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REVIEW 3 major objections 4 minor 31 references

Unweighted Gapped Clique Homology is $\mathsf{QMA}_1$-complete

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that deciding gapped clique homology on unweighted graphs is QMA1-complete, so the hard part is the spectral gap promise, not vertex weights.

desk verdict The blow-up construction is genuinely new and the Section 4 unweighting is sound, but the NO-case gap inherits King–Kohler imports that are only sketched, so the completeness claim needs referee checking before being treated as settled. read the letter →

arxiv 2608.02726 v1 pith:YOI6LSSW submitted 2026-08-03 quant-ph

Ryu Hayakawa This is my paper · ORCID
classification quant-ph MSC 68Q1255U1005C50
keywords unweightedcliquehomologyQMA1-completenessHodgeLaplacianspectralgappromiseblow-upconstructionsymmetricsectortopologicaldataanalysisHamiltoniancomplexity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that deciding whether the clique complex of an unweighted graph has nonzero homology in a given dimension, under an inverse-polynomial spectral gap promise on the Hodge Laplacian, is complete for the quantum complexity class $\mathsf{QMA}_1$ with a fixed universal gate set $G_2=\{X,\mathsf{CX},\mathsf{CCX},H\otimes H\}$. The known hardness proof for the gapped problem assigned tiny weights to certain vertices to create scale separation; the paper shows those weights can be removed entirely. The construction replaces each weighted vertex by a clique whose size encodes the weight, so that symmetric averages over the copies reproduce the weighted Hodge metric. A local averaging argument then proves that all copy-label-dependent directions have uniformly positive energy, so the blow-up adds no spurious low-energy states or homology. If correct, the result isolates the gap promise, not vertex weighting, as the source of hardness.

What carries the argument

The load-bearing object is the level-ratio blow-up (Definition 4.1): a base vertex of level $\ell$ is replaced by a clique of size $M/\rho^{\ell}$, with $M=\rho$ and only levels 0 and 1 used in the application. The argument rides on two identities: the symmetric reduction $\Phi \hat\Delta_k|_{\hat T_k}\Phi^{-1} = \Delta_{\hat w,k}$ (Lemma 4.2), which makes the symmetric sector carry exactly $\rho$ times the weighted Laplacian, and the asymmetric gap $\hat\Delta_k|_{\hat T_k^\perp} \succeq \min_v \hat f_v\, I$ (Theorem 4.3), which removes copy-label-dependent directions. Together they transfer the weighted spectral gap to an unweighted clique complex without spurious low-energy states or homology.

What would settle it

Compute the degree-$(2n-1)$ unweighted Hodge Laplacian of the blow-up of a single gadget with $\rho=4$ (so $\lambda=1/2$), and check the smallest eigenvalue on the orthogonal complement of the symmetric sector: if it is below $\min_v \hat f_v=1$, Theorem 4.3 is wrong. Alternatively, inspect the explicit weighting of the cited single-gadget construction: any gadget vertex whose amplitude weight differs from $\lambda=\rho^{-1/2}$ immediately breaks the binary-weight correspondence.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.2: for a sufficiently large fixed constant $c_{\mathrm{gap}}$, Unweighted Gapped Clique Homology is $\mathsf{QMA}_1^{G_2}$-complete, where $G_2=\{X,\mathsf{CX},\mathsf{CCX},H\otimes H\}$. The reduction takes an instance of the $G_2$ 4-QSAT problem with integer local projectors and constructs the weighted gadget complex from the cited gapped-clique-homology work, then blows it up: each register vertex becomes a clique $K_\rho$ and each gadget vertex stays a singleton, with $\rho=(C_{\mathrm{wt}}t q(n))^2$. Lemma 4.2 identifies the unweighted blow-up Laplacian on the symmetric one-copy-per-block sector with $\rho$ times the weighted Laplacian, so the weighted many-gadget bound applies at the exact parameter $\lambda=1/(C_{\mathrm{wt}}t q(n))$; Theorem 4.3 gives a uniform lower bound of $\min_v \hat f_v=1$ on the asymmetric sector. Combining the two bounds yields the promised $N^{-c_{\mathrm{gap}}}$ NO-case gap, while the YES case is preserved by weight-free gadget selectivity. Containment is proved by normalizing the sparse integer clique Laplacian and applying exact linear-combination-of-unitaries simulation.

