Convergence rates of Sum-of-Hermitian-Squares Hierarchies for the Pauli algebra

Ali Almasi; Cambyse Rouz\'e; D\'avid Bug\'ar; Peter Brown

arxiv: 2606.04940 · v1 · pith:XOYKD7JLnew · submitted 2026-06-03 · 🪐 quant-ph · math.OC

Convergence rates of Sum-of-Hermitian-Squares Hierarchies for the Pauli algebra

Ali Almasi , D\'avid Bug\'ar , Cambyse Rouz\'e , Peter Brown This is my paper

Pith reviewed 2026-06-28 05:57 UTC · model grok-4.3

classification 🪐 quant-ph math.OC

keywords convergence ratessum-of-hermitian-squarespauli algebranoncommutative polynomial optimizationkrawtchouk polynomialsmoment relaxationsquantum optimizationground state energy

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The pith

Convergence rates of Sum-of-Hermitian-Squares relaxations for Pauli algebra problems are bounded by the smallest roots of Krawtchouk polynomials.

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

The paper develops explicit bounds on how fast these moment relaxations approach the true optimum for noncommutative polynomial problems built from the Pauli algebra of n-qubit systems. This supplies the first quantitative guarantee on accuracy for methods already used to approximate ground-state energies and other quantum optimization tasks. A reader cares because the bounds turn previously heuristic approximations into ones whose error can be estimated in advance from known polynomial properties. The work focuses on the Pauli case because it directly models standard qubit Hamiltonians.

Core claim

For noncommutative polynomial optimization problems generated from the Pauli algebra, the rate of convergence of the Sum-of-Hermitian-Squares hierarchy can be bounded in terms of the smallest roots of a family of orthogonal polynomials known as Krawtchouk polynomials.

What carries the argument

The smallest roots of Krawtchouk polynomials, used to bound the gap between the hierarchy value at finite level and the true optimum.

If this is right

Error estimates for ground-state energy approximations of n-qubit systems become computable from tabulated Krawtchouk roots.
The hierarchy level needed for a target accuracy can be chosen in advance rather than by trial and error.
Quantitative rates now exist for the first time for any noncommutative polynomial problem whose algebra is the Pauli algebra.
The same bounding technique applies uniformly across all problems generated from this algebra, including many-body Hamiltonians.

Where Pith is reading between the lines

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

The Krawtchouk-root bound may let practitioners decide when to stop increasing the relaxation level for a given qubit number.
Analogous orthogonal-polynomial bounds could be sought for other finite-dimensional algebras that appear in quantum information.
The explicit rates open the possibility of comparing the practical cost of these relaxations against other approximation methods on the same Pauli problems.

Load-bearing premise

The problems being relaxed are exactly those whose variables and constraints come from the Pauli algebra on n qubits.

What would settle it

A concrete n-qubit Pauli optimization instance where the observed convergence gap shrinks slower than the rate predicted by the smallest Krawtchouk root at the corresponding hierarchy level.

read the original abstract

Moment/Sum-of-Hermitian-Squares relaxations for noncommutative polynomial optimization problems have become an important tool for analyzing problems within quantum theory. Despite their widespread success, little is known about their rate of convergence and, consequently, their accuracy. In this work, we develop explicit convergence rates for relaxations of noncommutative polynomial optimization problems generated from the Pauli algebra -- covering applications to the ground state energy problem for n-qubit systems. In particular, we show that the rate of convergence can be bounded in terms of the smallest roots of a family of orthogonal polynomials known as Krawtchouk polynomials. Our result represents the first quantitative analysis of the rate of convergence for relaxations of noncommutative polynomial optimization 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 paper supplies the first explicit convergence rates for SoHS hierarchies on the Pauli algebra using Krawtchouk polynomial roots.

read the letter

The punchline is that this paper gives the first quantitative convergence rates for the sum-of-Hermitian-squares hierarchy when the underlying algebra is the Pauli algebra on n qubits. They bound the rate using the smallest roots of Krawtchouk polynomials.

This is new because prior work on these moment relaxations in quantum information had mostly qualitative convergence results or none at all for the noncommutative case. The authors apply it directly to ground-state energy certification, which is a common use case.

What they do well is make the connection to classical orthogonal polynomials explicit and position the result as filling a gap in the literature on noncommutative polynomial optimization. The abstract suggests the bound comes from properties of these polynomials rather than any ad-hoc fitting.

The main soft spot is that without the full derivation visible in the abstract, it's hard to judge how sharp the bounds are or whether the constants make the result practical for moderate n. The paper will stand or fall on whether the error analysis holds up under scrutiny and if they provide any numerical checks.

This paper is aimed at researchers in quantum information theory who use SDP relaxations for qubit systems. A reader interested in convergence theory for hierarchies will get value from the explicit rates, even if they end up being conservative.

It deserves a serious referee because the claim is concrete and addresses a real open question in the area.

Referee Report

0 major / 2 minor

Summary. The paper claims to derive explicit convergence rates for Sum-of-Hermitian-Squares (SoHS) moment relaxations of noncommutative polynomial optimization problems over the Pauli algebra on n qubits. The central result is that these rates are bounded in terms of the smallest roots of Krawtchouk polynomials, providing the first quantitative error analysis for such hierarchies and their application to ground-state energy problems.

Significance. If the derivation holds, the result supplies the first explicit, non-asymptotic convergence bounds for SoHS hierarchies in a noncommutative quantum setting. The explicit link to the smallest roots of Krawtchouk polynomials is a concrete strength, as it yields computable, parameter-free bounds that can be compared directly with classical commutative hierarchies and used to certify approximation quality for n-qubit ground states.

minor comments (2)

The abstract states the main theorem but does not display the explicit bound or the precise statement involving the Krawtchouk roots; adding a displayed equation or theorem number in the introduction would improve readability.
Notation for the SoHS hierarchy levels (e.g., the degree or moment order) should be defined once at the beginning and used consistently throughout.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance as the first explicit non-asymptotic convergence bounds for SoHS hierarchies in the noncommutative setting, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation claims explicit convergence rates for SoHS hierarchies on the Pauli algebra, bounded via smallest roots of Krawtchouk polynomials. This rests on standard properties of orthogonal polynomials (external to the paper) rather than any self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The abstract positions the result as the first quantitative analysis without reducing the bound to the problem inputs by construction. No equations or steps in the provided text exhibit the enumerated circular patterns; the central claim remains independent of its own fitted values or prior author work invoked as uniqueness.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5664 in / 910 out tokens · 19886 ms · 2026-06-28T05:57:11.067249+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

