arxiv: 2604.25625 · v1 · submitted 2026-04-28 · ⚛️ physics.comp-ph

Recognition: unknown

Basic linear algebra methods for quantum problems

Aaron Dayton, Kiana Gallagher, Sarah E. Huber, Thomas E. Baker

Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:47 UTC · model grok-4.3

classification ⚛️ physics.comp-ph

keywords linear algebraeigenvalue problemsquantum systemsnumerical methodsmatrix decompositionsQR decompositionLU decompositioncomputational physics

0 comments

The pith

Basic linear algebra routines enable computers to solve quantum eigenvalue problems too complex for hand calculation.

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

The paper reviews standard linear algebra operations and their computer implementations to give readers a foundation for tackling quantum problems numerically. It emphasizes eigenvalue problems but notes the methods are generic, covering matrix forms common in quantum systems along with their solution strategies and computational costs. A sympathetic reader cares because these routines sit inside optimized libraries, so learning the underlying decompositions avoids the need to rebuild them from scratch for each new application. The discussion includes how free libraries implement eigenvalue, Schur, QR, LU, LDL, Cholesky, and singular-value methods efficiently.

Core claim

The paper establishes that reviewing elementary linear algebra solutions, computational complexity, and common decompositions supplies the practical knowledge needed to perform operations on quantum systems that cannot be completed with pen and paper alone. It shows that focusing on eigenvalue problems for quantum systems still yields broadly applicable guidance because the same matrix factorizations serve many other numerical tasks.

What carries the argument

Matrix decompositions (eigenvalue, Schur, QR, LU, LDL, Cholesky, and singular-value) that reduce quantum linear systems and eigenvalue problems to efficient numerical steps available in standard libraries.

If this is right

Quantum systems expressed as common matrix forms become solvable by selecting the matching decomposition strategy from available libraries.
Users avoid spending time re-implementing basic routines and instead use the most efficient existing functions for eigenvalue and related problems.
Awareness of computational complexity guides choices among decompositions when scaling to larger quantum models.

Where Pith is reading between the lines

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

The same foundation could help researchers quickly prototype numerical checks for new quantum models before investing in specialized code.
Readers might combine these basic tools with domain-specific quantum symmetries to reduce matrix size before decomposition.
The review suggests that teaching these routines early lowers the barrier for students entering computational quantum work.

Load-bearing premise

Readers need a review of these standard methods to access and apply computational linear algebra effectively to quantum problems.

What would settle it

A demonstration that typical quantum eigenvalue problems can be solved to the same accuracy and speed using only analytic pen-and-paper techniques, without any numerical matrix decompositions, would remove the need for the computational foundation the paper provides.

Figures

Figures reproduced from arXiv: 2604.25625 by Aaron Dayton, Kiana Gallagher, Sarah E. Huber, Thomas E. Baker.

**Figure 1.** Figure 1: FIG. 1: Visual representations of matrices: (a) an identity matrix, (b) an upper triangular matrix, (c) a lower view at source ↗

**Figure 2.** Figure 2: FIG. 2: A spin-1/2 XXZ Heisenberg model with 10 sites view at source ↗

**Figure 3.** Figure 3: FIG. 3: The Hamiltonian of a 10-site spin-half view at source ↗

read the original abstract

Making new methods for quantum problems often relies on using basic operations in linear algebra. Often these routines are hidden behind well-known libraries that have been optimized over decades. Attempting to improve on those basic routines would be highly time-consuming. We aim in this article to review those basic routines and provide a knowledge foundation for how to perform basic operations on a computer that would be inaccessible with pen and paper. Elementary details on the solutions to linear algebra problems and computational complexity are reviewed. The focus is on solving eigenvalue problems for quantum systems, but the discussion is generic to many other applications. Common matrix forms relevant to quantum systems and their solution strategies are covered. The discussion extends to computational numerical methods for which the most efficient functions exist in freely available libraries. These include eigenvalue, Schur, QR, LU, LDL, Cholesky, and singular value decompositions. The algorithms for obtaining some of these decompositions are discussed, with focus being placed on those used in modern libraries.

