{"id":"2dcd2cf8-f17a-48c1-89dc-7dbc7f1e6d27","arxiv_id":"2605.09017","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A dichotomy for path-containment problems shows some are solvable with linear queries while others are equivalent to cycle problems and admit a quantum-walk algorithm with query complexity Õ(n^{3/2 - α_k}) where α_k decays exponentially in k, plus a conditional lower bound.","lead":"This paper maps the quantum query complexity landscape for finding or detecting constant-length paths and cycles in graphs under the adjacency matrix model. It groups variants into equivalence classes via reductions and gives a new quantum-walk algorithm that improves the query bound for the harder class.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Randomized reductions may fail to preserve quantum query complexity up to constants for some variants","rationale":"The reader's weakest_assumption correctly isolates the reduction step as the point where the broad claim is most vulnerable. Because the novel algorithm is proven only for one problem in the class, everything else rests on those reductions preserving the quantum query complexity up to constants. No other internal inconsistency or unsupported assumption appears more load-bearing once the reductions are granted.","tokens_in":1866,"tokens_out":337,"duration_ms":30125,"concrete_test":"Take the reduction from the 'find path of length exactly k' problem to the 'detect path of length at most k' problem (or whichever pair is used to transfer the walk algorithm). Re-derive the query-complexity bound explicitly, tracking the precise constant factor and error amplification; if the resulting overhead exceeds a small constant (e.g., >10) or introduces an extra log n factor, the claimed equivalence fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim groups many path/cycle variants into equivalence classes via randomized reductions and transfers the new Õ(n^{3/2-α_k}) upper bound (from the quantum-walk algorithm on one representative) to the entire class. Randomized reductions in the quantum query model require careful control of error probabilities and oracle simulation overhead; if any reduction incurs super-constant factors or fails to compose cleanly with the adjacency-matrix oracle, the equivalence collapses and the broad applicability of the improved bound does not hold. The paper states the reductions preserve complexity up to constants, but this is the least-secured link in the argument chain.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript classifies quantum query complexities for path- and cycle-containment problems of constant length k in the adjacency-matrix model. Randomized reductions are used to group variants into equivalence classes, yielding a dichotomy for path problems (some linear-query solvable, others equivalent to each other and to certain cycle problems). For the non-trivial class, a new quantum-walk algorithm is given with query complexity Õ(n^{3/2-α_k}) where α_k ∈ Θ(c^{-k}) and c = √(3+√17)/2 ≈ 1.33, improving on the prior O(n^{3/2}) bound; a conditional lower bound is also derived from the graph-collision problem.","tokens_in":2013,"tokens_out":556,"duration_ms":43300,"significance":"If the reductions and algorithm hold, the work meaningfully advances quantum query complexity for subgraph problems by supplying both an explicit improved upper bound via quantum walks and a structural dichotomy via reductions. The exponential-in-k improvement in the exponent for fixed k and the conditional lower bound are concrete contributions that help map the complexity landscape.","major_comments":[{"comment":"Section 3 (Randomized Reductions and Equivalence Classes): The assertion that randomized reductions preserve quantum query complexity up to constant factors is load-bearing for the equivalence classes and for transferring the Õ(n^{3/2-α_k}) upper bound to all members of the class. The manuscript must supply explicit bounds on error-probability amplification, the number of oracle calls in the reduction, and the overhead of simulating the adjacency-matrix oracle on the reduced instances; without these, the claimed constant-factor equivalence cannot be verified.","section":"Section 3"},{"comment":"Section 4 (Quantum Walk Algorithm): The derivation of the exponent α_k ∈ Θ(c^{-k}) with the specific constant c = √(3+√17)/2 is central to the claimed improvement over O(n^{3/2}). The recurrence or closed-form analysis that produces this exponent (and confirms it is strictly positive for each fixed k) should be stated with all intermediate steps so that the query-complexity bound can be checked independently.","section":"Section 4"}],"minor_comments":[{"comment":"The abstract lists promise versions without enumerating them; a one-sentence clarification of the promise assumptions would improve readability.","section":"Abstract"},{"comment":"Notation for the Õ symbol and the precise definition of α_k should be repeated at the first use in the main text for consistency.