73 extracted references · 20 linked inside Pith

[1]

Quantitative semidefinite certificates for ground- state energies of pauli hamiltonians

Igor Klep, Nando Leijenhorst, and Victor Magron. “Quantitative semidefinite certificates for ground- state energies of pauli hamiltonians” (2026). arXiv:2605.29959

Pith/arXiv arXiv 2026
[2]

Quantum Phase Transitions

Subir Sachdev. “Quantum Phase Transitions”. Cambridge University Press. Cambridge (2011). 2nd edition

2011
[3]

Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory

Attila Szabo and Neil S. Ostlund. “Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory”. Courier Corporation. (1996)

1996
[4]

Classical and Quantum Computation

Alexei Kitaev, Alexander Shen, and Mikhail N. Vyalyi. “Classical and Quantum Computation”. Vol- ume 47 of Graduate Studies in Mathematics. American Mathematical Society. Providence, Rhode Is- land (2002)

2002
[5]

The Complexity of the Local Hamiltonian Problem

Julia Kempe, Alexei Kitaev, and Oded Regev. “The Complexity of the Local Hamiltonian Problem”. SIAM Journal on Computing35, 1070–1097 (2006). arXiv:quant-ph/0406180

Pith/arXiv arXiv 2006
[6]

Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems

Alastair Kay. “Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems”. Physical Review A76, 030307 (2007). arXiv:0704.3142

Pith/arXiv arXiv 2007
[7]

The complexity of quantum spin systems on a two- dimensional square lattice

Roberto Oliveira and Barbara M. Terhal. “The complexity of quantum spin systems on a two- dimensional square lattice”. Quantum Info. Comput.8, 900–924 (2008). arXiv:quant-ph/0504050

Pith/arXiv arXiv 2008
[8]

The Power of Quantum Systems on a Line

Dorit Aharonov, Daniel Gottesman, Sandy Irani, and Julia Kempe. “The Power of Quantum Systems on a Line”. Communications in Mathematical Physics287, 41–65 (2009). arXiv:0705.4077

Pith/arXiv arXiv 2009
[9]

Two-dimensional local Hamiltonian problem with area laws is QMA-complete

Yichen Huang. “Two-dimensional local Hamiltonian problem with area laws is QMA-complete”. Jour- nal of Computational Physics443, 110534 (2021). arXiv:1411.6614

arXiv 2021
[10]

Complexity Classification of Local Hamiltonian Problems

Toby Cubitt and Ashley Montanaro. “Complexity Classification of Local Hamiltonian Problems”. SIAM Journal on Computing45, 268–316 (2016). arXiv:1311.3161

Pith/arXiv arXiv 2016
[11]

The complexity of antiferromagnetic interactions and 2D lattices

Stephen Piddock and Ashley Montanaro. “The complexity of antiferromagnetic interactions and 2D lattices”. Quantum Info. Comput.17, 636–672 (2017). arXiv:1506.04014

Pith/arXiv arXiv 2017
[12]

Almost Optimal Classical Approximation Algorithms for a Quan- tum Generalization of Max-Cut

Sevag Gharibian and Ojas Parekh. “Almost Optimal Classical Approximation Algorithms for a Quan- tum Generalization of Max-Cut”. In Dimitris Achlioptas and L´aszl´o A. V´egh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Volume 145 of Leibniz International Proceedings in Informatics (...

arXiv 2019
[13]

Beyond Product State Approximations for a Quan- tum Analogue of Max Cut

Anurag Anshu, David Gosset, and Karen Morenz. “Beyond Product State Approximations for a Quan- tum Analogue of Max Cut”. In Steven T. Flammia, editor, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Volume 158 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1–7:15. Dagstuhl, Germany (2...

arXiv 2020
[14]

Approximation algorithms for quan- tum many-body problems

Sergey Bravyi, David Gosset, Robert K¨onig, and Kristan Temme. “Approximation algorithms for quan- tum many-body problems”. Journal of Mathematical Physics60, 032203 (2019). arXiv:1808.01734

arXiv 2019
[15]

An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem

Sean Hallgren, Eunou Lee, and Ojas Parekh. “An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem”. In Jarosław Byrka and Raghu Meka, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Volume 176 of Leibniz International Proceedings in Informatics (LIPIcs), pages 59:1–59:18. ...

2020
[16]

Beating Random Assignment for Approximating Quantum 2- Local Hamiltonian Problems

Ojas Parekh and Kevin Thompson. “Beating Random Assignment for Approximating Quantum 2- Local Hamiltonian Problems”. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms (ESA 2021). Volume 204 of Leibniz International Pro- ceedings in Informatics (LIPIcs), pages 74:1–74:18. Dagstuhl, Germany (2021). Schl...

arXiv 2021
[17]

Application of the Level-2 Quantum Lasserre Hierarchy in Quan- tum Approximation Algorithms

Ojas Parekh and Kevin Thompson. “Application of the Level-2 Quantum Lasserre Hierarchy in Quan- tum Approximation Algorithms”. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 102...

arXiv 2021
[18]

An Optimal Product-State Approximation for 2-Local Quantum Hamiltonians with Positive Terms

Ojas Parekh and Kevin Thompson. “An Optimal Product-State Approximation for 2-Local Quantum Hamiltonians with Positive Terms” (2022). arXiv:2206.08342

arXiv 2022
[19]

An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut

Jun Takahashi, Chaithanya Rayudu, Cunlu Zhou, Robbie King, Kevin Thompson, and Ojas Parekh. “An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut” (2023). arXiv:2307.15688

Pith/arXiv arXiv 2023
[20]

An Improved Approximation Algorithm for Quantum Max-Cut on Triangle-Free Graphs

Robbie King. “An Improved Approximation Algorithm for Quantum Max-Cut on Triangle-Free Graphs”. Quantum7, 1180 (2023). arXiv:2209.02589

arXiv 2023
[21]

Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators

Adam Bene Watts, Anirban Chowdhury, Aidan Epperly, J. William Helton, and Igor Klep. “Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators”. Quantum 8, 1352 (2024). arXiv:2307.15661

arXiv 2024
[22]

Second order cone relaxations for quantum Max Cut

Felix Huber, Kevin Thompson, and Sevag Gharibian. “Second order cone relaxations for quantum Max Cut” (2024). arXiv:2411.04120

arXiv 2024
[23]