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

A basic review of standard dense linear algebra decompositions for quantum eigenvalue problems that adds no new results and largely skips the sparsity and scaling realities of actual quantum Hamiltonians.

read the letter

This paper reviews common linear algebra routines such as QR, LU, Cholesky, SVD, Schur, and eigenvalue decompositions, along with their algorithms and complexities, and points to free libraries for quantum applications. The discussion stays generic while noting some matrix forms that appear in quantum systems. It aims to give readers a foundation for doing these operations on a computer rather than by hand. That is the entire contribution. Nothing in it is new; it organizes existing textbook material into one place with a quantum tilt. The writing is clear enough on the dense case and the library pointers are practical for beginners. The soft spot is the mismatch with real quantum work. Quantum Hamiltonians are almost always sparse and exponentially large, so dense O(n^3) methods stop being usable past small sizes. The paper does not spend meaningful time on iterative sparse eigensolvers like Lanczos or Arnoldi, nor on storage formats or scaling limits that matter once particle number grows. That leaves the claimed foundation thinner than advertised for the target domain. This is for graduate students or early-career researchers who want a compact refresher on the numerical building blocks before writing code. Experienced people in the area will not learn anything they do not already know. It does not rise to the level that deserves peer review as a research article; a journal might still consider it as a short tutorial if the authors add more on sparse methods and concrete quantum examples.

Referee Report

1 major / 2 minor

Summary. The manuscript reviews standard dense linear algebra routines (eigenvalue, Schur, QR, LU, LDL, Cholesky, and SVD decompositions) and their algorithms, with emphasis on computational complexity and applicability to eigenvalue problems in quantum systems. It positions these methods as a computer-based foundation for operations inaccessible by hand calculation, while noting that the discussion is generic across applications.

Significance. If the coverage is accurate and complete, the paper could function as a concise educational reference for students and researchers entering computational quantum mechanics who need to understand the numerical linear algebra underlying common libraries. It explicitly credits the decades of optimization in existing packages and avoids reinventing core algorithms.

major comments (1)

[Abstract, §1] Abstract and §1: the central claim that these dense decompositions provide a usable foundation for quantum eigenvalue problems is undercut by the absence of any treatment of sparsity, iterative eigensolvers (Lanczos/Arnoldi), or scaling limits. Quantum Hamiltonians are almost always sparse with dimension exponential in particle number; O(n³) dense methods become unusable beyond modest sizes, yet no explicit discussion of sparse formats or when dense routines cease to apply appears.

minor comments (2)

[Abstract] The abstract states that 'common matrix forms relevant to quantum systems' are covered, but without section numbers or examples (e.g., Hermitian, positive-definite, or block-structured cases) it is unclear how much domain-specific adaptation is actually provided.
Computational complexity statements should be accompanied by explicit big-O notation and references to standard texts (e.g., Golub & Van Loan) for each decomposition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments. We address the major comment below.

read point-by-point responses

Referee: [Abstract, §1] Abstract and §1: the central claim that these dense decompositions provide a usable foundation for quantum eigenvalue problems is undercut by the absence of any treatment of sparsity, iterative eigensolvers (Lanczos/Arnoldi), or scaling limits. Quantum Hamiltonians are almost always sparse with dimension exponential in particle number; O(n³) dense methods become unusable beyond modest sizes, yet no explicit discussion of sparse formats or when dense routines cease to apply appears.

Authors: We thank the referee for highlighting this point. The manuscript is intentionally scoped to review foundational dense linear algebra routines (eigenvalue, Schur, QR, LU, LDL, Cholesky, and SVD decompositions) and their algorithms, which remain directly applicable to quantum problems of modest Hilbert-space dimension and serve as the computational primitives underlying many libraries. We agree that the absence of any mention of sparsity, iterative solvers such as Lanczos/Arnoldi, or explicit scaling limits weakens the framing of these methods as a “usable foundation” for the broader class of quantum eigenvalue problems. To correct this, we will revise the abstract and §1 to state the intended scope, note the O(n³) complexity, and briefly indicate when dense methods cease to apply, thereby directing readers to sparse-matrix and iterative techniques for larger systems. revision: partial

Circularity Check

0 steps flagged

No circularity: review of standard linear algebra methods

full rationale

The paper is a review of known linear algebra routines (eigenvalue, Schur, QR, LU, Cholesky, SVD decompositions) and their computational complexity, aimed at providing a foundation for quantum eigenvalue problems. No new derivations, predictions, or first-principles results are claimed that could reduce to inputs by construction. The text explicitly positions itself as reviewing existing library methods without introducing fitted parameters, self-definitional claims, or load-bearing self-citations. The discussion is generic and explanatory, remaining self-contained against external benchmarks of standard numerical linear algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper with no new derivations, so the ledger contains no free parameters, axioms, or invented entities beyond standard background knowledge of linear algebra.

pith-pipeline@v0.9.0 · 5465 in / 978 out tokens · 54005 ms · 2026-05-07T13:47:06.402340+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

49 extracted references · 2 canonical work pages

[1]

