Approximate quantum 3-colorings of graphs and the quantum Max 3-Cut problem

Samuel J. Harris

REVIEW 2 major objections 4 minor 23 references

Approximate quantum 3-colorings of graphs and the quantum Max 3-Cut problem

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Approximate values of any synchronous nonlocal game are controlled by the 3-coloring game of an explicitly constructed graph, transferring errors with only a square-root loss.

desk verdict Two real contributions miss the advertised target: the quantitative preservation theorems are new and mostly sound, but the fixed-gap undecidability applications don't follow as written, and Lemma 5.7 has a counting gap. read the letter →

arxiv 2412.19405 v1 pith:NG4GZKSY submitted 2024-12-27 quant-ph math.OA

Samuel J. Harris This is my paper · ORCID

classification quant-phmath.OA MSC 81P6805C1546L1068Q1781P45

keywords quantum3-coloringsynchronousnon-localgamesnon-commutativeMax-3-Cutapproximatestrategiesgamealgebraundecidabilitygappedpromiseproblemgraphhomomorphism

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 proves a transfer theorem: for every synchronous non-local game $\mathcal{G}$ with $n$ questions and $m \geq 3$ answers there is an explicitly built graph $G_\lambda$ such that approximately winning synchronous strategies for $\mathcal{G}$ and for the $3$-coloring game of $G_\lambda$ are equivalent, up to a polynomial factor in $n$ and $2^m$ and a square-root loss in the error. If a strategy wins the $3$-coloring game on $G_\lambda$ with probability $1-\varepsilon$, then the same model yields a strategy for $\mathcal{G}$ winning with probability at least $1 - h(n,2^m)\varepsilon^{1/2}$; conversely, a $1-\varepsilon$ strategy for $\mathcal{G}$ gives a $1-\mathrm{poly}(n,m)\varepsilon$ strategy for the coloring game. The author uses this to show that the gapped promise problem for quantum $3$-coloring is undecidable, and that there is some $\alpha \in (0,1)$ for which approximating the non-commutative Max-$3$-Cut (and its commuting-operator version) within a factor $\alpha$ is uncomputable. These conclusions matter because they transfer the hardness of arbitrary synchronous games onto a single natural graph-coloring problem, answering a question raised in prior work on approximation algorithms for non-commutative constraint satisfaction problems without assuming the unique games conjecture.

What carries the argument

The carrying object is the graph $G_\lambda$, assembled from a control triangle $\Delta = \{A,B,C\}$, copies of $K_3 \times K_3$ written $R_{\alpha,x}$, triangular prisms $T_{\alpha,x}$, and orthogonality gadgets $Q_{a,b,x,y}$, with gluings that force a six-output PVM structure on the control triangle via the operators $S_{i,j,k} = P_{i,A}P_{j,B}P_{k,C}P_{j,B}P_{i,A}$ for $\{i,j,k\} = \{1,2,3\}$. The proof shows that an approximately winning $3$-coloring strategy makes these operators positive contractions that almost form a PVM, and a perturbation lemma converts them into a genuine PVM, producing a strategy for the original game $\mathcal{G}$. The error analysis tracks triangle defects $\zeta$, prism defects $\xi$, and edge defects $\theta$, all of which are controlled by $\varepsilon^{1/4}$ and summed over the edges of $G_\lambda$.

What would settle it

Exhibit a synchronous game $\mathcal{G}$ and an $\varepsilon>0$ for which the synchronous $t$-value of $\mathrm{Hom}(G_\lambda,K_3)$ under $\pi_{\mathrm{edges}}$ is at least $1-\varepsilon$ but the synchronous $t$-value of $\mathcal{G}$ under $\pi_u$ is below $1-h(n,2^m)\varepsilon^{1/2}$; or, for the hardness claim, show that any valid $\alpha$ in Theorem 6.3 must depend on $n$ and $m$, so no single fixed $\alpha$ exists.

Watch

Extended reading notes

Core claim

The central claim is a two-way transfer of approximate winning strategies between any synchronous game $\mathcal{G}$ and the $3$-coloring game on an associated graph $G_\lambda$. For each model $t \in \{\mathrm{loc}, q, \mathrm{qa}, \mathrm{qc}\}$, Theorem 5.13 states that if the synchronous $t$-value of $\mathrm{Hom}(G_\lambda,K_3)$ with respect to the uniform edge distribution is at least $1-\varepsilon$, then the synchronous $t$-value of $\mathcal{G}$ with respect to the uniform question distribution is at least $1 - h(n,2^m)\varepsilon^{1/2}$ for a polynomial $h$; Theorem 4.4 gives the converse with error $\mathrm{poly}(n,m)\varepsilon$. The paper then applies this transfer to prove that the gapped $(1,\alpha)$-promise problem for the synchronous $t$-value of the $3$-coloring game is undecidable for some $\alpha \in (0,1)$, and consequently that deciding whether the non-commutative Max-$3$-Cut of a graph equals $|E|$ or is less than $\alpha |E|$ is RE-hard, with analogous results for the commuting-operator version.

