{"id":"95c6018e-01c5-4df0-8ad6-f4cfe8c4c3b0","arxiv_id":"2504.12607","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Using variational imaginary time evolution with a Max-Cut-designed ansatz, the paper reports lower optimality gaps than VQE-style training on small simulated Multiple Knapsack instances.","lead":"This paper applies a quantum algorithm called variational imaginary time evolution, paired with a Max-Cut-specific ansatz, to small instances of the Multiple Knapsack Problem. It reports lower optimality gaps than standard variational quantum eigensolver methods on 68 simulated instances.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The mean optimality gap in Eq. (18) can be negative for infeasible bit-strings, so the reported QITE advantage may be an artifact of the gap convention; the rescaling scale d is also undisclosed.","rationale":"The paper's central claim is that VarQITE+iHVA achieves significantly lower mean optimality gaps than VQE methods on MKP instances. The reader identified ansatz expressivity as the weakest assumption. I agree that VarQITE's variational projection is an assumption, but the more immediate threat to the headline is the definition of the metric itself. Eq. (18) is applied to the sampled bit-string, and no feasibility restriction is stated. For a constrained problem, an infeasible bit-string can have a raw objective above the true optimum, producing negative gaps that would make a method look artificially good. The paper does not resolve this: the 'mean optimality gap' is defined over all trials, while Fig. 3(c) is plotted on a y-axis starting at 0.0. These two statements are inconsistent unless infeasible trials are excluded or clipped, which is never declared. This is load-bearing because the central comparison is a ranking of mean gaps; a convention change could reorder the methods. The undisclosed rescaling scale d compounds the issue: the claimed advantage of the rescaled variant depends on a hyperparameter that is not fixed or reported, and Sec. IV shows that different d values qualitatively change whether the optimum is reached. The reader's expressivity concern is worth checking, but for the tested small instances the empirical convergence itself suggests the ansatz is at least adequate; the metric and hyperparameter issues are necessary conditions for accepting any empirical claim. My recommendation remains CONDITIONAL, matching the reader's verdict, with the added explicit condition that the gap convention and d be specified and robustness tests provided. No change in verdict is needed unless the recomputation shows the ranking flips.","tokens_in":13989,"tokens_out":13944,"duration_ms":153313,"concrete_test":"Re-run the 68-instance benchmark and recompute Table I and Fig. 3(c) under three gap conventions: (a) exclude infeasible trials; (b) set gap = 1 for infeasible trials; (c) use Eq. (18) without clipping. Report, per method, the fraction of trials with C_VQA > C_opt and the mean gap restricted to feasible trials. If the ordering of mean/median gaps (rescaled QITE+iHVA < QITE+iHVA < HEA < ma-QAOA < iHVA) does not survive convention (a), the central claim is a metric artifact. Also disclose the d used for each instance in the rescaled column and rerun with a fixed rule, e.g., d = ||H|| or d = 10, to determine whether the Table I advantage depends on per-instance tuning.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 3 / Eq. (18) defines the optimality gap as 1 - C_VQA/C_opt without restricting C_VQA to feasible solutions. The protocol in Sec. III (item 6) says the sampled bit-string is substituted back into the original problem, but an infeasible MKP string can set x_ij = 1 for the same item in several knapsacks or exceed capacity, giving an objective larger than C_opt and hence a negative gap. If such trials enter the mean, a method that frequently outputs infeasible but high-value strings is scored as closer to optimal than a method that respects constraints. The paper never states whether infeasible trials are excluded, set to gap = 1, or clipped at 0; Fig. 3(c) starts its y-axis at 0, suggesting clipping or exclusion, but the text says the mean is over all trials. This convention directly controls Table I: rescaled QITE+iHVA has mean feasibility 86.7%, so 13.3% of its trials are infeasible; iHVA has 75.6% feasibility, so 24.4% are infeasible. If negative gaps are allowed, the comparison is biased in a method-dependent way. In addition, the rescaled headline uses an undisclosed value of d; Sec. IV only reports that d = 10 is best and d = 100 is cheap, and the scale is a free hyperparameter that can be tuned per instance. The reader's iHVA expressivity concern is plausible but secondary: for 10-13 qubit instances, the empirically reported success itself suggests the ansatz is adequate, whereas the gap convention affects every reported number.