Load-bearing premise

The NO-case gap transfer depends on an imported many-gadget spectral estimate holding with uniform constants at the exact value $\lambda=1/(C_{\mathrm{wt}}t q(n))$, and on the gadget vertex weights being exactly binary (weight 1 on register vertices, weight $\lambda$ on gadget vertices); if either fails, the symmetric sector no longer carries the intended weighted Laplacian and the promised $N^{-c_{\mathrm{gap}}}$ gap does not follow.

Editorial extensions

If this is right

  • The hardness of gapped clique homology is independent of vertex weights: every vertex can be given weight one without changing $\mathsf{QMA}_1^{G_2}$-completeness.
  • An inverse-polynomial spectral gap promise is itself enough to make the unweighted problem quantumly hard, so future separation arguments can target the promise rather than artificial weights.
  • The blow-up is polynomial-sized and dense; the output graph has $N\le R\rho$ vertices with $R=\mathrm{poly}(n)$, so the reduction is efficient.
  • No spurious homology is created: the harmonic chains of the blow-up coincide with those of the weighted base complex, so topological YES witnesses are preserved.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same multiplicity-as-weight mechanism may convert other weighted homological hardness results into unweighted ones, for example independence-complex Laplacians or persistence-style promises, since the mechanism is metric reconstruction plus redundancy.
  • Sparsifying the clique blocks is the natural stress test: the paper's own discussion predicts that expander-like replacements lose block symmetry and may create unintended homology, so a persistence-style promise on marked cycles would be a cleaner next problem.
  • One can read the construction as an error-correcting encoding of the logical chain space inside the symmetric sector, suggesting that unweighting may work wherever a Hamiltonian's parameters can be encoded as multiplicities of identical copies.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proves that the Unweighted Gapped Clique Homology problem, in which every vertex has weight one, is QMA_1^{G2}-complete for the gate set G2={X,CX,CCX,H⊗H}. The core technical contribution is a blow-up construction (Section 4) that replaces each vertex weight by the size of an expanded clique block. On the symmetric sector, the unweighted Hodge Laplacian is shown to be unitarily equivalent to M times the weighted Laplacian (Lemma 4.2); on the orthogonal complement, a local averaging argument gives a uniform spectral gap (Theorem 4.3). Hardness is obtained by applying this to the King-Kohler weighted construction (Sections 5-6), with the NO-case gap transferred via Theorem 6.3. Containment is proved in Section 7 using Rudolph's exact sparse Hamiltonian simulation. The paper is transparent about importing several technical estimates from [KK24] and [Rud25]; Appendix A gives only a proof sketch of the many-gadget estimate.

Significance. If the imported estimates hold, the result is significant: it shows that the inverse-polynomial gap promise, rather than the vertex weighting, is the source of QMA_1-hardness in gapped clique homology. The blow-up and symmetric-sector mechanism is a clean and potentially transferable technique, and Section 4 is self-contained and appears correct. The paper gives a useful survey of related weighted and filtered homology complexity results. Its main weakness is that the central NO-case gap rests on unproven imported bounds and an unverified binary-weight assertion, so the present manuscript is conditional rather than fully self-contained.