Improved approximation ratios for the Quan- tum Max-Cut problem on general, triangle-free and bipartite graphs

Sander Gribling, Lennart Sinjorgo, and Renata Sotirov. “Improved approximation ratios for the Quan- tum Max-Cut problem on general, triangle-free and bipartite graphs” (2025). arXiv:2504.11120

arXiv 2025
[24]

Improved Approximation Algorithms for the EPR Hamiltonian

Nathan Ju and Ansh Nagda. “Improved Approximation Algorithms for the EPR Hamiltonian”. In Alina Ene and Eshan Chattopadhyay, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Volume 353 of Leibniz In- ternational Proceedings in Informatics (LIPIcs), pages 24:1–24:9. Dagstuhl, Germany (20...

arXiv 2025
[25]

Improved Algorithms for Quantum MaxCut via Partially Entangled Matchings

Anuj Apte, Eunou Lee, Kunal Marwaha, Ojas Parekh, and James Sud. “Improved Algorithms for Quantum MaxCut via Partially Entangled Matchings”. In Anne Benoit, Haim Kaplan, Sebastian Wild, and Grzegorz Herman, editors, 33rd Annual European Symposium on Algorithms (ESA 2025). Vol- ume 351 of Leibniz International Proceedings in Informatics (LIPIcs), pages 101...

arXiv 2025
[26]

A Lov ´asz theta lower bound on Quantum Max Cut

Felix Huber. “A Lov ´asz theta lower bound on Quantum Max Cut” (2025). arXiv:2512.20326

Pith/arXiv arXiv 2025
[27]

Projected entangled states: Properties and applications

Frank Verstraete, Michael Wolf, David P ´erez-Garc´ıa, and Juan I. Cirac. “Projected entangled states: Properties and applications”. International Journal of Modern Physics B20, 5142–5153 (2006)

2006
[28]

Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems

Frank Verstraete, Valentin Murg, and Juan I. Cirac. “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems”. Advances in Physics57, 143–224 (2008). arXiv:0907.2796

Pith/arXiv arXiv 2008
[29]

Quantum Monte Carlo and Related Approaches

Brian M. Austin, Dmitry Yu. Zubarev, and William A. Lester. “Quantum Monte Carlo and Related Approaches”. Chemical Reviews112, 263–288 (2012)

2012
[30]

From architectures to applications: A review of neural quantum states

Hannah Lange, Anka Van De Walle, Atiye Abedinnia, and Annabelle Bohrdt. “From architectures to applications: A review of neural quantum states”. Quantum Science and Technology9, 040501 (2024). arXiv:2402.09402

arXiv 2024
[31]

A convergent hierarchy of semidefinite pro- grams characterizing the set of quantum correlations

Miguel Navascu ´es, Stefano Pironio, and Antonio Ac´ın. “A convergent hierarchy of semidefinite pro- grams characterizing the set of quantum correlations”. New Journal of Physics10, 073013 (2008). arXiv:0803.4290

Pith/arXiv arXiv 2008
[32]

Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables

S. Pironio, M. Navascu ´es, and A. Ac´ın. “Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables”. SIAM Journal on Optimization20, 2157–2180 (2010). arXiv:0903.4368

Pith/arXiv arXiv 2010
[33]

Global Optimization with Polynomials and the Problem of Moments

Jean B. Lasserre. “Global Optimization with Polynomials and the Problem of Moments”. SIAM Journal on Optimization11, 796–817 (2001)

2001
[34]

Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robust- ness and Optimization

Pablo A. Parrilo. “Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robust- ness and Optimization”. PhD thesis. California Institute of Technology. Pasadena, California (2000)

2000
[35]

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Monique Laurent. “Sums of Squares, Moment Matrices and Optimization Over Polynomials”. In Mihai Putinar and Seth Sullivant, editors, Emerging Applications of Algebraic Geometry. Volume 149, pages 157–270. Springer New York, New York, NY (2009)

2009
[36]

A positivstellensatz for non-commutative polynomials

J. Helton and Scott McCullough. “A positivstellensatz for non-commutative polynomials”. Transac- tions of the American Mathematical Society356, 3721–3737 (2004)

2004
[37]

An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs

Jean B. Lasserre. “An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs”. In Gerhard Goos, Juris Hartmanis, Jan Van Leeuwen, Karen Aardal, and Bert Gerards, editors, Integer Programming and Combinatorial Optimization. Volume 2081, pages 293–303. Springer Berlin Heidelberg, Berlin, Heidelberg (2001)

2081
[38]

Semidefinite pro- gramming relaxations for quantum correlations

Armin Tavakoli, Alejandro Pozas-Kerstjens, Peter Brown, and Mateus Ara ´ujo. “Semidefinite pro- gramming relaxations for quantum correlations”. Reviews of Modern Physics96, 045006 (2024). arXiv:2307.02551

arXiv 2024
[39]

Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm

Maho Nakata, Hiroshi Nakatsuji, Masahiro Ehara, Mitsuhiro Fukuda, Kazuhide Nakata, and Katsuki Fujisawa. “Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm”. The Journal of Chemical Physics114, 8282–8292 (2001)

2001
[40]

Lower Bounds for Ground States of Condensed Matter Systems

Tillmann Baumgratz and Martin B. Plenio. “Lower Bounds for Ground States of Condensed Matter Systems”. New Journal of Physics14, 023027 (2012). arXiv:1106.5275

Pith/arXiv arXiv 2012
[41]

Solving Condensed-Matter Ground-State Problems by Semidef- inite Relaxations

Thomas Barthel and Robert H ¨ubener. “Solving Condensed-Matter Ground-State Problems by Semidef- inite Relaxations”. Physical Review Letters108, 200404 (2012). arXiv:1106.4966

Pith/arXiv arXiv 2012
[42]

Quantum Many-body Bootstrap

Xizhi Han. “Quantum Many-body Bootstrap” (2020). arXiv:2006.06002

arXiv 2020
[43]

Quantum Many-Body Theory from a Solution of theN-representability Prob- lem

David A. Mazziotti. “Quantum Many-Body Theory from a Solution of theN-representability Prob- lem”. Physical Review Letters130, 153001 (2023). arXiv:2304.08570. 17

arXiv 2023
[44]

