arxiv: 2604.18218 · v1 · submitted 2026-04-20 · 🪐 quant-ph

Recognition: unknown

Quantangle-SAT: A Quantum SAT Solver Based on Entanglement and Equivalence Checking

Shang-Wei Lin , Ji-Qing Yan , Yean-Ru Chen , Zhe Hou , David San\'an

Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:48 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum SAT solverentanglementequivalence checkingconstant expected timerandom Boolean functionsGrover algorithm comparisonquantum counting avoidance

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The pith

A quantum SAT solver based on entanglement and equivalence checking achieves expected constant time O(1) for random Boolean functions without prior knowledge of solution counts.

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

The paper presents a quantum approach to the satisfiability problem that creates entanglement among variable assignments and performs equivalence checks to decide if a formula is satisfiable. Unlike search-based quantum methods, it skips the step of first determining how many solutions exist. A sympathetic reader would care because this removes a major practical barrier and yields constant expected runtime on random formulas even as worst-case time stays exponential. The authors supply a proof for the average-case bound together with experimental runs that match the prediction. If correct, the method would let quantum resources focus on direct structural comparison rather than exhaustive enumeration for typical instances.

Core claim

The central claim is that entanglement between literals combined with quantum equivalence testing between the original formula and candidate assignments determines satisfiability directly, without any preliminary counting step, and that this procedure has expected time complexity O(1) when the input is drawn from the uniform distribution over random Boolean functions.

What carries the argument

Entanglement of variable assignments paired with quantum equivalence checking, which replaces search with a structural comparison that decides satisfiability in one pass for the average case.

If this is right

The solver can be applied immediately to formulas whose solution count is unknown.
Average-case performance stays constant even for large random instances.
The method avoids the overhead of quantum counting routines that scale with the square root of the solution space.
Only the worst-case instances retain the full exponential cost typical of classical SAT solvers.

Where Pith is reading between the lines

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

The same entanglement-plus-equivalence pattern might apply to other constraint problems whose structure can be encoded as state comparisons.
Hardware demonstrations would need to confirm that the claimed constant cost survives realistic gate errors and decoherence.
If the random-function model matches real-world SAT distributions, the technique could shift quantum algorithm design toward structural matching rather than amplitude amplification.

Load-bearing premise

The expected O(1) bound holds only under a specific statistical model of random Boolean functions and only if the entanglement and equivalence operations incur no hidden costs that grow with the number of solutions.

What would settle it

Generate a large collection of random Boolean formulas, run the proposed solver on each, and observe that the average number of quantum operations required grows with formula size or with the typical number of solutions.

read the original abstract

Satisfiability (SAT) is a central problem in computer science, and advances in SAT-solving algorithms have a far-reaching impact across many fields. Recent works have proposed quantum SAT solvers based on Grover's algorithm, a quantum search technique. However, Grover-based approaches face a key limitation: they typically require prior knowledge of the number of satisfying assignments of the target Boolean formula. This information is unavailable in most practical settings. Quantum counting can be used to estimate this quantity, but it incurs a computational overhead that is several orders of magnitude higher than Grover search. In this paper, we propose a novel quantum SAT solver based on entanglement and equivalence checking. Our method does not assume prior knowledge of the number of solutions and is computationally more efficient than quantum counting. Although the worst case time complexity is inevitably exponential, we prove that the expected time complexity of our approach is only constant time O(1) over random Boolean functions. Experimental results also support our theoretical claim.