major comments (3)
  1. [Section 5.3, Definition 5.3] The binary weight assignment is load-bearing and is not verified in the manuscript. The claim that every register vertex has weight 1 and every added gadget vertex has weight λ is cited to [KK24, Sec. 8], but the construction is not reproduced. If any gadget vertex carries a different weight, Lemma 4.2 identifies the symmetric sector with M times a weighted Laplacian that differs from the one for which the imported many-gadget bound (Theorem 6.2) applies, and the promised gap N^{-c_gap} would not follow. The paper should provide a detailed derivation of the vertex weights for the concrete gadget family used (including the Rudolph integer-projector embedding), or a rigorous verification from [KK24].
  2. [Section 6.2 and Appendix A] The NO-case gap transfer in Theorem 6.3 depends on the imported many-gadget estimate [KK24, Thm. 10.1] and the single-gadget spectral decomposition [KK24, Lems. 9.1, 10.1]. Appendix A is only a proof sketch: the key estimates (1)-(3) and the operator identity Σ_i bΦ_i^⊥ − (t−1)Π_0 = Π_H − H − (t−1)(Π_0 − Π_H) + O(λ t) are stated as imports from [KK24, Lems. 10.2–10.4] without proof. The manuscript should either give a complete proof of these estimates in the present notation, or state explicitly that the main theorem is conditional on [KK24] and verify that all hypotheses (uniform constants, t-dependent λ) are satisfied at the chosen λ = 1/(C_wt t q(n)). As written, the central NO-case claim is not self-contained.
  3. [Section 5.4, Lemma 5.4] The weight-free selectivity statement is also imported. Lemma 5.4 asserts that each gadget kills exactly the intended encoded cycle and creates no additional target-degree class, citing [KK24, Lem. 8.4] and [Rud25, App. D.1]. This is load-bearing for the YES-case direction (perfect completeness) and for the absence of spurious homology. The paper should either provide a proof of this selectivity for the specific integer-projector gadgets used, or clearly state the conditions under which the cited lemmas apply and explain why they hold for the present fixed finite family.
minor comments (4)
  1. [Title] The title states 'is QMA1-complete' while the abstract and Theorem 3.2 state QMA_1^{G2}-complete; the title should be qualified to avoid overstatement.
  2. [Definition 5.3] In the table, the row `amplitude weight w(v) q(bf_v/ρ)` appears to have a typo: `q` should be `\sqrt` (the surrounding text uses `\sqrt{bf_v/ρ}`).
  3. [Abstract and Section 2.4] The notation `QMA_1^{g_2}` in the abstract uses lowercase g_2 while the text uses uppercase G_2; please unify.
  4. [Section 7] The containment proof relies on Rudolph's Exact Sparse Hamiltonian problem [Rud25, Prob. 2.9] and lemmas [Rud25, Lem. 6.1, 6.2]; the paper would benefit from stating the relevant definitions so that the containment argument is self-contained.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the proof imports the weighted King–Kohler gap bound and Rudolph's QMA1^g2 machinery, but neither input assumes the unweighted theorem being proved.

full rationale

The central derivation chain is independent of its conclusion. Lemma 4.2 and Theorem 4.3 are proved in the text: the symmetric reduction is an explicit isometry computation (Phi bDelta_k|_{bT_k} Phi^{-1} = Delta_w,k), and the asymmetric bound is proved by a local join and complete-graph spectral argument. The NO-case gap transfer in Theorem 6.3 combines these self-contained lemmas with the cited weighted many-gadget bound [KK24, Thm. 10.1]; the paper explicitly chooses lambda = c_wt p_H / t so that the parameter matches the imported theorem rather than fitting the unweighted conclusion. The binary-weights assertion in Definition 5.3 is an external assumption about the King–Kohler construction, not a re-use of the unweighted result being proved. Self-citations such as [Hay22], [GSK+26], [HGYRD25], and [LKBH26] occur only in motivational or discussion passages and carry no load in the proof of Theorem 3.2. No equation or construction in the paper reduces by definition to the claimed QMA1^g2-completeness, so the central claim has independent grounding.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The central claim rests on the cited King-Kohler and Rudolph constructions. The paper's original contribution is the blow-up and symmetric-sector reduction, which is self-contained; but the NO-case gap and the source Hamiltonian depend on external theorems whose proofs are only sketched or cited.

