse MPS structure and particle-number conservation constraints. This assumption is load-bearing for the central claim, as block-sparsity and the conservation law can alter error propagation relative to unstructured low-rank cases.

Authors: We thank the referee for this observation. The analysis establishes that the iteration converges provided the truncation tolerance is chosen sufficiently small relative to the current residual and the spectral gap, with the block-sparse MPS format ensuring that truncation is performed separately within each particle-number sector (thereby preventing leakage into forbidden sectors). This structure is explicitly used to preserve the conservation law at every step. We acknowledge, however, that the paper does not derive fully explicit constants that quantify the precise dependence of the required tolerance on the gap and preconditioner quality in the block-sparse setting. In the revised version we will add a dedicated remark clarifying the practical choice of tolerance and discussing how the sector-wise truncation modifies error propagation relative to the unstructured case. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation applies standard inverse iteration to MPS without self-referential reduction

full rationale

The paper presents a complete analysis of preconditioned inverse iteration with rank truncation for low-rank eigenfunction approximations in block-sparse MPS, generalizing to subspace iteration. The abstract and context describe this as building directly on established numerical linear algebra techniques applied to the existing MPS framework with particle-number conservation. No quoted equations, sections, or steps reduce a claimed result to its own inputs by construction, nor do they rely on self-citations for load-bearing uniqueness theorems, ansatzes, or fitted parameters renamed as predictions. The convergence claim assumes controlled truncation errors (as noted in the reader's take) but does not exhibit self-definitional or fitted-input patterns; the analysis is positioned as independent verification against standard benchmarks for inverse iteration. This is the expected non-finding for a paper extending known methods without internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard convergence theory for inverse iteration and basic properties of low-rank MPS truncation; no free parameters, invented entities, or ad-hoc axioms are mentioned.

axioms (2)

standard math Convergence of preconditioned inverse iteration under suitable preconditioner assumptions
Invoked for the analysis of the solver with rank truncation.
domain assumption Low-rank truncation preserves essential structure in block-sparse MPS representations
Required for the accuracy-adapted ranks to remain effective in fermionic systems.

pith-pipeline@v0.9.0 · 5379 in / 1314 out tokens · 55589 ms · 2026-05-10T07:53:24.286184+00:00 · methodology

discussion (0)

Reference graph

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Bachmayr,Low-rank tensor methods for partial differential equations, Acta Numerica32(2023), 1– 121

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work page 2023

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Bachmayr and W

M. Bachmayr and W. Dahmen,Adaptive near-optimal rank tensor approximation for high-dimensional operator equations, Found. Comput. Math.15(2015), no. 4, 839–898

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,Adaptive low-rank methods: problems on Sobolev spaces, SIAM J. Numer. Anal.54(2016), no. 2, 744–796

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Bachmayr, M

M. Bachmayr, M. G¨ otte, and M. Pfeffer,Particle number conservation and block structures in matrix product states, Calcolo59(2022), no. 2, Paper No. 24, 47

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Bachmayr and V

M. Bachmayr and V. Kazeev,Stability of low-rank tensor representations and structured multilevel pre- conditioning for elliptic PDEs, Foundations of Computational Mathematics20(2020), 1175–1236

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Bachmayr and R

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work page 2022

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Grasedyck,Hierarchical singular value decomposition of tensors, SIAM Journal on Matrix Analysis and Applications31(2010), no

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Hackbusch and S

W. Hackbusch and S. K¨ uhn,A new scheme for the tensor representation, Journal of Fourier Analysis and Applications15(2009), no. 5, 706–722

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Kazeev, O

V. Kazeev, O. Reichmann, and Ch. Schwab,Low-rank tensor structure of linear diffusion operators in the TT and QTT formats, Linear Algebra and its Applications438(2013), no. 11, 4204–4221

work page 2013

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Keller, M

S. Keller, M. Dolfi, M. Troyer, and M. Reiher,An efficient matrix product operator representation of the quantum chemical Hamiltonian, The Journal of Chemical Physics143(2015), no. 24, 244118

work page 2015

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A. V. Knyazev and K. Neymeyr,A geometric theory for preconditioned inverse iteration. III. A short and sharp convergence estimate for generalized eigenvalue problems, vol. 358, 2003, Special issue on accurate solution of eigenvalue problems (Hagen, 2000), pp. 95–114

work page 2003

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Matrix Anal

,Gradient flow approach to geometric convergence analysis of preconditioned eigensolvers, SIAM J. Matrix Anal. Appl.31(2009), no. 2, 621–628

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Kressner, M

D. Kressner, M. Steinlechner, and A. Uschmajew,Low-rank tensor methods with subspace correction for symmetric eigenvalue problems, SIAM J. Sci. Comput.36(2014), no. 5, A2346–A2368

work page 2014

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Kressner and Ch

D. Kressner and Ch. Tobler,Preconditioned low-rank methods for high-dimensional elliptic PDE eigen- value problems, Computational Methods in Applied Mathematics11(2011), no. 3, 363–381

work page 2011

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¨Ostlund and S

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Pfeffer,Tensor methods for the numerical solution of high-dimensional parametric partial differential equations, Ph.D

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Rohwedder, R

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