{"id":"393a3ee3-3cc3-4e8c-a657-d8973e14e238","arxiv_id":"2608.05394","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A column-wise swap mixer for QAOA-based TSP and VRP is proposed, but its claimed feasibility guarantee is contradicted by the paper's own inter-vehicle swap equations.","lead":"This paper proposes a quantum search mixer that is supposed to keep candidate delivery routes valid at all times, and compares it with penalty-based quantum routing on small TSP and VRP instances. It matters because if the design worked, quantum optimizers could avoid wasting samples on impossible routes, but the central proof has a flaw.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1's inter-vehicle XY terms do not preserve feasibility: swapping a customer bit into a vehicle-position already occupied by another customer creates a two-customer block, so the central feasibility guarantee fails as stated.","rationale":"The reader's weakest_assumption identifies the same load-bearing defect: the proof of Proposition 1 assumes every XY exchange term maps feasible basis states to feasible basis states, but the inter-vehicle terms in Eq. 18 do not. The paper's own example encoding allows a vehicle-position to be empty, but the formal Omega_F requires exactly one customer per vehicle-position; for the experimental VRP instance, that formal set is empty. Because the feasibility-preserving guarantee is the paper's central claimed contribution, and the numerical comparison is interpreted through that guarantee, the invalid Proposition 1 is sufficient to reject the central claim as stated. I see no independent support that repairs this: there is no machine-checked proof, no released code, and the reported success of QAOA+ on the VRP would need to be re-examined under a corrected mixer. The paper is clearly written and its scope is modest, but the central mathematical claim is not supported. The appropriate verdict is REJECT, matching the reader's assessment.","tokens_in":24080,"tokens_out":4484,"duration_ms":43546,"concrete_test":"Use a 4-qubit statevector simulator to apply the inter-vehicle term g_{a1,c,k;a2,c,k} = X*X + Y*Y from Eq. 18 to a basis state with bits set as |1>_{a1,c,k}|0>_{a2,c,k}|1>_{a2,c',k} (with c' != c) and all other qubits chosen to satisfy the paper's Omega_F. Compute the resulting state; it will have a nonzero amplitude 2 on the basis state with both customer c and customer c' set to 1 in vehicle a2 at position k, which violates sum_c x_{a2,k,c} = 1. Separately, enumerate all bitstrings satisfying the two constraints defining Omega_F for the paper's VRP instance (|A|=2, K=3, |C|=3); the count is 0 because there are 6 exactly-one-hot slots but only 3 customers, showing that the formal feasibility set cannot be the encoding used in Section 3.3. Either outcome settles that Proposition 1 is false as stated.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim is Proposition 1, which asserts that the column-wise swap mixer preserves the feasible subspace H_F. The proof fails at part (i). Eq. 18's inter-vehicle term couples the same customer index c across vehicles at the same position k. Acting on a feasible basis state in which vehicle a1 has customer c at position k and vehicle a2 has a different customer c' at position k, the XY term g_{a1,c,k;a2,c,k} maps |10> to 2|01> on the two c-qubits. The resulting state has vehicle a2 containing both c and c' at the same position, violating the one-hot constraint sum_c x_{a2,k,c} = 1, while vehicle a1 is left empty at that position. Thus the proof's assertion that each g_pq maps feasible basis states to feasible basis states is false; leakage occurs even under the Proposition's own exactly-one-hot definition. Separately, the formal definition of Omega_F as sum_c x_{a,c,k} = 1 for all a,k conflicts with Section 3.3's at-most-one-hot encoding with all-zero unused blocks. For the paper's VRP instance, |A|*K = 6 slots but |C| = 3 customers, so Omega_F contains no bitstrings at all. The numerical comparison cannot rescue the claim because the alleged feasibility-preserving mixer is not proven to preserve feasibility, and the empirical feasibility rates would need to be recomputed once the mixer is corrected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a quantum alternating operator ansatz (QAOA+) framework for TSP and VRP in which feasibility is enforced by a custom 'column-wise swap' mixer rather than by QUBO penalty terms alone. The authors introduce a jumper mixer (full XY swaps) and a search mixer (partial XY swaps), and they compare three architectures—penalty-based QUBO-QAOA, penalty-free QAOA+, and Hybrid QAOA+—on small instances using a classical simulator. The central theoretical claim is Proposition 1: the column-wise swap mixer preserves the feasible Hilbert space H_F. I find that the proof of Proposition 1 is incorrect, and the formal definition of the feasible set conflicts with the encoding used in the experiments; these are load-bearing defects that invalidate the paper's main claim as stated.","tokens_in":24373,"tokens_out":2669,"duration_ms":26028,"significance":"If Proposition 1 were correct, the paper would make a useful methodological contribution by transferring feasibility-preserving