Certifying Ground-State Properties of Many-Body Systems

Jie Wang, Jacopo Surace, Ir ´en´ee Fr ´erot, Benoˆıt Legat, Marc-Olivier Renou, Victor Magron, and An- tonio Ac ´ın. “Certifying Ground-State Properties of Many-Body Systems”. Physical Review X14, 031006 (2024). arXiv:2310.05844

arXiv 2024
[45]

Scal- able Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polyno- mial Optimization

Jie Wang, David Jansen, Ir ´en´ee Frerot, Marc-Olivier Renou, Victor Magron, and Antonio Ac´ın. “Scal- able Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polyno- mial Optimization” (2026). arXiv:2604.01555

arXiv 2026
[46]

Product-state approximations to quantum ground states

Fernando G.S.L. Brandao and Aram W. Harrow. “Product-state approximations to quantum ground states”. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing. Pages 871–880. STOC ’13New York, NY, USA (2013). Association for Computing Machinery. arXiv:1310.0017

Pith/arXiv arXiv 2013
[47]

On the complexity of Putinar’s Positivstellensatz

Jiawang Nie and Markus Schweighofer. “On the complexity of Putinar’s Positivstellensatz”. Journal of Complexity23, 135–150 (2007). arXiv:0812.2657

Pith/arXiv arXiv 2007
[48]

Convergence rates of moment-sum-of-squares hierarchies for optimal control problems

Milan Korda, Didier Henrion, and Colin N. Jones. “Convergence rates of moment-sum-of-squares hierarchies for optimal control problems”. Systems & Control Letters100, 1–5 (2017). arXiv:1609.02762

Pith/arXiv arXiv 2017
[49]

Convergence rates of moment-sum-of-squares hierarchies for vol- ume approximation of semialgebraic sets

Milan Korda and Didier Henrion. “Convergence rates of moment-sum-of-squares hierarchies for vol- ume approximation of semialgebraic sets”. Optimization Letters12, 435–442 (2018). arXiv:1612.04146

Pith/arXiv arXiv 2018
[50]

The sum-of-squares hierarchy on the sphere and applications in quan- tum information theory

Kun Fang and Hamza Fawzi. “The sum-of-squares hierarchy on the sphere and applications in quan- tum information theory”. Mathematical Programming190, 331–360 (2021). arXiv:1908.05155

arXiv 2021
[51]

Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel–Darboux Kernel

Lucas Slot. “Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel–Darboux Kernel”. SIAM Journal on Optimization32, 2612–2635 (2022). arXiv:2111.04610

arXiv 2022
[52]

An effective version of Schm ¨udgen’s Positivstellensatz for the hypercube

Monique Laurent and Lucas Slot. “An effective version of Schm ¨udgen’s Positivstellensatz for the hypercube”. Optimization Letters17, 515–530 (2023). arXiv:2109.09528

arXiv 2023
[53]

On the effective Putinar’s Positivstellensatz and moment ap- proximation

Lorenzo Baldi and Bernard Mourrain. “On the effective Putinar’s Positivstellensatz and moment ap- proximation”. Mathematical Programming200, 71–103 (2023). arXiv:2111.11258

arXiv 2023
[54]

Degree Bounds for Putinar’s Positivstellensatz on the Hypercube

Lorenzo Baldi and Lucas Slot. “Degree Bounds for Putinar’s Positivstellensatz on the Hypercube”. SIAM Journal on Applied Algebra and Geometry8, 1–25 (2024). arXiv:2302.12558

arXiv 2024
[55]

Convergence rates for the moment- sos hierarchy

Corbinian Schlosser, Matteo Tacchi-B ´enard, and Alexey Lazarev. “Convergence rates for the moment- sos hierarchy”. Numerical Algebra, Control and Optimization16, 105–156 (2026). arXiv:2402.00436

arXiv 2026
[56]

Specialized Effective Positivstellens ¨atze for Improved Con- vergence Rates of the Moment-SOS Hierarchy

Corbinian Schlosser and Matteo Tacchi. “Specialized Effective Positivstellens ¨atze for Improved Con- vergence Rates of the Moment-SOS Hierarchy”. IEEE Control Systems Letters8, 2319–2324 (2024). arXiv:2403.05934

arXiv 2024
[57]

On Łojasiewicz inequalities and the effec- tive Putinar’s Positivstellensatz

Lorenzo Baldi, Bernard Mourrain, and Adam Parusi ´nski. “On Łojasiewicz inequalities and the effec- tive Putinar’s Positivstellensatz”. Journal of Algebra662, 741–767 (2025). arXiv:2212.09551

arXiv 2025
[58]

Convergence rates for sums-of- squares hierarchies with correlative sparsity

Milan Korda, Victor Magron, and Rodolfo R ´ıos-Zertuche. “Convergence rates for sums-of- squares hierarchies with correlative sparsity”. Mathematical Programming209, 435–473 (2025). arXiv:2303.14824

arXiv 2025
[59]

Convergence rates for polynomial optimization on set products

Victor Magron. “Convergence rates for polynomial optimization on set products” (2025). arXiv:2505.18580

Pith/arXiv arXiv 2025
[60]

Sum-of-squares hierarchies for binary polynomial optimization

Lucas Slot and Monique Laurent. “Sum-of-squares hierarchies for binary polynomial optimization”. Mathematical Programming197, 621–660 (2023). arXiv:2011.04027

arXiv 2023
[61]

An Overview of Convergence Rates for Sum of Squares Hierarchies in Polynomial Optimization

Monique Laurent and Lucas Slot. “An Overview of Convergence Rates for Sum of Squares Hierarchies in Polynomial Optimization”. In Shin’ichi Oishi, Hisashi Okamoto, and Ken Hayami, editors, Recent Developments in Industrial and Applied Mathematics. Pages 149–176. Singapore (2026). Springer Nature. arXiv:2408.04417. 18

arXiv 2026
[62]

Orthogonal Polynomials

G. Szeg ˝o. “Orthogonal Polynomials”. Volume 23 of Colloquium Publications. American Mathematical Society. Providence, Rhode Island (1939)

1939
[63]

Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces

Vladimir I. Levenshtein. “Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces”. IEEE Transactions on Information Theory41, 1303–1321 (1995)

1995
[64]

Sur une g ´en´eralisation des polyn ˆomes d’Hermite

Mykha ˘ılo Krawtchouk. “Sur une g ´en´eralisation des polyn ˆomes d’Hermite”. Comptes Rendus Heb- domadaires des S´eances de l’Acad´emie des Sciences189, 620–622 (1929). url:https://gallica.bnf. fr/ark:/12148/bpt6k3142j.f620