Load-bearing premise

The undecidability applications depend on the threshold $\alpha$ being a fixed constant independent of the original game, whereas the proof supplies an $\alpha$ that depends on the game's question and answer counts.

Editorial extensions

If this is right

If the main theorem is correct, the gapped promise problem for quantum $3$-coloring is undecidable: there is an $\alpha \in (0,1)$ such that deciding whether the synchronous value is $1$ or at most $\alpha$ is RE-hard.
Consequently, approximating the non-commutative Max-$3$-Cut within some constant factor $\alpha$ is uncomputable, giving a sharp computability gap between the known $0.864$ polynomial-time approximation and a factor below which hardness appears.
The transfer works in all four synchronous correlation models (local, quantum, quantum approximate, and quantum commuting), so the hardness conclusions apply to each model simultaneously.
The reduction avoids the unique games conjecture, making the complexity-theoretic consequences unconditional in the standard computability setting.
Small errors in the $3$-coloring game correspond to at most square-root-larger errors in the original game, so hardness of approximation transfers between the two problems with a modest loss in precision.

Reading between the lines

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

The instance-dependent threshold $\alpha$ in the proof of Theorem 6.3 means the paper establishes hardness for a family of promise problems whose parameters depend on the original game size; a uniform fixed $\alpha$ for all graphs would require a stronger reduction than the one given here.
If the perturbation question posed as Problem 6.7 has a positive answer, the $2^m$ dependence in the main theorem could be replaced by a polynomial in $m$, improving the transfer rate and potentially strengthening the hardness statements.
The construction likely adapts to any prior distribution on question pairs that is positive on all edges, not just the uniform distribution, at the cost of more complicated constants in the error bounds.
A natural test of the method is whether a similar approximate-strategy-preserving equivalence can be built for $k$-coloring games with $k \geq 4$; the paper leaves this as an explicit open problem.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here's my read. The genuinely new thing is the quantitative back-and-forth between synchronous games and 3-coloring games: Theorem 4.4 (G to Hom(Gλ,K3)) and Theorem 5.13 (reverse). The exact equivalence was in the author's earlier AHP paper; this version tracks errors through the construction and obtains explicit polynomial bounds in n and 2^m. That is a solid tool, and the style is careful: the perturbation machinery (Lemmas 2.6, 2.7, 2.9) is standard but applied cleanly to control the 2-norm losses. Section 6 also poses a reasonable open problem (6.7) about POVM perturbation.

The soft spots are in the applications. Theorem 6.3 claims a fixed α ∈ (0,1) for the (1,α)-promise problem of 3-coloring. The proof sets α = 1−(ε/h(n,2^m))², which depends on the original game's question and answer counts. That's an instance-dependent threshold, so the reduction does not target the promised fixed-gap problem. The advertised RE-hardness of NC-Max-3-Cut at fixed gap is not established as stated.

Worse, I think Lemma 5.7 in the proof of Theorem 5.11 has a genuine counting error. The proof claims every edge appears only a constant number of times in the sums ζ(T), ξ(R), θ(e) and hence the total is O(|E(Gλ)|)·ε^{1/4}. But the edge (A,B) of the control triangle contributes to ζ(Δ), which is charged for each x and each a in Lemmas 5.4–5.6. That multiplicity grows like n·m, not constant. So the bound in Lemma 5.7 does not follow from (5.1). This gap sits in the derivation of the main transfer theorem; it may be repairable by keeping polynomial factors n,m, but as written the proof is incomplete.

The debt to [8] is disclosed and the earlier result is independently published, so I don't count that as circular. No parameter-free magic; all constants are explicit.

Who should read it? Specialists in nonlocal games, quantum chromatic number, and NC-Max-k-Cut. The construction and the questions are worth their time even if the undecidability claims are overstated. It deserves a serious referee, not a desk reject, with instructions to check Lemma 5.7 and the α dependence carefully.

Recommendation: engage, but with a referee who is willing to dig into the counts. If Lemma 5.7 pans out and the α issue is fixed with a uniform threshold (or the claim weakened), the paper has a valuable core.

Referee Report

2 major / 4 minor

Summary. The paper defines, for each synchronous non-local game G=(I,O,λ) with |I|=n and |O|=m≥3, a graph G_λ and proves approximate strategy-preservation results between G and the 3-coloring game Hom(G_λ,K_3). The main technical theorem (Theorem 5.13) states that a 1−ε winning synchronous strategy for Hom(G_λ,K_3) yields a 1−h(n,2^m) ε^{1/2} winning synchronous strategy for G, with the converse (Theorem 4.4) having only linear loss in ε. These estimates are applied in Section 6 to claim undecidability of a gapped promise problem for quantum 3-coloring and of approximating non-commutative and qc Max-3-Cut within some constant factor α.