There are a few gen- eral properties of any Hamiltonian in quantum mechan- ics

Hamiltonian of a Spin Matrix A Hamiltonian in quantum mechanics encapsulates the behavior of the system in question. There are a few gen- eral properties of any Hamiltonian in quantum mechan- ics. For what we consider, the Hamiltonian is Hermitian. The Hamiltonian describing a 10-site spin-1/2 XXZ Heisenberg model in the absence of an external magnetic fi...
[2]

Matrix partition- ing plays a key role in some algorithms, often towards improving efficiency

Block Matrices Square and rectangular matrices can be partitioned into block matrices arbitrarily [25]. Matrix partition- ing plays a key role in some algorithms, often towards improving efficiency. Fig. 3 exhibits block structure. Be- low is an example of how some3×3matrix might be partitioned. ˆH=   a b c d e f g h i j k l m n o p   (33) ˆH= (...
[3]

A banded matrix has upper and lower bandwidths according to aij = 0∀(j >i+q)∨(i>j+p)(35) whereqis the upper bandwidth andpis the lower band- width [21]

Banded Matrices A banded matrix is a sparse matrix with non-zero en- tries only near the main diagonal. A banded matrix has upper and lower bandwidths according to aij = 0∀(j >i+q)∨(i>j+p)(35) whereqis the upper bandwidth andpis the lower band- width [21]. Banded matrix structure can be leveraged in computational linear algebra problems for efficient com-...
[4]

Any alternative storage method requires modifications to linear algebra programs such that the benefits may be reaped

Storage Formats Sparsity can be leveraged such that less memory is con- sumed through clever storage methods. Any alternative storage method requires modifications to linear algebra programs such that the benefits may be reaped. If done well, these methods will often cost fewer operations. Themostobviousformatistostoreelementsasalistof 3-tuples where each...
[5]

Notice that all unitary matrices, which have unit de- terminant and a unitary inverse, must then have a con- dition number of 1. Forlargeconditionnumbers, smallperturbations(from rounding, numerical errors, problem formulation approx- imations, etc.) in the input vector which the matrix is applied to can result in relatively large deviations in the result...
[6]

Unshifted QR Algorithm The basic unshifted QR algorithm is best encapsulated by Ref. 29 in the following steps: Algorithm 2Unshifted QR method 1:Initialize ˆA1 = ˆA 2:whileA k is not in (approximate) upper triangular form do 3:Perform the decomposition ˆAk = ˆQk ˆRk 4:Take ˆAk+1 = ˆRk ˆQk 5:end while 6:return the final ˆAk The essential idea of this algor...
[7]

It can be shown that finding the QR decomposition of a Hessenberg ma- trix can be done inO(N2)rather thanO(N 3)[29]

Utilizing Hessenberg Form This brings us to why so much attention has been fo- cussed on Hessenberg form matrices. It can be shown that finding the QR decomposition of a Hessenberg ma- trix can be done inO(N2)rather thanO(N 3)[29]. Addi- tionally, inHessenbergformfeweriterationsoftheQRal- gorithm are required. This makes intuitive sense as Hes- senberg ma...
[8]

Shifted QR Algorithm The rate of convergence for the QR algorithm when applied to a Hessenberg matrix may be improved by in- corporating a shift at each step. Given matrixˆAin Hes- 15 senberg form, the steps of the shifted QR algorithm are as follows [21]: Algorithm 3Shifted QR method 1:Initialize ˆA1 = ˆA 2:whileA k is not in (approximate) upper triangul...
[9]

It is therefore necessary to define an extended algorithm capable of handling complex values

Double Shifted QR Algorithm In the case of complex numbers along the diagonal of the Hessenberg form matrix, the single shift algorithm will not converge. It is therefore necessary to define an extended algorithm capable of handling complex values. This can be done by introducing a second shift byσkIat each step. The double shifted QR algorithm takes a He...

