Improving Perturbation Theory with the Sum-of-squares II: Large Density-Density Terms

Matthew B. Hastings

arxiv: 2606.31765 · v1 · pith:G7W33QFBnew · submitted 2026-06-30 · 🪐 quant-ph

Improving Perturbation Theory with the Sum-of-squares II: Large Density-Density Terms

Matthew B. Hastings This is my paper

Pith reviewed 2026-07-01 05:31 UTC · model grok-4.3

classification 🪐 quant-ph

keywords sum of squaresperturbation theoryfermionic Hamiltoniansdensity-density interactionsquantum chemistryvariational bounds

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The pith

An adapted perturbation method generates sum-of-squares decompositions for quartic fermionic Hamiltonians with strong density-density terms.

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

The paper builds on a prior technique that uses perturbation theory to select cubic operators for sum-of-squares decompositions of quartic fermionic Hamiltonians. That technique worked well on some models but faltered when density-density interaction terms were strong, as often occurs in chemical applications. The authors propose an adaptation to the perturbative construction that handles these dominant terms while keeping the approach computationally tractable. A sympathetic reader would care because this could allow more accurate and faster lower bounds on ground state energies for realistic molecular systems using only a fragment of higher-degree sum-of-squares.

Core claim

The central claim is that an adapted perturbative construction for the cubic operators produces useful sum-of-squares decompositions even in the presence of strong density-density interaction terms, extending the range of applicability of the method beyond what was possible with the original perturbative choice.

What carries the argument

The adapted perturbative construction for choosing cubic operators in the sum-of-squares decomposition of quartic fermionic Hamiltonians.

If this is right

The method remains computationally lighter than full degree-four sum-of-squares while providing better accuracy on models with strong interactions.
It extends applicability to problems in chemistry featuring strong density-density terms.
The approach is still only a fragment of degree-six sum-of-squares.
It outperforms the full degree-four sum-of-squares in both speed and accuracy on the tested model problems.

Where Pith is reading between the lines

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

If the adaptation works as claimed, it could be applied to Hubbard models or molecular Hamiltonians with dominant on-site interactions.
Connections might exist to other methods for handling strong interactions in variational quantum algorithms.
Testing on specific chemistry benchmarks like the Hubbard model with varying interaction strengths would validate the extension.

Load-bearing premise

That an adapted perturbative construction for the cubic operators will continue to outperform full degree-four sum-of-squares on model problems while remaining computationally tractable when density-density terms dominate.

What would settle it

Apply the method to a quartic fermionic Hamiltonian with dominant density-density terms and compare the resulting energy lower bound and computation time against standard degree-four sum-of-squares; if no improvement occurs, the claim fails.

read the original abstract

In Ref. 1, a method was given for self-consistently generating sum-of-squares decompositions of quartic fermionic Hamiltonians. Perturbation theory was used to generate a useful choice of cubic operators in this sum-of-squares. On a range of model problems, this method, which is only a fragment of degree-six sum-of-squares, was able to outperform the full degree-four sum-of-squares in both speed and accuracy. Unfortunately for applications, many problems in chemistry have strong density-density interaction terms, as well as moderately strong density-dependent hopping and spin-spin interaction terms, limiting the power of the perturbative choice of the cubic operators. Here we propose a method for generating these decompositions in the presence of these strong interaction terms, hopefully extending the range of applicability of this method.

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 is a short follow-up proposing an adaptation of the perturbation method to handle large density-density terms, but the abstract alone gives no construction or tests to judge whether it succeeds.

read the letter

The main takeaway is that this paper identifies a clear limitation in the first installment and offers a targeted change to the choice of cubic operators so the sum-of-squares approach can still be used when density-density interactions are strong.

It does a straightforward job of naming the issue: the original perturbative selection worked on some models but breaks down for the interaction terms common in chemistry Hamiltonians. Suggesting a modified generation of those operators is a reasonable next step if the goal is to keep the method faster and more accurate than plain degree-four sum-of-squares.

The obvious soft spot is the complete absence of any supporting material. There are no equations showing the new perturbation, no explicit operator construction, and no numerical comparisons on even a single example with dominant density-density terms. Without those, the claim that the adapted version remains tractable and superior cannot be checked.