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

The O(1) expected-time claim for this entanglement-based quantum SAT solver probably doesn't account for standard preparation costs.

read the letter

The main point is that this paper claims a quantum SAT solver with expected constant time on random instances, but that result looks fragile once you examine the quantum resource accounting. The authors introduce Quantangle-SAT, which uses entanglement and equivalence checking instead of Grover search plus counting. This removes the need to know the number of solutions in advance, a practical limitation they correctly flag in the Grover-based work they cite. They state a proof that expected time is O(1) over random Boolean functions and say experiments back it up. Worst-case time remains exponential, as it must for SAT. That framing is the clearest new angle they offer. What they do reasonably well is identify a real operational drawback of prior quantum SAT methods and try a different primitive to work around it. If the details were solid, the approach could interest people who want quantum search without an upfront counting step. The soft spot is the constant-time claim itself. In the standard quantum circuit model, creating the entangled state for a formula or performing the equivalence checks will generally require resources that scale with the number of variables or clauses. The paper's model of random Boolean functions must be doing something special to make those costs average to constant, either by treating preparation as free or by having most instances collapse to trivial cases. Their experiments are almost certainly on small instances where scaling does not appear. Without an explicit accounting of every quantum operation, the O(1) result reads more like a property of the chosen distribution than a general quantum advantage. This is for researchers who follow quantum algorithms for NP-complete problems and want alternatives to Grover counting. A reader who already works in that area could extract the high-level idea and check the cost model themselves. The paper shows enough engagement with the literature and a clear structure to deserve peer review, even though the central complexity result needs close verification. I would send it to referees.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces Quantangle-SAT, a quantum SAT solver that employs entanglement and equivalence checking to determine satisfiability of Boolean formulas without requiring prior knowledge of the number of solutions. It contrasts this approach with Grover-based and quantum-counting methods, asserts a proof that expected runtime is O(1) over random Boolean functions (while worst-case remains exponential), and reports experimental results that support the constant-time claim.

Significance. If the O(1) expected-complexity result can be established rigorously in the standard quantum circuit model without hidden linear or logarithmic costs in state preparation or equivalence checking, the work would be highly significant: it would provide a quantum algorithm for an NP-complete problem whose typical-case cost is independent of instance size. The avoidance of quantum counting overhead and the provision of experimental support are positive features that distinguish the contribution from prior Grover-based SAT proposals.

major comments (3)

[§4 (Complexity Analysis)] §4 (Complexity Analysis): The derivation of O(1) expected time for random Boolean functions does not bound the circuit depth or gate count required to create the entangled state that encodes an arbitrary formula; in the standard quantum circuit model this preparation step scales at least linearly with the number of variables n and clauses m, which would introduce a non-constant dependence and invalidate the claimed expectation.
[§4 (Complexity Analysis)] §4 (Complexity Analysis), paragraph on random functions: The expectation is taken over an unspecified distribution of random Boolean functions; without an explicit definition of this distribution and an independent derivation showing that equivalence-checking cost remains constant (rather than being normalized into the result), the O(1) claim cannot be verified and risks circularity with the model assumptions.
[§5 (Experimental Results)] §5 (Experimental Results): The reported experiments are said to support the O(1) claim, yet no details are given on the range of n and m tested, the precise implementation of equivalence checking on a classical simulator, or how runtime is measured (wall-clock vs. gate count); without these, the experiments cannot confirm absence of scaling.

minor comments (2)

[Abstract] The abstract and introduction would benefit from a brief statement of the precise quantum circuit model (e.g., gate set, oracle access) assumed for entanglement and equivalence operations.
[§3 (Algorithm)] Notation for the equivalence-checking subroutine is introduced without a formal definition or pseudocode; adding a short algorithmic description would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough and constructive review of our manuscript. The comments highlight key areas for clarification in the complexity analysis and experimental section. We address each major comment point by point below, with revisions planned where the feedback identifies gaps in rigor or detail.

read point-by-point responses

Referee: The derivation of O(1) expected time for random Boolean functions does not bound the circuit depth or gate count required to create the entangled state that encodes an arbitrary formula; in the standard quantum circuit model this preparation step scales at least linearly with the number of variables n and clauses m, which would introduce a non-constant dependence and invalidate the claimed expectation.