neighborhood logic from classical routing heuristics into quantum mixer design, and the controlled comparison of three constraint-handling architectures would be a valuable empirical reference. The paper is clearly written, the experimental setup is transparent, and the authors are appropriately cautious about scalability claims. However, the central feasibility-preservation guarantee fails under the paper's own definitions, and the numerical results therefore do not provide evidence for the claimed architectural benefit. The core idea may be salvageable with a redesigned mixer, but the present manuscript does not establish it.","major_comments":[{"comment":"Part (i) of the proof is incorrect. The inter-vehicle term of H_jump couples the same customer index c across two vehicles at the same position k: X_{a1,c,k}X_{a2,c,k} + Y_{a1,c,k}Y_{a2,c,k}. On a feasible basis state in which vehicle a1 has customer c at position k and vehicle a2 has a different customer c' at the same position, this term acts on the two c-qubits as |10> -> 2|01>, producing a state in which vehicle a2 contains both c and c' at position k. This violates the one-hot constraint sum_c x_{a2,c,k} = 1, even under the exact-one-hot definition used in Proposition 1. The assertion that each g_pq maps feasible basis states to feasible basis states is therefore false, and the power-series argument for U_jump does not go through. Leakage occurs for the very inter-vehicle moves that the mixer is designed to permit.","section":"§3.5, Proposition 1 proof, Eqs. (18) and (19)"},{"comment":"The formal feasible set Ω_F requires sum_c x_{a,c,k} = 1 for every vehicle-position pair (a,k), i.e., exactly one customer per position. Section 3.3, however, explicitly defines the block encoding as at-most-one-hot, with the all-zero block representing an unused route position. For the VRP instance used in Section 4 (|A|=2, K=3, |C|=3), there are 6 vehicle-position slots but only 3 customers, so no bitstring can satisfy the exact-one-hot condition; Ω_F is empty. This makes the formal feasibility-preservation claim vacuous under its own definition, while the numerical experiments and the notion of 'feasible routes' in Section 3.6 rely on a different, at-most-one-hot feasibility notion that is never formalized. The inconsistency is load-bearing because Proposition 1 is stated for an empty subspace.","section":"§3.5, definition of Ω_F, vs. §3.3 encoding"},{"comment":"Part (ii) is circular. N_local is defined as the set of index pairs whose exchange preserves the one-hot and permutation constraints, so H_search is feasibility-preserving by construction; the proof merely restates the definition. The substantive claim therefore rests entirely on the jumper mixer, whose proof fails as noted in the first comment. Moreover, if Ω_F is empty, the condition 'exchanging ... in any feasible assignment yields another feasible assignment' is vacuously satisfied by all pairs, making N_local ill-defined with respect to the actual encoding used in the experiments.","section":"§3.5, Eq. (20) and Proposition 1 part (ii)"}],"minor_comments":[{"comment":"The sentence 'using column-wise swap moves, it restricts evolution to feasible configurations' is a comma splice; it should be rephrased, e.g., 'which restricts evolution to feasible configurations.'","section":"Abstract"},{"comment":"The text introduces the binary variable as x_{a,i,k} but the rest of the paper uses x_{a,c,k}; the index naming should be made consistent.","section":"§3.3"},{"comment":"For TSP QAOA+ at p=3, the table reports 'Failed*' for the cost ratio but also reports a near-optimal sampling probability of 0.011; the footnote should clarify under what decoding rule a run can be 'failed' while still producing near-optimal samples.","section":"Table 6"},{"comment":"The reference to Hansen and Mladenović is missing the character 'ć' in the author name and the title of the 2001 work is incomplete.","section":"References"},{"comment":"The penalty weight λ1 is set to 60 for QUBO, 5 for Hybrid QAOA+, and 0 for QAOA+, but the text does not justify the factor-of-12 difference between QUBO and Hybrid; adding a sentence on how these values were calibrated would improve reproducibility.","section":"§3.7, Table 5"}],"recommendation":"reject","confidential_remarks":"The manuscript's central theoretical guarantee—Proposition 1—is false under the paper's own definitions, and the formal feasible subspace is empty for the experimental instance. This is not a matter of presentation or missing references; it is a load-bearing error that affects the interpretation of all numerical results. The authors could potentially redesign the mixer (e.g., by using full SWAP operations between positions that are guaranteed to exchange feasible blocks, or by explicitly enforcing at-most-one-hot in the mixer definition) and re-run the experiments, but as submitted the paper does not meet the bar for publication. A major rewrite and new experiments would be needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the central theorem is false. The inter-vehicle term in Eq. 18 swaps a single customer bit between two vehicles at the same position, not the whole column. If vehicle a1 has customer c at position k and vehicle a2 has a different customer c' at position k, the XY term on the c-qubits maps |10> to |01>; vehicle a2 now has both c and c' at one position. So the proof's claim that each g_pq maps feasible basis states to feasible basis states fails. The 'jumper' mixer does not preserve feasibility.