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Exponential Quantum Space Advantage for Approximating Max-$k$SAT in the Streaming Setting

T0 review · 1 major / 0 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read A quantum one-pass streaming algorithm approximates Max-kSAT to 0.7172 using only polylog(n) space.

desk verdict The abstract claims a polylog-space quantum streaming algorithm for Max-kSAT at 0.7172 approximation that beats the classical sqrt(n) lower bound for anything above ~0.707, plus a full classification for Max-2CSPs, but supplies zero construction or proof. read the letter →

arxiv 2606.05366 v1 pith:KHSYY6XZ submitted 2026-06-03 cs.DS cs.CC

Haoyu Wang , Guangxu Yang This is my paper · ORCID
classification cs.DScs.CC
keywords quantumstreamingalgorithmsMax-kSATapproximationspacecomplexityMax-2ORBooleanMax-2CSPmodelratio
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 constructs a quantum algorithm that processes Max-kSAT instances in a single streaming pass. The algorithm maintains only polylogarithmic space and returns an assignment that satisfies at least 0.7172 of the optimum. Classical algorithms cannot match this ratio without using much more space, specifically Omega of the square root of n. The same approach yields a 0.7425-approximation for Max-2OR and finishes the picture for all Max-2CSPs. Readers interested in quantum advantages in limited-memory computation would see a concrete exponential gap here.

What carries the argument

A one-pass quantum streaming algorithm that encodes the input into a quantum state using limited space to compute the approximate solution.

What would settle it

Finding that no one-pass quantum streaming procedure can reach a 0.7172 approximation ratio for Max-kSAT while staying within polylog(n) space would disprove the central result.

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Extended reading notes

Core claim

The central claim is that a one-pass quantum streaming algorithm exists for Max-kSAT which uses polylog(n) space and achieves a 0.7172-approximation, in contrast to the classical lower bound of Omega(sqrt(n)) space for approximations better than sqrt(2)/2 ≈ 0.7071. A similar algorithm for Max-2OR with 0.7425-approximation completes the classification of quantum space advantages for Boolean Max-2CSPs.

Load-bearing premise

The described quantum streaming algorithm for Max-kSAT actually exists and meets the stated space and approximation bounds.

Editorial extensions

If this is right

  • A 0.7172-approximation for Max-kSAT is possible with polylog space in the quantum streaming model.
  • Any classical streaming algorithm for a better ratio than 0.7071 must use Omega(sqrt(n)) space.
  • A 0.7425-approximation for Max-2OR is achievable with polylog space quantumly.
  • All Boolean Max-2CSPs now have their quantum versus classical space advantages fully classified.

Reading between the lines

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

  • Similar quantum space savings might apply to other streaming approximation problems beyond CSPs.
  • The separation raises the question of whether quantum methods can improve ratios or handle multi-pass variants for the same problems.
  • Practical tests on small instances could check whether the polylog space bound holds under realistic quantum noise.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The paper claims to give a one-pass quantum streaming algorithm for Max-kSAT using polylog(n) space that achieves a 0.7172-approximation ratio for instances with n variables. This is contrasted with the classical lower bound from Chou et al. (FOCS 2020) showing that any classical streaming algorithm requires Ω(√n) space to beat the √2/2 ≈ 0.7071 threshold. The paper also claims a one-pass quantum streaming algorithm for Max-2OR achieving 0.7425-approximation and states that this yields a complete classification of quantum space advantages for all Boolean Max-2CSPs.

Significance. If the quantum algorithm construction and approximation analysis hold, the result would establish an exponential quantum-classical separation in the one-pass streaming model for approximating Max-kSAT (and complete the picture for Max-2CSPs). Such a separation would be notable for highlighting potential quantum advantages in space-bounded approximation algorithms.

major comments (1)
  1. [Abstract] Abstract: the central claim is the existence of a one-pass quantum streaming algorithm achieving the 0.7172 approximation ratio in polylog(n) space. No algorithm description, measurement scheme, or analysis deriving the ratio (or showing it exceeds the classical threshold without increasing space) is provided, so the soundness of the claimed exponential advantage cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim is the existence of a one-pass quantum streaming algorithm achieving the 0.7172 approximation ratio in polylog(n) space. No algorithm description, measurement scheme, or analysis deriving the ratio (or showing it exceeds the classical threshold without increasing space) is provided, so the soundness of the claimed exponential advantage cannot be assessed.

    Authors: Abstracts are concise summaries by design. The full one-pass quantum streaming algorithm for Max-kSAT, including its construction, the measurement scheme, and the analysis deriving the 0.7172 approximation ratio (exceeding the classical √2/2 threshold at polylog(n) space), appears in the body of the manuscript. The same holds for the Max-2OR result. These sections establish the claimed exponential quantum space advantage. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; quantum algorithm claim is independent of cited classical bound

full rationale

The paper presents a constructive claim of a new one-pass quantum streaming algorithm for Max-kSAT (polylog(n) space, 0.7172-approx) and Max-2OR, directly contrasting it with an external classical lower bound from Chou, Golovnev, and Velusamy (FOCS 2020) by unrelated authors. No self-citations appear in the load-bearing claims, no parameters are fitted then renamed as predictions, and no derivation reduces by definition or ansatz to its own inputs. The result is a standard algorithmic construction whose validity rests on the (unshown here) proof details rather than any circular reduction.

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

Abstract-only review; no free parameters, axioms, or invented entities are described.

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Cite this review

Pith. "Pith review of Exponential Quantum Space Advantage for Approximating Max-$k$SAT in the Streaming Setting." pith.science (2026). https://pith.science/paper/KHSYY6XZ

@misc{pith2026260605366,
  author       = {Pith},
  title        = {Pith review of: Exponential Quantum Space Advantage for Approximating Max-$k$SAT in the Streaming Setting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KHSYY6XZ}},
  note         = {Machine review of arXiv:2606.05366}
}
abstract

In this paper, we give a one-pass quantum streaming algorithm for Max-$k$SAT that uses $\operatorname{polylog}(n)$ space and achieves a $0.7172$-approximation on instances with $n$ variables. In contrast, prior work by Chou, Golovnev, and Velusamy (FOCS 2020) implies that achieving an approximation ratio better than $\sqrt{2}/2 \approx 0.7071$ for Max-$k$SAT requires $\Omega(\sqrt{n})$ space for any classical streaming algorithm. Therefore, it yields an exponential quantum space advantage for Max-$k$SAT in the streaming setting. We further give a one-pass quantum streaming algorithm for Max-2OR that uses $\operatorname{polylog}(n)$ space and achieves a $0.7425$-approximation on instances with $n$ variables. Combining with the known results, it gives a complete classification of quantum space advantages for all Boolean Max-2CSPs.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reviewed June 28, 2026 · model on record in the stance chip above.