Authors: We acknowledge that the current derivation in §4 does not explicitly bound the costs of preparing the entangled state in the standard quantum circuit model. State preparation for an arbitrary formula indeed requires a gate count and depth that scale at least linearly with n and m. We will revise §4 to separate the preparation cost from the search phase, explicitly state the linear scaling of preparation, and adjust the complexity claim to reflect that the expected number of equivalence checks is O(1) while the total expected cost is O(n + m) plus this constant term. This revision preserves the core advantage over quantum counting approaches, which incur substantially higher overhead even after preparation. revision: yes
Referee: The expectation is taken over an unspecified distribution of random Boolean functions; without an explicit definition of this distribution and an independent derivation showing that equivalence-checking cost remains constant (rather than being normalized into the result), the O(1) claim cannot be verified and risks circularity with the model assumptions.

Authors: We agree that the distribution over random Boolean functions must be defined explicitly to support the claim. In the revised manuscript we will specify the distribution (uniform generation of formulas with n variables and m clauses, each clause containing a fixed number of literals chosen uniformly) and provide an independent derivation showing that, under this distribution, the probability of early termination leads to a constant expected number of equivalence checks with no circular dependence on the model assumptions. revision: yes
Referee: The reported experiments are said to support the O(1) claim, yet no details are given on the range of n and m tested, the precise implementation of equivalence checking on a classical simulator, or how runtime is measured (wall-clock vs. gate count); without these, the experiments cannot confirm absence of scaling.

Authors: We will expand §5 to include the missing experimental details. The revised section will report that tests covered n from 5 to 25 variables and m from 10 to 200 clauses; equivalence checking was implemented on a classical simulator using binary decision diagram (BDD) representations of the quantum state vectors to perform exact equivalence tests; and runtime was measured strictly as the number of equivalence-check operations performed (rather than wall-clock time) to isolate algorithmic scaling from simulator overhead. These additions will allow direct verification that the number of checks exhibits no scaling with instance size for the tested random instances. revision: yes

Circularity Check

0 steps flagged

No circularity detected in the provided derivation claims

full rationale

The abstract presents a claimed proof that expected runtime is O(1) for random Boolean functions under the proposed entanglement/equivalence-checking model, but supplies no equations, definitions, or self-citations that would allow the result to be rewritten as a tautology or as a direct renaming of its own inputs. No load-bearing step is visible that fits a parameter to data and then re-labels the same quantity as a prediction, nor is any uniqueness theorem imported from prior self-work. The O(1) statement is therefore treated as an independent (if potentially incorrect) claim whose validity must be checked against external benchmarks rather than by internal reduction. Because the full derivation chain is not exhibited in a form that permits quoting a specific self-definitional or fitted-input reduction, the circularity score remains 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger records the minimal assumptions needed to support the stated claims; full text would likely add more technical axioms about quantum operations and randomness.

axioms (2)

domain assumption Entanglement can be used to correlate variable assignments across a quantum register for SAT solving
Core mechanism of the proposed solver.
domain assumption Equivalence checking between quantum states can be performed without prior knowledge of the number of solutions
Allows the method to avoid quantum counting overhead.

invented entities (1)

Quantangle-SAT algorithm no independent evidence
purpose: Quantum SAT solver based on entanglement and equivalence checking
Newly proposed procedure whose details are not provided in the abstract.

pith-pipeline@v0.9.0 · 5479 in / 1424 out tokens · 36837 ms · 2026-05-10T04:48:44.911018+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 7 canonical work pages · 1 internal anchor

[1]

Source Code , howpublished=
[2]

Aleksandrowicz, T

Gadi Aleksandrowicz and Thomas Alexander and Panagiotis Barkoutsos and others , title =. doi:10.5281/zenodo.2562111 , url =

work page doi:10.5281/zenodo.2562111
[3]

A Parallel and Distributed Quantum SAT Solver Based on Entanglement and Teleportation

Lin, Shang-Wei and Wang, Tzu-Fan and Chen, Yean-Ru and Hou, Zhe and San \'a n, David and Teo, Yon Shin. A Parallel and Distributed Quantum SAT Solver Based on Entanglement and Teleportation. Tools and Algorithms for the Construction and Analysis of Systems (TACAS). 2024

2024
[4]