\n\nThe paper is not sloppy overall. It asks a good question—can routing feasibility be embedded in a QAOA+ mixer rather than penalized—and the three-way architecture comparison is a sensible experimental design. The encoding and Hamiltonian are laid out readably, and the discussion of how feasible-space topology might change the value of penalty guidance is worth reading. The writing is clear and the literature is engaged.\n\nThe other problems are minor in comparison but real. Omega_F is defined as exactly-one-hot per vehicle-position, which contradicts the at-most-one-hot encoding in Section 3.3 and is actually empty for the 2-vehicle, 3-customer VRP instance (6 slots, 3 customers). The numerical section uses single runs, no seeds, hand-tuned lambdas, and no code, so none of the empirical claims can be checked—and once the mixer is corrected, the feasibility rates in Table 6 would need to be recomputed anyway.\n\nThe right fix is conceptually clear: the inter-vehicle move should swap full position columns between vehicles, which means acting on all customer qubits in the block together, not on single c-qubits. With that correction, the framework might hold up and the TSP/VRP contrast could be informative. As written, the load-bearing guarantee fails, so I would not cite this version. If it reaches my desk, I would reject, but a referee round is defensible because the idea is serious and the error is contained.","headline":"The central feasibility-preservation theorem is false: the inter-vehicle XY term swaps a single customer bit, not the whole column, so the proposed QAOA+ mixer can leak into two-customers-per-position states.","tokens_in":24929,"tokens_out":3664,"would_cite":false,"duration_ms":34050,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A column-wise swap mixer keeps quantum routing search inside the feasible solution space, and on small TSP and VRP instances that architectural choice determines how much probability lands on usable low-cost routes.","keywords":["Vehicle Routing Problem","Traveling Salesperson Problem","Constrained optimization","Quantum optimization","Quantum Alternating Operator Ansatz","Feasibility-preserving mixer","QUBO"],"falsifier":"Enumerate all feasible basis states for the paper's two-vehicle, three-customer, three-position encoding, apply $U_{\\mathrm{jump}}(\\pi/4)$ and $U_{\\mathrm{search}}(\\beta)$ to each, and check whether any output state has a vehicle with two customers in the same position; one such transition falsifies Proposition 1 under the at-most-one-hot encoding used in the experiments.","tokens_in":1701,"feed_emoji":"🚚","tokens_out":4548,"duration_ms":122650,"temperature":0.7,"pith_summary":"This paper tries to show that for constrained transportation routing, feasibility can be built into quantum search instead of only patched on with penalty terms. It introduces a column-wise swap mixer for the Quantum Alternating Operator Ansatz, QAOA+, and proves in Proposition 1 that this mixer never moves a feasible route assignment into an assignment-infeasible one. On small TSP and VRP instances, it compares penalty-based QUBO-QAOA with penalty-free QAOA+ and a hybrid version; the mixer-based variants concentrate more sampled probability on feasible low-cost routes. If the claim holds, constraint handling in near-term quantum optimization becomes a question of designing admissible transitions, not of calibrating penalties.","feed_headline":"Swap-based mixer keeps quantum routing search feasible","feed_subtitle":"Penalty-free QAOA+ samples low-cost valid routes where QUBO-QAOA puts probability on infeasible ones.","key_machinery":"The load-bearing object is the column-wise swap mixer $U_M = U_{\\mathrm{search}}(\\beta)\\,U_{\\mathrm{jump}}(\\pi/4)$, assembled from two-qubit XY-exchange terms $X_p X_q + Y_p Y_q$ acting on the vehicle–position–customer encoding. A column is a route-position block in one vehicle's assignment table, and a swap exchanges customer assignments either within a vehicle (resequencing a route) or across vehicles (reassigning customers). On basis states, an XY term sends $|01\\rangle \\leftrightarrow |10\\rangle$ and vanishes on $|00\\rangle$ and $|11\\rangle$, so the mixer only pivots between assignments that differ by exchanging two one-hot entries. The proof of Proposition 1 relies on this support property to conclude that both the full-swap jumper and the partial-swap search ingredient preserve the one-hot and customer-count constraints, and therefore preserve $\\mathcal{H}_F$.","core_discovery":"On the paper's own terms, the central discovery is that feasibility can be made a property of the search operator rather than the objective. Under a vehicle–position–customer bit encoding, the proposed column-wise swap mixer is built from XY-exchange couplings; it has a jumper component that performs full column swaps at a fixed $\\pi/4$ angle and a search component that