2021
[10]

Edward Anderson, Zhaojun Bai, Christian Bischof, L Su- san Blackford, James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum, Sven Hammarling, Alan McKenney,et al., LAPACK users’ guide(SIAM, 1999)

1999
[11]

Cosmological baryon and lepto n number in the presence of electroweak fermion number violation,

Richard Cleve, Robin Kothari, Dominic W. Berry, Andrew M. Childs and Rolando D. Somma, “Simu- lating hamiltonian dynamics with a truncated Taylor Series,” Phys. Rev. Lett. 114 (2015), 10.1103/Phys- RevLett.114.090502

work page doi:10.1103/phys- 2015
[12]

John S Townsend,A modern approach to quantum me- chanics(University Science Books, 2000)

2000
[13]

Diprima William E

Richard C. Diprima William E. Boyce and Douglas B. 19 Meade, Elementary Differential Equations and Boundary ValueProblems-11thEdition(JohnWiley&Sons, 2017) p. 303

2017
[14]

Méthodes de calcul avec réseaux de tenseurs en physique,

Thomas E Baker, Samuel Desrosiers, Maxime Trem- blay, and Martin P Thompson, “Méthodes de calcul avec réseaux de tenseurs en physique,” Canadian Journal of Physics99, 4 (2021); “Basic tensor network computa- tions in physics,” arXiv:1911.11566, p. 20 (2019)

work page arXiv 2021
[15]

Goodrich and Roberto TamassiaAlgorithm Design - Foundations, Analysis, and Internet Examples - First Edition(John Wiley & Sons, 2002) pp

Michael T. Goodrich and Roberto TamassiaAlgorithm Design - Foundations, Analysis, and Internet Examples - First Edition(John Wiley & Sons, 2002) pp. 13-20

2002
[16]

Chapra and Raymond P

Steven C. Chapra and Raymond P. Canale,Numerical Methods for Engineers - Eighth Edition(McGraw Hill,
[17]

180–183, 249–274

pp. 180–183, 249–274
[18]

Trefethen and David Bau III,Numerical Linear Algebra(Society for Industrial and Applied Mathematics,

Lloyd N. Trefethen and David Bau III,Numerical Linear Algebra(Society for Industrial and Applied Mathematics,
[19]

34, 94–95, 172–177, 187–188

pp. 34, 94–95, 172–177, 187–188
[20]

Shafarevich and Alexey O

Igor R. Shafarevich and Alexey O. RemizovLinear Alge- bra and Geometry(Springer Science & Business Media,
[21]

Application of the Q-Less QR Factorization to Resolve Sparse Linear Over-Constraints

Cho, Jeong-Rae and Cho, Keunhee and Yoon, Hyejin and Rhee, Dong Sop and Lee, Jin Ho, "Application of the Q-Less QR Factorization to Resolve Sparse Linear Over-Constraints" Applied Sciences,15, 13059 (2025)

2025
[22]

A high-performance ma- trix–matrix multiplication methodology for cpu and gpu architectures,

Iosif Mporas Vasilios Kelefouras, Angeliki Kritikakou, and Vasilios Kolonias, “A high-performance ma- trix–matrix multiplication methodology for cpu and gpu architectures,” The Journal of Supercomputing72, 804–844 (2016)

2016
[23]

An overview of cache optimization techniques and cache-aware numeri- cal algorithms,

Markus Kowarschik and Christian Weiß, “An overview of cache optimization techniques and cache-aware numeri- cal algorithms,” Algorithms for memory hierarchies: ad- vanced lectures, 213–232 (2003)

2003
[24]

Minimizing devel- opment and maintenance costs in supporting persistently optimized blas,

R Clint Whaley and Antoine Petitet, “Minimizing devel- opment and maintenance costs in supporting persistently optimized blas,” Software: Practice and Experience 35, 101–121 (2005)

2005
[25]

GEMM- based level 3 BLAS: High-performance model implemen- tations and performance evaluation benchmark,

Bo Kågström, Per Ling, and Charles van Loan, “GEMM- based level 3 BLAS: High-performance model implemen- tations and performance evaluation benchmark,” ACM Trans. Math. Softw. 24, 268–302 (1998)

1998
[26]

Robert van de Geijn and Kazushige Goto, BLAS (Basic Linear Algebra Subprograms), edited by David Padua (Springer US, 2011) pp. 157–164

2011
[27]

Ascher and Chen Greif,A First Course on Numerical Methods(Society for Industrial and Applied Mathematics, 2011) pp

Uri M. Ascher and Chen Greif,A First Course on Numerical Methods(Society for Industrial and Applied Mathematics, 2011) pp. 117–121

2011
[28]

Conjugate gradient methods for Toeplitz systems

Chan, Raymond H.; Ng, Michael K. Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38 (1996), no. 3, 427–482

1996
[29]

Iterative methods for Toeplitz sys- tems.Numerical Mathematics and Scientific Computa- tion.(Oxford University Press, New York, 2004)

Ng, Michael K. Iterative methods for Toeplitz sys- tems.Numerical Mathematics and Scientific Computa- tion.(Oxford University Press, New York, 2004)

2004
[30]

Garoni, Carlo; Serra-Capizzano, Stefano.Generalized lo- cally Toeplitz sequences: theory and applications. Vol. I. (Springer, Cham, 2017)