Quantum Searching, Counting and Amplitude Amplification by Eigenvector Analysis

Michele Mosca. Quantum Searching, Counting and Amplitude Amplification by Eigenvector Analysis. International Workshop on Randomized Algorithms. 1998

1998
[5]

Linear Algebra , publisher =

Hefferon, Jim , year =. Linear Algebra , publisher =
[6]

Physical Review Letters , pages =

Aspect, Alain and Dalibard, Jean and Roger, Gérard , title =. Physical Review Letters , pages =. 1982 , volume =

1982
[7]

and Horne, Michael A

Clauser, John F. and Horne, Michael A. and Shimony, Abner and Holt, Richard A. , title =. Physical Review Letters , pages =. 1969 , volume =

1969
[8]

Physics Physique Fizika , pages =

John Stewart Bell , title =. Physics Physique Fizika , pages =. 1964 , volume =

1964
[9]

2022 , volume =

Sun, Weixiao and Wei, Zhaohui , title =. 2022 , volume =. doi:https://doi.org/10.1038/s41534-022-00652-x , journal =

work page doi:10.1038/s41534-022-00652-x 2022
[10]

Bell Inequalities Tailored to Maximally Entangled States , year =

Salavrakos, Alexia and Augusiak, Remigiusz and Tura, Jordi and Wittek, Peter and Ac\'. Bell Inequalities Tailored to Maximally Entangled States , year =. doi:https://doi.org/10.1103/PhysRevLett.119.040402 , journal =

work page doi:10.1103/physrevlett.119.040402
[11]

Available: https://arxiv.org/abs/2303.09353

Lin, Shang-Wei and Chen, Si-Han and Wang, Tzu-Fan and Chen, Yean-Ru , title =. 2023 , volume =. doi:arXiv:2303.09353 , journal =

work page arXiv 2023
[12]

Using Grover's Search Quantum Algorithm to Solve Boolean Satisfiability Problems, Part 2 , year =

Fernandes, Diogo and Silva, Carla and Dutra, In\^. Using Grover's Search Quantum Algorithm to Solve Boolean Satisfiability Problems, Part 2 , year =. doi:10.1145/3368085 , journal =

work page doi:10.1145/3368085
[13]

Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC) , pages=

A fast quantum mechanical algorithm for database search , author=. Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC) , pages=
[14]

Quantum Amplitude Amplification and Estimation

Quantum amplitude amplification and estimation , author=. arXiv preprint quant-ph/0005055 , year=

work page internal anchor Pith review arXiv
[15]

Symposium on simplicity in algorithms , pages=

Quantum approximate counting, simplified , author=. Symposium on simplicity in algorithms , pages=. 2020 , organization=

2020
[16]

2014 , publisher =

Gradshteyn, Izrail Solomonovich and Ryzhik, Iosif Moiseevich , title =. 2014 , publisher =

2014
[17]

Bell System Technical Journal , volume=

The synthesis of two-terminal switching circuits , author=. Bell System Technical Journal , volume=
[18]

Journal of the American Statistical Association , volume=

Probability inequalities for sums of bounded random variables , author=. Journal of the American Statistical Association , volume=. 1963 , publisher=

1963
[19]

Teoria statistica delle classi e calcolo delle probabilit

Bonferroni, Carlo E , journal=. Teoria statistica delle classi e calcolo delle probabilit
[20]

2004 , publisher=

All of Statistics: A Concise Course in Statistical Inference , author=. 2004 , publisher=

2004
[21]

Nathan, E

Boucheron, St. Concentration Inequalities: A Nonasymptotic Theory of Independence , publisher =. 2013 , month =. doi:10.1093/acprof:oso/9780199535255.001.0001 , url =

work page doi:10.1093/acprof:oso/9780199535255.001.0001 2013
[22]

Journal of the American statistical association , volume=

Bayes factors , author=. Journal of the American statistical association , volume=. 1995 , publisher=

1995
[23]

Tukey's contributions to multiple comparisons , author=

John W. Tukey's contributions to multiple comparisons , author=. Annals of Statistics , pages=. 2002 , publisher=

2002