performs partial swaps with a tunable angle. Proposition 1 asserts that both components map the feasible Hilbert space into itself, so a QAOA+ circuit that starts in a feasible state never acquires amplitude on assignment-infeasible states. The numerical comparison on small instances then shows that this architectural choice changes sampling behavior: QUBO-QAOA spreads probability over feasible and infeasible high-energy states, while QAOA+ and Hybrid QAOA+ concentrate more mass on feasible low-cost routes. The paper presents the difference between the TSP result, where the hybrid's added penalty guidance concentrates probability near the optimum, and the VRP result, where penalty-free QAOA+ preserves exploration and reaches the optimum at both tested depths.","pith_inferences":["Since the proof's feasible set is defined with exactly one customer per position while the experiments use at-most-one-hot blocks with empty slots, a natural extension is to carry the invariance argument through the at-most-one-hot case; the paper leaves that derivation to follow-up work.","The TSP/VRP reversal suggests a design rule the authors do not state: for compact, symmetric feasible regions use mild penalty guidance, and for multi-vehicle decomposable regions prefer pure mixer feasibility preservation.","A testable next step is to add capacity or time-window constraints and define the allowed swap set so both resulting routes remain resource-feasible; this would test whether the mechanism extends beyond one-hot structure."],"forward_implications":["A QAOA+ run initialized in a feasible superposition keeps zero amplitude on assignment-infeasible route configurations, eliminating the need for assignment penalties in the mixer-based variants.","On the tested small instances, penalty-based QUBO-QAOA spends a large part of its measurement shots on infeasible states; the mixer-based variants obtain substantially higher feasible-route sampling fractions.","In the compact TSP feasible space, adding a small penalty guidance to the feasibility-preserving mixer concentrates about 45% of sampled probability within 5% of the optimum at depth $p=2$.","In the multi-vehicle VRP instance, penalty-free QAOA+ finds the optimal cost 30.0 at both $p=2$ and $p=3$, whereas deeper hybrid penalty guidance converges to a suboptimal feasible route at cost 33.0."],"supporting_citations":[{"why":"Supplies the Quantum Alternating Operator Ansatz framework and problem-specific mixers that this paper's column-wise swap mixer extends.","marker":"Hadfield et al., 2019"},{"why":"Defines the standard QAOA algorithm that serves as the baseline architecture for the penalty-based QUBO formulation.","marker":"Farhi et al., 2014"},{"why":"Gives the Ising/QUBO encoding of combinatorial problems, the basis for the penalty-term representation of TSP and VRP constraints.","marker":"Lucas, 2014"},{"why":"Provides the constraint-preserving mixer concept for one-hot constraints that motivates and supports the proposed feasibility-preserving design.","marker":"Fuchs et al., 2022"},{"why":"Represents the existing quantum approximate optimization approach to the VRP that the paper contrasts with its QAOA+ formulation.","marker":"Azad et al., 2023"},{"why":"Shows the QUBO modeling of routing and ride-hailing problems, the penalty-based approach that the paper compares against.","marker":"Cattelan and Yarkoni, 2024"},{"why":"Informs the calibration of penalty weights in QUBO permutation problems, used in the numerical parameter choices.","marker":"Ayodele, 2022"}],"fun_headline_variants":["Swap mixer keeps quantum routing search on valid routes","Feasibility-preserving mixer for quantum routing search","Quantum search for routing stays feasible with swap moves","Swap-based quantum search enforces routing feasibility"],"cache_read_input_tokens":26880,"weakest_assumption_plain":"The guarantee rests on the assumption that every elementary XY swap sends every feasible assignment to another feasible assignment, where the proof uses exactly-one-hot states even though the experiments run on at-most-one-hot blocks with empty positions allowed.","fun_headline_variants_meta":{"raw":{"variants":["Swap mixer keeps quantum routing search on valid routes","Feasibility-preserving mixer for quantum routing search","Quantum search for routing stays feasible with swap moves","Swap-based quantum search enforces routing feasibility"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00021,"raw_usage":{"total_tokens":1454,"prompt_tokens":1030,"completion_tokens":424,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":365}},"tokens_in":646,"tokens_out":424,"duration_ms":3882,"temperature":1.0,"reasoning_tokens":365,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T13:59:46.566651+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all feasible basis states for the paper's two-vehicle, three-customer, three-position encoding, apply $U_{\\mathrm{jump}}(\\pi/4)$ and $U_{\\mathrm{search}}(\\beta)$ to each, and check whether any output state has a vehicle with two customers in the same position; one such transition falsifies Proposition 1 under the at-most-one-hot encoding used in the experiments.","supporting_citations":[],"review_version":1}