free parameters (1)
  • c_gap = unspecified; 'sufficiently large' fixed constant
    Definition 3.1 fixes the promise gap p(N)=N^{-c_gap}, and Theorem 3.2 only proves existence of a large enough c_gap. The central completeness claim is parameterized by this unquantified constant.
assumptions (8)
  • standard math Hodge decomposition: ker ∆_k ≅ H_k for finite clique complexes
    Invoked in Section 2.1 to express the promise problem spectrally and to identify harmonic chains with homology.
  • standard math Join formula for augmented Hodge Laplacians (Horak-Jost, King-Kohler)
    Lemma 2.1, used for the qubit-register join and the star decomposition in Theorem 4.3.
  • standard math Complete-complex spectrum: ∆_i(Cl(K_f))=f I and augmented version
    Lemma 2.2, cited from Gundert-Wagner and used in Theorem 4.3 to lower-bound the asymmetric sector.
  • domain assumption King-Kohler weighted gadget construction with binary weights
    Section 5.3 asserts all register vertices have weight 1 and gadget vertices weight λ; the blow-up transfer depends on this exact weighting.
  • domain assumption King-Kohler single-gadget spectral decomposition and O(λ)-perturbation bounds
    Lemma 6.1 imports [KK24, Lems. 9.1, 10.1]; used to set the penalty and bulk scales in the NO-case analysis.
  • domain assumption King-Kohler many-gadget weighted energy bound (Theorem 10.1)
    Theorem 6.2 is the external lower bound transferred to the symmetric sector; only a proof sketch is given in Appendix A.
  • domain assumption Rudolph's G2 4-QSAT source Hamiltonian with integer local projectors and gadget selectivity
    Section 2.4 and Lemma 5.4 rely on [Rud25, Thm 4.2, Thm 6.8, App. D.1] for the source family and for the fact that the attached gadgets make exactly the projected cycles null-homologous.
  • domain assumption Rudolph's exact sparse Hamiltonian simulation is in QMA1^{g2}
    Proposition 7.1 uses [Rud25, Lems. 6.1 and 6.2] to contain the constructed sparse integer Laplacian in QMA1^{g2}.

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Cite this review

Pith. "Pith review of Unweighted Gapped Clique Homology is $\mathsf{QMA}_1$-complete." pith.science (2026). https://pith.science/paper/YOI6LSSW

@misc{pith2026260802726,
  author       = {Pith},
  title        = {Pith review of: Unweighted Gapped Clique Homology is $\mathsfQMA_1$-complete},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOI6LSSW}},
  note         = {Machine review of arXiv:2608.02726}
}
abstract

Deciding whether the clique complex of a given graph has nontrivial homology in a given dimension, under vertex-product weighting and an inverse-polynomial spectral gap promise on the combinatorial Hodge Laplacian, is known to be $\mathsf{QMA}_1$-hard and contained in $\mathsf{QMA}$ by King and Kohler (FOCS 2024). The vertex weights are essential in the known proof, where they provide the scale separation needed for the spectral gap analysis. We prove that, when every vertex has weight one, the problem remains $\mathsf{QMA}_1^{g_2}$-hard and is contained in $\mathsf{QMA}_1^{g_2}$, where $\mathsf{QMA}_1^{g_2}$ is $\mathsf{QMA}_1$ with the universal gate set $g_2=\{\mathsf{X},\mathsf{CX},\mathsf{CCX},H\otimes H\}$. The construction replaces weight by expansion: each vertex of the weighted complex is blown up into a clique whose size encodes its weight. The block sizes are chosen so that symmetric averages reproduce the weighted Hodge metric. The symmetric sector therefore carries the weighted Laplacian up to a common scalar factor, while a local averaging argument gives a uniform lower bound on the orthogonal complement to the symmetric sector. Hence the weighted gap analysis transfers to an unweighted clique complex without introducing either additional low-energy states or spurious homology. Containment in $\mathsf{QMA}_1^{g_2}$ follows from Rudolph's exact linear combination of unitaries simulation of sparse integer clique Laplacians. The result shows that the gap promise, rather than vertex weighting, is the source of the complexity of gapped clique homology.

Figures

Figures reproduced from arXiv: 2608.02726 by the authors.

Figure 1
Figure 1. The blow-up correspondence. A weighted base complex is replaced by an unweighted [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. The qubit-register blow-up. A one-qubit bowtie has two harmonic cycles representing [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. A schematic single-projector filling gadget. The boundary cycle is filled by an interior [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

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