1929
[65]

Association schemes and coding theory

Philippe Delsarte and Vladimir I. Levenshtein. “Association schemes and coding theory”. IEEE Trans- actions on Information Theory44, 2477–2504 (1998)

1998
[66]

Algebraic Combinatorics

Eiichi Bannai, Etsuko Bannai, Tatsuro Ito, and Rie Tanaka. “Algebraic Combinatorics”. De Gruyter. (2021)

2021
[67]

Quantum Analog of the MacWilliams Identities for Classical Coding Theory

Peter Shor and Raymond Laflamme. “Quantum Analog of the MacWilliams Identities for Classical Coding Theory”. Physical Review Letters78, 1600–1602 (1997). arXiv:quant-ph/9610040

Pith/arXiv arXiv 1997
[68]

SDP bounds on quantum codes

Gerard Angl `es Munn ´e, Andrew Nemec, and Felix Huber. “SDP bounds on quantum codes” (2025). arXiv:2408.10323

arXiv 2025
[69]

The Christoffel–Darboux Kernel for Data Analysis

Jean Bernard Lasserre, Edouard Pauwels, and Mihai Putinar. “The Christoffel–Darboux Kernel for Data Analysis”. Cambridge University Press. (2022). 1st edition

2022
[70]

Noncommutative Christoffel-Darboux ker- nels

Serban Belinschi, Victor Magron, and Victor Vinnikov. “Noncommutative Christoffel-Darboux ker- nels”. Transactions of the American Mathematical Society376, 181–230 (2022). arXiv:2106.06212

arXiv 2022
[71]

Interacting electrons and quantum magnetism

Assa Auerbach. “Interacting electrons and quantum magnetism”. Graduate Texts in Contemporary Physics. Springer New York. New York, NY (1994)

1994
[72]

Exact semidefinite programming re- laxations with truncated moment matrix for binary polynomial optimization problems

Shinsaku Sakaue, Akiko Takeda, Sunyoung Kim, and Naoki Ito. “Exact semidefinite programming re- laxations with truncated moment matrix for binary polynomial optimization problems”. SIAM Journal on Optimization27, 565–582 (2017)

2017
[73]

“Algebra”

Serge Lang. “Algebra”. Volume 211 of Graduate Texts in Mathematics. Springer New York. New York, NY (2002). A Proof of Lemma 2 Proof.For an arbitraryi∈[n], considerP ◦ def=C⟨X i,Y i,Z i⟩/I◦, whereI ◦ is the two-sided ideal generated by polynomials X2 i −1,Y 2 i −1,Z 2 i −1,X iYi −iZ i,Y iZi −iX i,Z iXi −iY i. Using these relations, any polynomial inX i,Y ...

2002

[1] [1]

Quantitative semidefinite certificates for ground- state energies of pauli hamiltonians

Igor Klep, Nando Leijenhorst, and Victor Magron. “Quantitative semidefinite certificates for ground- state energies of pauli hamiltonians” (2026). arXiv:2605.29959

Pith/arXiv arXiv 2026

[2] [2]

Quantum Phase Transitions

Subir Sachdev. “Quantum Phase Transitions”. Cambridge University Press. Cambridge (2011). 2nd edition

2011

[3] [3]

Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory

Attila Szabo and Neil S. Ostlund. “Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory”. Courier Corporation. (1996)

1996

[4] [4]

Classical and Quantum Computation

Alexei Kitaev, Alexander Shen, and Mikhail N. Vyalyi. “Classical and Quantum Computation”. Vol- ume 47 of Graduate Studies in Mathematics. American Mathematical Society. Providence, Rhode Is- land (2002)

2002

[5] [5]

The Complexity of the Local Hamiltonian Problem

Julia Kempe, Alexei Kitaev, and Oded Regev. “The Complexity of the Local Hamiltonian Problem”. SIAM Journal on Computing35, 1070–1097 (2006). arXiv:quant-ph/0406180

Pith/arXiv arXiv 2006

[6] [6]

Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems

Alastair Kay. “Quantum-Merlin-Arthur-complete translationally invariant Hamiltonian problem and the complexity of finding ground-state energies in physical systems”. Physical Review A76, 030307 (2007). arXiv:0704.3142

Pith/arXiv arXiv 2007

[7] [7]

The complexity of quantum spin systems on a two- dimensional square lattice

Roberto Oliveira and Barbara M. Terhal. “The complexity of quantum spin systems on a two- dimensional square lattice”. Quantum Info. Comput.8, 900–924 (2008). arXiv:quant-ph/0504050

Pith/arXiv arXiv 2008

[8] [8]

The Power of Quantum Systems on a Line

Dorit Aharonov, Daniel Gottesman, Sandy Irani, and Julia Kempe. “The Power of Quantum Systems on a Line”. Communications in Mathematical Physics287, 41–65 (2009). arXiv:0705.4077

Pith/arXiv arXiv 2009

[9] [9]

Two-dimensional local Hamiltonian problem with area laws is QMA-complete

Yichen Huang. “Two-dimensional local Hamiltonian problem with area laws is QMA-complete”. Jour- nal of Computational Physics443, 110534 (2021). arXiv:1411.6614

arXiv 2021

[10] [10]

Complexity Classification of Local Hamiltonian Problems

Toby Cubitt and Ashley Montanaro. “Complexity Classification of Local Hamiltonian Problems”. SIAM Journal on Computing45, 268–316 (2016). arXiv:1311.3161

Pith/arXiv arXiv 2016

[11] [11]

The complexity of antiferromagnetic interactions and 2D lattices

Stephen Piddock and Ashley Montanaro. “The complexity of antiferromagnetic interactions and 2D lattices”. Quantum Info. Comput.17, 636–672 (2017). arXiv:1506.04014

Pith/arXiv arXiv 2017

[12] [12]

Almost Optimal Classical Approximation Algorithms for a Quan- tum Generalization of Max-Cut

Sevag Gharibian and Ojas Parekh. “Almost Optimal Classical Approximation Algorithms for a Quan- tum Generalization of Max-Cut”. In Dimitris Achlioptas and L´aszl´o A. V´egh, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Volume 145 of Leibniz International Proceedings in Informatics (...

arXiv 2019

[13] [13]

Beyond Product State Approximations for a Quan- tum Analogue of Max Cut

Anurag Anshu, David Gosset, and Karen Morenz. “Beyond Product State Approximations for a Quan- tum Analogue of Max Cut”. In Steven T. Flammia, editor, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Volume 158 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1–7:15. Dagstuhl, Germany (2...

arXiv 2020

[14] [14]

Approximation algorithms for quan- tum many-body problems

Sergey Bravyi, David Gosset, Robert K¨onig, and Kristan Temme. “Approximation algorithms for quan- tum many-body problems”. Journal of Mathematical Physics60, 032203 (2019). arXiv:1808.01734

arXiv 2019

[15] [15]

An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem

Sean Hallgren, Eunou Lee, and Ojas Parekh. “An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem”. In Jarosław Byrka and Raghu Meka, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Volume 176 of Leibniz International Proceedings in Informatics (LIPIcs), pages 59:1–59:18. ...