2017
[31]

Garoni, Carlo; Serra-Capizzano, Stefano.Generalized lo- cally Toeplitz sequences: theory and applications. Vol. II. (Springer, Cham, 2018)

2018
[32]

Golub and Charles F

Gene H. Golub and Charles F. van Loan, Matrix Com- putations - Third Edition (The Johns Hopkins Univer- sity Press, 1996) pp. 16, 80–82, 89–98, 135–143, 223–233, 352–361, 442–449, 470–488

1996
[33]

LeslieHogben,Elementary Linear Algebra - First Edition (West Publishing Company, 1987) p. 130

1987
[34]

Horn,Matrix Mathematics - A Second Course in Linear Algebra(Cam- bridge University Press, 2017)

Stephan Ramon Garcia and Roger A. Horn,Matrix Mathematics - A Second Course in Linear Algebra(Cam- bridge University Press, 2017)

2017
[35]

Hall,Lie Groups, Lie Algebras, and Represen- tations - An Elementary Introduction - Second Edition (Springer US, 2015) p

Brian C. Hall,Lie Groups, Lie Algebras, and Represen- tations - An Elementary Introduction - Second Edition (Springer US, 2015) p. 27

2015
[36]

30–31, 237–238

Howard Anton and Chris Rorres,Elementary Linear Al- gebra - Applications version - 11th Edition(John Wiley & Sons, 2014) pp. 30–31, 237–238

2014
[37]

Strassen’s algorithm reloaded,

Greg Henry Jianyu Huang, Tyler Smith and Robert van de Geijn, “Strassen’s algorithm reloaded,” (IEEE Press,
[38]

Yousef Saad,Iterative Methods for Sparse Linear Sys- tems - Second Edition(Society for Industrial and Applied Mathematics, 2003) pp. 92–95

2003
[39]

Watkins,Fundamentals of Matrix Computations- Second Edition(John Wiley & Sons,

David S. Watkins,Fundamentals of Matrix Computations- Second Edition(John Wiley & Sons,
[40]

14, 17, 18

William Ford,Numerical Linear Algebra with Applica- tions - Using MATLAB(Academic Press, 2015) Chap. 14, 17, 18

2015
[41]

Research and imple- mentation of SVD in machine learning,

Yongchang Wang and Ligu Zhu, “Research and imple- mentation of SVD in machine learning,” (IEEE Press,
[42]

Mary L Boas,Mathematical methods in the physical sci- ences(Wiley, 2006)

2006
[43]

On the fast reduction of a quasi-separable matrix to Hes- senberg and tridiagonal forms,

Israel Gohberg Yuli Eidelman and Luca Gemignani, “On the fast reduction of a quasi-separable matrix to Hes- senberg and tridiagonal forms,”Linear Algebra and its Applications420, 86–101 (2007)

2007
[44]

The mul- tishift QR algorithm. part i: Maintaining well-focused shifts and level 3 performance,

Ralph Byers Karen Braman and Roy Mathias, “The mul- tishift QR algorithm. part i: Maintaining well-focused shifts and level 3 performance,” SIAM Journal on Ma- trix Analysis and Applications23, 929–947 (2002)

2002
[45]

The mul- tishift QR algorithm. part ii: Aggressive early deflation,

Ralph Byers Karen Braman and Roy Mathias, “The mul- tishift QR algorithm. part ii: Aggressive early deflation,” SIAM Journal on Matrix Analysis and Applications23, 948–973 (2002)

2002
[46]

Matrix Anal

Bogoya, Manuel; Grudsky, Sergei M.; Serra-Capizzano, StefanoFast non-Hermitian Toeplitz eigenvalue compu- tations, joining matrixless algorithms and FDE approxi- mation matrices.SIAM J. Matrix Anal. Appl.45(2024), no. 1, 284–305

2024
[47]

Matrix Anal

Serra-Capizzano, Stefano.Jordan canonical form of the Google matrix: a potential contribution to the PageRank computation.SIAM J. Matrix Anal. Appl.27(2005), no. 2, 305–312

2005
[48]

The PageR- ank vector: properties, computation, approximation, and acceleration

Brezinski, Claude; Redivo-Zaglia, Michela "The PageR- ank vector: properties, computation, approximation, and acceleration". SIAMJ.MatrixAnal.Appl.28(2006), no. 2, 551–575

2006
[49]

TENPACK,

Thomas E. Baker, “TENPACK,” https://github.com/bakerte/TensorPACK.jl (2020)

2020