Significance. If the main approximation theorem is correct, it is a substantial contribution: it extends the author's earlier ∗-equivalence result to approximate strategies in all four synchronous models (loc, q, qa, qc), with explicit polynomial dependence, and it does not rely on the unique games conjecture. The paper also gives a clear reduction from synchronous games to 3-coloring games, which is a natural and potentially useful tool. However, the advertised undecidability applications in Section 6 depend on a fixed gap α, and the proof as written provides only an instance-dependent threshold. This is a load-bearing gap in the paper's central claims as stated.

major comments (2)

[Theorem 6.3] The reduction in Theorem 6.3 sets α = 1 − (ε/h(n,2^m))^2, where n=|I| and m=|O| are the question and answer counts of the source game G. This threshold depends on the input instance, so the reduction is not a many-one reduction to the fixed (1,α)-promise problem for quantum 3-coloring. Since h(n,2^m) grows without bound, α can be arbitrarily close to 1, and no fixed α0 ∈ (0,1) can serve all instances: for a source game with α_G > α0, the target value could lie in (α0, α_G] and would satisfy neither branch of the (1,α0)-promise problem. Consequently, the claimed RE-hardness of a fixed-gap promise problem for quantum 3-coloring, and the derived statements in Theorem 6.4 about NC-Max-3-Cut and qc-Max-3-Cut, are not established as stated. The core approximation theorem 5.13 is not affected by this issue, but the undecidability applications require either a genuinely fixed-α construction or a reformulation with instance-dependent thresholds.
[Section 5, Lemma 5.7] The proof of Lemma 5.7 contains an unsupported counting claim. It asserts that in the sums bounding the commutators, each edge of G_λ appears only a finite number of times independent of n, m, and λ. This appears false: in Lemma 5.5, the quantity ζ(Δ) appears for each a = 2,…,m−1 with coefficient 24, so the edge (A,B) is counted with multiplicity growing with m. Similarly, the proof of part (2) says commutator terms are counted once per element of λ^{-1}({0}), but the contribution from a fixed (a,x) is weighted by the number of (b,y) with (a,b,x,y) losing, which can depend on n and m. The final constant β in Lemma 5.7 is therefore not justified. Since Lemma 5.7 feeds directly into Lemmas 5.8–5.11 and hence Theorem 5.13, this proof gap is load-bearing for the main approximation theorem. The gap may be repairable because the final h(n,2^m) in Theorem 5.11 is allowed to depend polynomially on n and 2^m, but the proof as written does not provide the needed argument.

minor comments (4)

[Remark 4.1] There is a typo: 'sitll' should be 'still'.
[Theorem 6.4] There is a typo: 'undedicable' should be 'undecidable'.
[Abstract] The abstract says 'the gapped promise problem for quantum 3-coloring is undecidable' without specifying that the gap α is not a fixed constant in the proof; this should be clarified or corrected.
[Lemma 5.7] The notation in the statement of Lemma 5.7 combines the fixed {i,j,k} and the tuple (a,b,x,y) in a way that is easy to misread; a short explanation that S = S_{i,j,k} is fixed throughout the summation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the approximate transfer theorem is proven from explicit estimates, and reliance on the author's prior equivalence is disclosed, independent support.

full rationale

The central derivation is not circular. Theorem 5.13 is proved from scratch in Sections 3–5: the graph Gλ is explicitly constructed (modeled on prior work [8]), and the estimates are self-contained lemmas about approximations in tracial von Neumann algebras (Propositions 2.1–2.5, Lemmas 2.6–2.9, Lemmas 5.2–5.10). The only direct use of the author's earlier ∗-equivalence [8] is to assert that perfect winning strategies transfer in the undecidability reduction (Theorem 6.3), and that result is an independently published theorem, not an unverified premise of this paper. The undecidability source is MIP*=RE [14], an external result. The instance-dependent α in Theorem 6.3 (α = 1 − (ε/h(n,2^m))^2) is a genuine gap issue in the promise-problem reduction, but it is a correctness/threshold defect, not a definitional or self-citational circle: the transfer theorem itself does not assume what it proves. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors to forbid alternatives. I therefore find no circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters and no ad hoc model entities. It relies on standard operator-algebraic background, the cited representation theorem for synchronous correlations, the author's earlier exact equivalence, and the external MIP*=RE result.

assumptions (4)

domain assumption Synchronous correlations admit tracial von Neumann algebra representations (cited [20], [12]).
Used throughout Section 4 and 5 to convert strategies into PVM/tracial form.
domain assumption RE-hardness of the gapped (1,1-epsilon)-promise problem for quantum values of synchronous games, from Ji et al. [14].
Basis for the undecidability reductions in Theorem 6.3.
domain assumption Exact star-equivalence between G and Hom(G_lambda,K_3) from the author's prior work [8].
Used for the perfect-strategy directions and as the starting graph construction.
standard math Lemma 2.6 from [12]: a positive contraction almost a projection is close to its spectral projection onto [1/2,1].
Used in Lemma 2.7 and 2.9 for perturbations.