2020

[16] [16]

Beating Random Assignment for Approximating Quantum 2- Local Hamiltonian Problems

Ojas Parekh and Kevin Thompson. “Beating Random Assignment for Approximating Quantum 2- Local Hamiltonian Problems”. In Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, editors, 29th Annual European Symposium on Algorithms (ESA 2021). Volume 204 of Leibniz International Pro- ceedings in Informatics (LIPIcs), pages 74:1–74:18. Dagstuhl, Germany (2021). Schl...

arXiv 2021

[17] [17]

Application of the Level-2 Quantum Lasserre Hierarchy in Quan- tum Approximation Algorithms

Ojas Parekh and Kevin Thompson. “Application of the Level-2 Quantum Lasserre Hierarchy in Quan- tum Approximation Algorithms”. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Volume 198 of Leibniz International Proceedings in Informatics (LIPIcs), pages 102...

arXiv 2021

[18] [18]

An Optimal Product-State Approximation for 2-Local Quantum Hamiltonians with Positive Terms

Ojas Parekh and Kevin Thompson. “An Optimal Product-State Approximation for 2-Local Quantum Hamiltonians with Positive Terms” (2022). arXiv:2206.08342

arXiv 2022

[19] [19]

An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut

Jun Takahashi, Chaithanya Rayudu, Cunlu Zhou, Robbie King, Kevin Thompson, and Ojas Parekh. “An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut” (2023). arXiv:2307.15688

Pith/arXiv arXiv 2023

[20] [20]

An Improved Approximation Algorithm for Quantum Max-Cut on Triangle-Free Graphs

Robbie King. “An Improved Approximation Algorithm for Quantum Max-Cut on Triangle-Free Graphs”. Quantum7, 1180 (2023). arXiv:2209.02589

arXiv 2023

[21] [21]

Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators

Adam Bene Watts, Anirban Chowdhury, Aidan Epperly, J. William Helton, and Igor Klep. “Relaxations and Exact Solutions to Quantum Max Cut via the Algebraic Structure of Swap Operators”. Quantum 8, 1352 (2024). arXiv:2307.15661

arXiv 2024

[22] [22]

Second order cone relaxations for quantum Max Cut

Felix Huber, Kevin Thompson, and Sevag Gharibian. “Second order cone relaxations for quantum Max Cut” (2024). arXiv:2411.04120

arXiv 2024

[23] [23]

Improved approximation ratios for the Quan- tum Max-Cut problem on general, triangle-free and bipartite graphs

Sander Gribling, Lennart Sinjorgo, and Renata Sotirov. “Improved approximation ratios for the Quan- tum Max-Cut problem on general, triangle-free and bipartite graphs” (2025). arXiv:2504.11120

arXiv 2025

[24] [24]

Improved Approximation Algorithms for the EPR Hamiltonian

Nathan Ju and Ansh Nagda. “Improved Approximation Algorithms for the EPR Hamiltonian”. In Alina Ene and Eshan Chattopadhyay, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Volume 353 of Leibniz In- ternational Proceedings in Informatics (LIPIcs), pages 24:1–24:9. Dagstuhl, Germany (20...

arXiv 2025

[25] [25]

Improved Algorithms for Quantum MaxCut via Partially Entangled Matchings

Anuj Apte, Eunou Lee, Kunal Marwaha, Ojas Parekh, and James Sud. “Improved Algorithms for Quantum MaxCut via Partially Entangled Matchings”. In Anne Benoit, Haim Kaplan, Sebastian Wild, and Grzegorz Herman, editors, 33rd Annual European Symposium on Algorithms (ESA 2025). Vol- ume 351 of Leibniz International Proceedings in Informatics (LIPIcs), pages 101...

arXiv 2025

[26] [26]

A Lov ´asz theta lower bound on Quantum Max Cut

Felix Huber. “A Lov ´asz theta lower bound on Quantum Max Cut” (2025). arXiv:2512.20326

Pith/arXiv arXiv 2025

[27] [27]

Projected entangled states: Properties and applications

Frank Verstraete, Michael Wolf, David P ´erez-Garc´ıa, and Juan I. Cirac. “Projected entangled states: Properties and applications”. International Journal of Modern Physics B20, 5142–5153 (2006)

2006

[28] [28]

Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems

Frank Verstraete, Valentin Murg, and Juan I. Cirac. “Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems”. Advances in Physics57, 143–224 (2008). arXiv:0907.2796

Pith/arXiv arXiv 2008

[29] [29]

Quantum Monte Carlo and Related Approaches

Brian M. Austin, Dmitry Yu. Zubarev, and William A. Lester. “Quantum Monte Carlo and Related Approaches”. Chemical Reviews112, 263–288 (2012)

2012

[30] [30]

From architectures to applications: A review of neural quantum states

Hannah Lange, Anka Van De Walle, Atiye Abedinnia, and Annabelle Bohrdt. “From architectures to applications: A review of neural quantum states”. Quantum Science and Technology9, 040501 (2024). arXiv:2402.09402

arXiv 2024

[31] [31]

A convergent hierarchy of semidefinite pro- grams characterizing the set of quantum correlations

Miguel Navascu ´es, Stefano Pironio, and Antonio Ac´ın. “A convergent hierarchy of semidefinite pro- grams characterizing the set of quantum correlations”. New Journal of Physics10, 073013 (2008). arXiv:0803.4290

Pith/arXiv arXiv 2008

[32] [32]

Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables

S. Pironio, M. Navascu ´es, and A. Ac´ın. “Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables”. SIAM Journal on Optimization20, 2157–2180 (2010). arXiv:0903.4368

Pith/arXiv arXiv 2010

[33] [33]