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their insightful comments on our manuscript. We address the two major comments below and will incorporate the necessary clarifications and details in the revised version.","responses":[{"response":"We agree that explicit bounds are required to rigorously establish constant-factor preservation of quantum query complexity. In the revised manuscript we will add a dedicated paragraph (or short appendix) that (i) specifies the error-amplification procedure, showing that O(1) independent repetitions suffice to boost success probability to 2/3 while incurring only constant overhead, (ii) bounds the number of reduction oracle calls by a k-dependent constant, and (iii) proves that each adjacency-matrix query on the reduced instance is simulated by at most a constant number of original-oracle calls. These additions will confirm the claimed equivalence classes.","revision_made":"yes","referee_comment":"[Section 3] Section 3 (Randomized Reductions and Equivalence Classes): The assertion that randomized reductions preserve quantum query complexity up to constant factors is load-bearing for the equivalence classes and for transferring the Õ(n^{3/2-α_k}) upper bound to all members of the class. The manuscript must supply explicit bounds on error-probability amplification, the number of oracle calls in the reduction, and the overhead of simulating the adjacency-matrix oracle on the reduced instances; without these, the claimed constant-factor equivalence cannot be verified."},{"response":"We thank the referee for this observation. The exponent α_k is obtained from the spectral analysis of a quantum walk on the Johnson graph whose marked set encodes paths of length k. In the revision we will expand the relevant subsection to include the full derivation: we first recall the general quantum-walk query bound, then derive the recurrence relating the improvement factor at step k to the eigenvalue gap and marked-set density at step k-1, solve the resulting linear recurrence to obtain the closed form α_k = Θ(c^{-k}) with the stated constant c, and verify that α_k > 0 for every finite k. All algebraic steps will be shown explicitly.","revision_made":"yes","referee_comment":"[Section 4] Section 4 (Quantum Walk Algorithm): The derivation of the exponent α_k ∈ Θ(c^{-k}) with the specific constant c = √(3+√17)/2 is central to the claimed improvement over O(n^{3/2}). The recurrence or closed-form analysis that produces this exponent (and confirms it is strictly positive for each fixed k) should be stated with all intermediate steps so that the query-complexity bound can be checked independently."}],"tokens_in":1624,"tokens_out":536,"duration_ms":39633,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key thing to know is that this work carves out which path-containment variants are equivalent under randomized reductions and then beats the prior O(n^{3/2}) upper bound on one of those classes with a quantum-walk algorithm that reaches Õ(n^{3/2 - α_k}) where α_k decays exponentially as roughly 1.33^{-k}. They also tie the class to a conditional lower bound from graph collision, so the linear-query regime is ruled out unless that other problem improves dramatically. That combination of an explicit better exponent and the full partition of variants is the actual advance here; it is not just a rephrasing of earlier subgraph results.","headline":"The paper gives a concrete improvement to the quantum query bound for an equivalence class of path and cycle problems via a new quantum walk, plus a clean dichotomy, but the constant-factor claim on the randomized reductions is the part that needs the closest look.","tokens_in":2487,"tokens_out":229,"would_cite":true,"duration_ms":23634,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Quantum query algorithms for paths/cycles use walks and reductions unrelated to J-cost or φ-ladder","alignment":"orthogonal","rationale":"Paper centers on MNRS quantum walks, randomized reductions via color-coding/layer insertion, and recurrence solving for exponents α_k ~ c^{-k} (c≈1.33) in query complexity for DirPath=k etc. No J(x)=½(x+x^{-1})-1, no φ, no 8-tick periodicity, no parameter-free constant derivation, no recognition-cost forcing. Domain is complexity theory; RS framework (reality_from_one_distinction, J-uniqueness via Aczél, AlexanderDuality for D=3) has no opinion on subgraph query bounds.","tokens_in":62069,"confidence":"high","tokens_out":168,"duration_ms":23554,"cache_read_input_tokens":32896,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-05-12T02:15:46.902250+00:00","model_set":{"reader":"grok-4.3"},"falsifier":null,"supporting_citations":[],"review_version":1}