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Cite this review

Pith. "Pith review of Approximate quantum 3-colorings of graphs and the quantum Max 3-Cut problem." pith.science (2026). https://pith.science/paper/NG4GZKSY

@misc{pith2026241219405,
  author       = {Pith},
  title        = {Pith review of: Approximate quantum 3-colorings of graphs and the quantum Max 3-Cut problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NG4GZKSY}},
  note         = {Machine review of arXiv:2412.19405}
}

abstract

We prove that, to each synchronous non-local game $\mathcal{G}=(I,O,\lambda)$ with $|I|=n$ and $|O|=m \geq 3$, there is an associated graph $G_{\lambda}$ for which approximate winning strategies for the game $\mathcal{G}$ and the $3$-coloring game for $G_{\lambda}$ are preserved. That is, using a similar graph to previous work of the author (Ann. Henri Poincar\'{e}, 2024), any synchronous strategy for $\text{Hom}(G_{\lambda},K_3)$ that wins the game with probability $1-\varepsilon$ with respect to the uniform probability distribution on the edges, yields a strategy in the same model that wins the game $\mathcal{G}$ with respect to the uniform distribution with probability at least $1-h(n,m)\varepsilon^{\frac{1}{2}}$, where $h$ is a polynomial in $n$ and $2^m$. As an application, we prove that the gapped promise problem for quantum $3$-coloring is undecidable. Moreover, we prove that there exists an $\alpha \in (0,1)$ for which determining whether the non-commutative Max-$3$-Cut of a graph is $|E|$ or less than $\alpha |E|$ is RE-hard, thus giving a positive answer to a problem posed by Culf, Mousavi and Spirig (arXiv:2312.16765), along with evidence for a sharp computability gap in the non-commutative Max-$3$-Cut problem. We also prove that there is some $\alpha \in (0,1)$ such that determining the non-commutative (respectively, commuting operator framework) versions of the Max-$3$-Cut of a graph within a factor of $\alpha$ is uncomputable. All of these results avoid use of the unique games conjecture.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [8]

APPROXIMATE QUANTUM 3-COLORINGS OF GRAPHS 37

, Universality of graph homomorphism games and the quantum co loring problem, Annales Henri Poincar´ e (2024). APPROXIMATE QUANTUM 3-COLORINGS OF GRAPHS 37

work page 2024

[2] [1]

E. Culf, H. Mousavi, and T. Spirig, Approximation algorithms for noncommutative constraint s atisfac- tion problems, preprint (arXiv:2312.16765) (2023)

work page Pith review arXiv 2023

[3] [2]

Dykema, V.I

K. Dykema, V.I. Paulsen, and J. Prakash, Non-closure of the set of quantum correlations via graphs , Communications in Mathematical Physics 365 (2019), 1125–1142

work page 2019

[4] [3]

Frieze and M

A.M. Frieze and M. Jerrum, Improved approximation algorithms for MAX k-CUT and MAX BIS EC- TION, In Conference on Integer Programming and Combinatorial Optimiz ation (1995)

work page 1995

[5] [4]

Goemans and D

M. Goemans and D. Williamson, Improved approximation algorithms for maximum cut and sati sﬁability problems using semideﬁnite programming , Journal of the ACM 42 (1995), 1115—1145. [5] , Approximation algorithms for max-3-cut and other problems via complex semideﬁnite program- ming, Journal of Computer and System Sciences 68 (2004), no. 2, 442–470

work page 1995

[6] [6]

Goldberg, Synchronous linear constraint system games , Journal of Mathematical Physics 62 (2021), 032201

A. Goldberg, Synchronous linear constraint system games , Journal of Mathematical Physics 62 (2021), 032201

work page 2021

[7] [7]

Harris, Synchronous games with ∗-isomorphic game algebras , Quantum Information and Computa- tion 22 (2022), no

S.J. Harris, Synchronous games with ∗-isomorphic game algebras , Quantum Information and Computa- tion 22 (2022), no. 11&12, 924–946

work page 2022

[8] [9]

Helton, K.P

J.W. Helton, K.P. Meyer, V.I. Paulsen, and M. Satriano, Algebras, synchronous games, and chromatic numbers of graphs , New York Journal of Mathematics 25 (2019), 328–361

work page 2019