Global Optimization with Polynomials and the Problem of Moments

Jean B. Lasserre. “Global Optimization with Polynomials and the Problem of Moments”. SIAM Journal on Optimization11, 796–817 (2001)

2001

[34] [34]

Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robust- ness and Optimization

Pablo A. Parrilo. “Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robust- ness and Optimization”. PhD thesis. California Institute of Technology. Pasadena, California (2000)

2000

[35] [35]

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Monique Laurent. “Sums of Squares, Moment Matrices and Optimization Over Polynomials”. In Mihai Putinar and Seth Sullivant, editors, Emerging Applications of Algebraic Geometry. Volume 149, pages 157–270. Springer New York, New York, NY (2009)

2009

[36] [36]

A positivstellensatz for non-commutative polynomials

J. Helton and Scott McCullough. “A positivstellensatz for non-commutative polynomials”. Transac- tions of the American Mathematical Society356, 3721–3737 (2004)

2004

[37] [37]

An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs

Jean B. Lasserre. “An Explicit Exact SDP Relaxation for Nonlinear 0-1 Programs”. In Gerhard Goos, Juris Hartmanis, Jan Van Leeuwen, Karen Aardal, and Bert Gerards, editors, Integer Programming and Combinatorial Optimization. Volume 2081, pages 293–303. Springer Berlin Heidelberg, Berlin, Heidelberg (2001)

2081

[38] [38]

Semidefinite pro- gramming relaxations for quantum correlations

Armin Tavakoli, Alejandro Pozas-Kerstjens, Peter Brown, and Mateus Ara ´ujo. “Semidefinite pro- gramming relaxations for quantum correlations”. Reviews of Modern Physics96, 045006 (2024). arXiv:2307.02551

arXiv 2024

[39] [39]

Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm

Maho Nakata, Hiroshi Nakatsuji, Masahiro Ehara, Mitsuhiro Fukuda, Kazuhide Nakata, and Katsuki Fujisawa. “Variational calculations of fermion second-order reduced density matrices by semidefinite programming algorithm”. The Journal of Chemical Physics114, 8282–8292 (2001)

2001

[40] [40]

Lower Bounds for Ground States of Condensed Matter Systems

Tillmann Baumgratz and Martin B. Plenio. “Lower Bounds for Ground States of Condensed Matter Systems”. New Journal of Physics14, 023027 (2012). arXiv:1106.5275

Pith/arXiv arXiv 2012

[41] [41]

Solving Condensed-Matter Ground-State Problems by Semidef- inite Relaxations

Thomas Barthel and Robert H ¨ubener. “Solving Condensed-Matter Ground-State Problems by Semidef- inite Relaxations”. Physical Review Letters108, 200404 (2012). arXiv:1106.4966

Pith/arXiv arXiv 2012

[42] [42]

Quantum Many-body Bootstrap

Xizhi Han. “Quantum Many-body Bootstrap” (2020). arXiv:2006.06002

arXiv 2020

[43] [43]

Quantum Many-Body Theory from a Solution of theN-representability Prob- lem

David A. Mazziotti. “Quantum Many-Body Theory from a Solution of theN-representability Prob- lem”. Physical Review Letters130, 153001 (2023). arXiv:2304.08570. 17

arXiv 2023

[44] [44]

Certifying Ground-State Properties of Many-Body Systems

Jie Wang, Jacopo Surace, Ir ´en´ee Fr ´erot, Benoˆıt Legat, Marc-Olivier Renou, Victor Magron, and An- tonio Ac ´ın. “Certifying Ground-State Properties of Many-Body Systems”. Physical Review X14, 031006 (2024). arXiv:2310.05844

arXiv 2024

[45] [45]

Scal- able Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polyno- mial Optimization

Jie Wang, David Jansen, Ir ´en´ee Frerot, Marc-Olivier Renou, Victor Magron, and Antonio Ac´ın. “Scal- able Ground-State Certification of Quantum Spin Systems via Structured Noncommutative Polyno- mial Optimization” (2026). arXiv:2604.01555

arXiv 2026

[46] [46]

Product-state approximations to quantum ground states

Fernando G.S.L. Brandao and Aram W. Harrow. “Product-state approximations to quantum ground states”. In Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing. Pages 871–880. STOC ’13New York, NY, USA (2013). Association for Computing Machinery. arXiv:1310.0017

Pith/arXiv arXiv 2013

[47] [47]

On the complexity of Putinar’s Positivstellensatz

Jiawang Nie and Markus Schweighofer. “On the complexity of Putinar’s Positivstellensatz”. Journal of Complexity23, 135–150 (2007). arXiv:0812.2657

Pith/arXiv arXiv 2007

[48] [48]

Convergence rates of moment-sum-of-squares hierarchies for optimal control problems

Milan Korda, Didier Henrion, and Colin N. Jones. “Convergence rates of moment-sum-of-squares hierarchies for optimal control problems”. Systems & Control Letters100, 1–5 (2017). arXiv:1609.02762

Pith/arXiv arXiv 2017

[49] [49]

Convergence rates of moment-sum-of-squares hierarchies for vol- ume approximation of semialgebraic sets

Milan Korda and Didier Henrion. “Convergence rates of moment-sum-of-squares hierarchies for vol- ume approximation of semialgebraic sets”. Optimization Letters12, 435–442 (2018). arXiv:1612.04146

Pith/arXiv arXiv 2018

[50] [50]

The sum-of-squares hierarchy on the sphere and applications in quan- tum information theory

Kun Fang and Hamza Fawzi. “The sum-of-squares hierarchy on the sphere and applications in quan- tum information theory”. Mathematical Programming190, 331–360 (2021). arXiv:1908.05155

arXiv 2021

[51] [51]

Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel–Darboux Kernel

Lucas Slot. “Sum-of-Squares Hierarchies for Polynomial Optimization and the Christoffel–Darboux Kernel”. SIAM Journal on Optimization32, 2612–2635 (2022). arXiv:2111.04610

arXiv 2022

[52] [52]

An effective version of Schm ¨udgen’s Positivstellensatz for the hypercube

Monique Laurent and Lucas Slot. “An effective version of Schm ¨udgen’s Positivstellensatz for the hypercube”. Optimization Letters17, 515–530 (2023). arXiv:2109.09528

arXiv 2023

[53] [53]

On the effective Putinar’s Positivstellensatz and moment ap- proximation

Lorenzo Baldi and Bernard Mourrain. “On the effective Putinar’s Positivstellensatz and moment ap- proximation”. Mathematical Programming200, 71–103 (2023). arXiv:2111.11258

arXiv 2023

[54] [54]