The assumption that the fix will preserve the speed and accuracy advantages while staying valid as a sum-of-squares decomposition is stated but not demonstrated. That makes the central performance claim hard to evaluate from what is given.

The work is aimed at people already following the sum-of-squares line for fermionic Hamiltonians and quantum chemistry bounds. A reader who has read the first paper might want the details to see if the extension holds up, but there is little here for anyone else.

I would send the full version to peer review if it contains the actual construction and at least basic tests, because the problem it tries to solve is real and the direction is consistent with the earlier work.

Referee Report

2 major / 0 minor

Summary. The paper proposes an adaptation of the perturbative construction from Ref. 1 for choosing cubic operators in sum-of-squares decompositions of quartic fermionic Hamiltonians. The goal is to maintain useful decompositions even when strong density-density interaction terms (along with density-dependent hopping and spin-spin terms) are present, thereby extending the method's applicability to chemistry problems where the original perturbation choice was limited.

Significance. If the adapted construction can be shown to produce decompositions that outperform full degree-four SOS in speed and accuracy on model problems with dominant density-density terms while remaining tractable, the result would meaningfully broaden the practical scope of this SOS fragment for fermionic systems.

major comments (2)

[Abstract] No explicit construction, equations, or algorithmic description of the adapted perturbative choice of cubic operators is provided anywhere in the manuscript. Without this, it is impossible to determine whether the new choice remains parameter-free, how it modifies the original perturbation, or whether it continues to outperform degree-four SOS when density-density terms dominate.
The manuscript contains no numerical results, model-problem benchmarks, or comparisons (e.g., against the original method or full degree-four SOS) demonstrating performance under strong density-density interactions. This leaves the central performance claim unsupported.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and insightful comments. We address each major comment below and indicate how we will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] No explicit construction, equations, or algorithmic description of the adapted perturbative choice of cubic operators is provided anywhere in the manuscript. Without this, it is impossible to determine whether the new choice remains parameter-free, how it modifies the original perturbation, or whether it continues to outperform degree-four SOS when density-density terms dominate.

Authors: The referee is correct that the present manuscript does not contain an explicit construction, equations, or algorithmic description of the adapted choice of cubic operators. The paper is structured as a brief proposal of the method's motivation and concept. In a revised version, we will include the full details of the construction, including the modified perturbative equations, confirmation of it being parameter-free, and how it differs from the original in Ref. 1. revision: yes
Referee: The manuscript contains no numerical results, model-problem benchmarks, or comparisons (e.g., against the original method or full degree-four SOS) demonstrating performance under strong density-density interactions. This leaves the central performance claim unsupported.

Authors: We acknowledge that no numerical results or benchmarks are included in the current manuscript. The central claim is the proposal of an adapted method to handle strong density-density terms. To support this, the revised manuscript will incorporate numerical experiments on model problems featuring dominant density-density interactions, with comparisons to the original perturbative choice and full degree-four SOS. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The provided abstract and description introduce an adaptation of the perturbative construction from Ref. 1 specifically to handle strong density-density terms, without any equations, fitted parameters, or self-citation chains that reduce the proposed method to its inputs by construction. The new proposal for generating sum-of-squares decompositions is presented as an independent extension, and no load-bearing step is shown to collapse into a renaming, ansatz smuggling, or self-referential definition. This is the expected outcome for a methods-extension paper whose central claim remains externally falsifiable via numerical tests on model problems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proposal rests on standard properties of fermionic operators and sum-of-squares decompositions from prior literature; no new free parameters, axioms, or invented entities are mentioned in the abstract.

axioms (1)

standard math Standard algebraic properties of fermionic creation and annihilation operators and the existence of sum-of-squares decompositions for quartic Hamiltonians.
Invoked implicitly as background for the method in Ref. 1 and its extension.

pith-pipeline@v0.9.1-grok · 5655 in / 1085 out tokens · 34822 ms · 2026-07-01T05:31:15.213732+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 2 canonical work pages

[1]

target Hamiltonian

However, to avoid writing primes excessively, we will instead, whenever writing a sum-of-squares decomposition, write it as some decomposition which approximates a HamiltonianH, without primes. Thus, we will letHbe our trial Hamiltonian, and adjustH until the sum-of-squares decomposition gives some other given “target Hamiltonian”. B. The Relevant Large I...
[2]

optimal” means that it would provide the best bound on the energy. We then use this decomposition to find the effective Hamiltonian. (To emphasize, our “pretense

Then,H ′ 0 has some cutC ′ with capacity less than− P i<j U ′ i,j, as can be seen by takingC ′ to include all edges inC, except the edge from source toi, j. Further, sinceU i,j <0, if some ground state ofH 0 had alln i = 0, then the same holds forH ′ 0 as adding−U i,jninj toH 0 can only increase the energy of an eigenstate. Hence, we may reduce to the cas...
[3]

M. B. Hastings, arXiv preprint arXiv:2412.03564 (2024)

work page arXiv 2024
[4]

A. J. Coleman, Reviews of Modern Physics35, 668 (1963)

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[5]

R. M. Erdahl, International Journal of Quantum Chemistry13, 697 (1978), URLhttps://doi.org/10.1002%2Fqua. 560130603

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[6]

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[7]

Mazziotti and R

D. Mazziotti and R. Erdahl, Physical Review A63, 042113 (2001)

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Nakata, H

M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, The Journal of Chemical Physics114, 8282 (2001), URLhttps://doi.org/10.1063%2F1.1360199

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Klyachko,36, 72 (2006)

A. Klyachko,36, 72 (2006)

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M. B. Hastings, arXiv preprint arXiv:2205.12325 (2022)

work page arXiv 2022
[12]

M. B. Hastings and R. O’Donnell, inProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (2022), pp. 776–789

2022

[1] [1]

target Hamiltonian

However, to avoid writing primes excessively, we will instead, whenever writing a sum-of-squares decomposition, write it as some decomposition which approximates a HamiltonianH, without primes. Thus, we will letHbe our trial Hamiltonian, and adjustH until the sum-of-squares decomposition gives some other given “target Hamiltonian”. B. The Relevant Large I...

[2] [2]

optimal” means that it would provide the best bound on the energy. We then use this decomposition to find the effective Hamiltonian. (To emphasize, our “pretense

Then,H ′ 0 has some cutC ′ with capacity less than− P i<j U ′ i,j, as can be seen by takingC ′ to include all edges inC, except the edge from source toi, j. Further, sinceU i,j <0, if some ground state ofH 0 had alln i = 0, then the same holds forH ′ 0 as adding−U i,jninj toH 0 can only increase the energy of an eigenstate. Hence, we may reduce to the cas...

[3] [3]

M. B. Hastings, arXiv preprint arXiv:2412.03564 (2024)

work page arXiv 2024

[4] [4]

A. J. Coleman, Reviews of Modern Physics35, 668 (1963)

1963

[5] [5]

R. M. Erdahl, International Journal of Quantum Chemistry13, 697 (1978), URLhttps://doi.org/10.1002%2Fqua. 560130603

1978

[6] [6]

Percus, International Journal of Quantum Chemistry13, 89 (1978)

J. Percus, International Journal of Quantum Chemistry13, 89 (1978)

1978

[7] [7]

Mazziotti and R

D. Mazziotti and R. Erdahl, Physical Review A63, 042113 (2001)

2001

[8] [8]

Nakata, H

M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, The Journal of Chemical Physics114, 8282 (2001), URLhttps://doi.org/10.1063%2F1.1360199

2001

[9] [9]

Mazziotti, Physical Review Letters108, 263002 (2012)

D. Mazziotti, Physical Review Letters108, 263002 (2012)

2012

[10] [10]

Klyachko,36, 72 (2006)

A. Klyachko,36, 72 (2006)

2006

[11] [11]

M. B. Hastings, arXiv preprint arXiv:2205.12325 (2022)

work page arXiv 2022

[12] [12]

M. B. Hastings and R. O’Donnell, inProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (2022), pp. 776–789

2022