Degree Bounds for Putinar’s Positivstellensatz on the Hypercube

Lorenzo Baldi and Lucas Slot. “Degree Bounds for Putinar’s Positivstellensatz on the Hypercube”. SIAM Journal on Applied Algebra and Geometry8, 1–25 (2024). arXiv:2302.12558

arXiv 2024

[55] [55]

Convergence rates for the moment- sos hierarchy

Corbinian Schlosser, Matteo Tacchi-B ´enard, and Alexey Lazarev. “Convergence rates for the moment- sos hierarchy”. Numerical Algebra, Control and Optimization16, 105–156 (2026). arXiv:2402.00436

arXiv 2026

[56] [56]

Specialized Effective Positivstellens ¨atze for Improved Con- vergence Rates of the Moment-SOS Hierarchy

Corbinian Schlosser and Matteo Tacchi. “Specialized Effective Positivstellens ¨atze for Improved Con- vergence Rates of the Moment-SOS Hierarchy”. IEEE Control Systems Letters8, 2319–2324 (2024). arXiv:2403.05934

arXiv 2024

[57] [57]

On Łojasiewicz inequalities and the effec- tive Putinar’s Positivstellensatz

Lorenzo Baldi, Bernard Mourrain, and Adam Parusi ´nski. “On Łojasiewicz inequalities and the effec- tive Putinar’s Positivstellensatz”. Journal of Algebra662, 741–767 (2025). arXiv:2212.09551

arXiv 2025

[58] [58]

Convergence rates for sums-of- squares hierarchies with correlative sparsity

Milan Korda, Victor Magron, and Rodolfo R ´ıos-Zertuche. “Convergence rates for sums-of- squares hierarchies with correlative sparsity”. Mathematical Programming209, 435–473 (2025). arXiv:2303.14824

arXiv 2025

[59] [59]

Convergence rates for polynomial optimization on set products

Victor Magron. “Convergence rates for polynomial optimization on set products” (2025). arXiv:2505.18580

Pith/arXiv arXiv 2025

[60] [60]

Sum-of-squares hierarchies for binary polynomial optimization

Lucas Slot and Monique Laurent. “Sum-of-squares hierarchies for binary polynomial optimization”. Mathematical Programming197, 621–660 (2023). arXiv:2011.04027

arXiv 2023

[61] [61]

An Overview of Convergence Rates for Sum of Squares Hierarchies in Polynomial Optimization

Monique Laurent and Lucas Slot. “An Overview of Convergence Rates for Sum of Squares Hierarchies in Polynomial Optimization”. In Shin’ichi Oishi, Hisashi Okamoto, and Ken Hayami, editors, Recent Developments in Industrial and Applied Mathematics. Pages 149–176. Singapore (2026). Springer Nature. arXiv:2408.04417. 18

arXiv 2026

[62] [62]

Orthogonal Polynomials

G. Szeg ˝o. “Orthogonal Polynomials”. Volume 23 of Colloquium Publications. American Mathematical Society. Providence, Rhode Island (1939)

1939

[63] [63]

Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces

Vladimir I. Levenshtein. “Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces”. IEEE Transactions on Information Theory41, 1303–1321 (1995)

1995

[64] [64]

Sur une g ´en´eralisation des polyn ˆomes d’Hermite

Mykha ˘ılo Krawtchouk. “Sur une g ´en´eralisation des polyn ˆomes d’Hermite”. Comptes Rendus Heb- domadaires des S´eances de l’Acad´emie des Sciences189, 620–622 (1929). url:https://gallica.bnf. fr/ark:/12148/bpt6k3142j.f620

1929

[65] [65]

Association schemes and coding theory

Philippe Delsarte and Vladimir I. Levenshtein. “Association schemes and coding theory”. IEEE Trans- actions on Information Theory44, 2477–2504 (1998)

1998

[66] [66]

Algebraic Combinatorics

Eiichi Bannai, Etsuko Bannai, Tatsuro Ito, and Rie Tanaka. “Algebraic Combinatorics”. De Gruyter. (2021)

2021

[67] [67]

Quantum Analog of the MacWilliams Identities for Classical Coding Theory

Peter Shor and Raymond Laflamme. “Quantum Analog of the MacWilliams Identities for Classical Coding Theory”. Physical Review Letters78, 1600–1602 (1997). arXiv:quant-ph/9610040

Pith/arXiv arXiv 1997

[68] [68]

SDP bounds on quantum codes

Gerard Angl `es Munn ´e, Andrew Nemec, and Felix Huber. “SDP bounds on quantum codes” (2025). arXiv:2408.10323

arXiv 2025

[69] [69]

The Christoffel–Darboux Kernel for Data Analysis

Jean Bernard Lasserre, Edouard Pauwels, and Mihai Putinar. “The Christoffel–Darboux Kernel for Data Analysis”. Cambridge University Press. (2022). 1st edition

2022

[70] [70]

Noncommutative Christoffel-Darboux ker- nels

Serban Belinschi, Victor Magron, and Victor Vinnikov. “Noncommutative Christoffel-Darboux ker- nels”. Transactions of the American Mathematical Society376, 181–230 (2022). arXiv:2106.06212

arXiv 2022

[71] [71]

Interacting electrons and quantum magnetism

Assa Auerbach. “Interacting electrons and quantum magnetism”. Graduate Texts in Contemporary Physics. Springer New York. New York, NY (1994)

1994

[72] [72]

Exact semidefinite programming re- laxations with truncated moment matrix for binary polynomial optimization problems

Shinsaku Sakaue, Akiko Takeda, Sunyoung Kim, and Naoki Ito. “Exact semidefinite programming re- laxations with truncated moment matrix for binary polynomial optimization problems”. SIAM Journal on Optimization27, 565–582 (2017)

2017

[73] [73]

“Algebra”

Serge Lang. “Algebra”. Volume 211 of Graduate Texts in Mathematics. Springer New York. New York, NY (2002). A Proof of Lemma 2 Proof.For an arbitraryi∈[n], considerP ◦ def=C⟨X i,Y i,Z i⟩/I◦, whereI ◦ is the two-sided ideal generated by polynomials X2 i −1,Y 2 i −1,Z 2 i −1,X iYi −iZ i,Y iZi −iX i,Z iXi −iY i. Using these relations, any polynomial inX i,Y ...

2002