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proposes a variational quantum imaginary time evolution (VarQITE) framework for the Multiple Knapsack Problem (MKP). The MKP is converted to an unbalanced QUBO and then to a Max-Cut instance so that a Max-Cut-tailored imaginary Hamiltonian variational ansatz (iHVA) can be used. The authors compare QITE+iHVA with iHVA, ma-QAOA, and HEA optimized by VQE on 68 small MKP instances (10-13 qubits), reporting feasibility rates, optimality rates, and mean optimality gaps. They find that QITE+iHVA, especially with rescaled Hamiltonian coefficients, achieves lower mean optimality gaps, and they discuss how scaling the Hamiltonian by 1/d allows larger effective time steps and fewer QITE iterations.","tokens_in":14379,"tokens_out":6854,"duration_ms":68819,"significance":"The paper addresses a timely problem: applying VarQITE to constrained combinatorial optimization, specifically MKP. If the reported results hold, the work would provide one of the first demonstrations that variational imaginary-time evolution can beat conventional VQE/QAOA variants on constrained QUBOs, and the Hamiltonian-rescaling observation in Eq. (22) is a useful practical idea for reducing QITE simulation cost. The benchmark is externally anchored to Gurobi optima, several ansatze are compared under a common workflow, and the rescaling algebra is internally consistent. However, the numerical support is currently weakened by the undefined treatment of infeasible trials in the mean optimality gap and by the undisclosed scale factor d; the experiments are also limited to 10-13 qubits. The iHVA expressivity concern raised by the stress-test note is plausible but secondary on these small instances. With the main reproducibility issues resolved, the qualitative conclusion that QITE+iHVA is competitive with or better than VQE baselines on small MKP instances would be credible.","major_comments":[{"comment":"The mean optimality gap is not well-defined for infeasible solutions. The definition substitutes the sampled bit-string back into the original constrained problem; an infeasible string can produce an objective value larger than C_opt, making the gap negative. The text says the average is taken over all trials, while Fig. 3(c) starts its y-axis at 0, so it is not clear whether infeasible trials are excluded, clipped, or counted at face value. Since Table I and Fig. 3(c) are the primary evidence for the central claim, the authors must state the convention explicitly and recompute the mean gaps under a fair convention, such as feasible trials only or mapping infeasible trials to a fixed gap.","section":"Section III, Definition 3, Eq. (18)"},{"comment":"The scale factor d used for the 'rescaled QITE+iHVA' results is not reported anywhere in the manuscript. Section IV only says d=10 is best for solution quality and d=100 is best for cost on the six instances in Fig. 4, and the main experiments in Table I and Fig. 3 do not state which d was used or whether it was chosen per instance. Because d is a tunable hyperparameter, the improved mean gap of the rescaled method in Table I is not reproducible and may reflect tuning on the benchmark instances. Please disclose d for the main experiments and provide a sensitivity analysis over d.","section":"Section IV, Eq. (21), Table I, Fig. 3"},{"comment":"The main QITE experiments do not report the total evolution time tau and the number of time steps N_tau. The experimental settings in Section III list the optimizer, ansatz, and penalty coefficients, but omit these two parameters, although Eq. (19) identifies them as controlling accuracy and cost, and Fig. 4 fixes tau=10. Without these values, the QITE results in Table I cannot be reproduced, and the comparison with VQE methods is under-specified. Please add these settings to Section III and state whether they were fixed across instances.","section":"Section II.C, Eq. (17) and Section IV, Eq. (19)"}],"minor_comments":[{"comment":"The expressivity assumption of iHVA is stated but not empirically checked; a sentence acknowledging that iHVA may not track the imaginary-time-evolved state for larger instances would help scope the claims.","section":"Section II.C, Section III"},{"comment":"The text says 'the mean optimality gap of QITE is drastically lower than those of the classical VQE methods,' but the table reports 0.64 for QITE+iHVA and 0.31 for rescaled QITE+iHVA; please clarify which version is meant in each statement.","section":"Section III, Fig. 3(c) and Table I"},{"comment":"The column headers 'Feasibility' and 'Optimality' would be clearer if they repeated 'best out of 5' in both the header row and the caption, since the table uses two different notions of feasibility/optimality rates.","section":"Table I caption"},{"comment":"There is no data or code availability statement; providing the 68 MKP instances and the implementation would significantly improve reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The central comparison in Table I is likely salvageable, but the negative-gap ambiguity and the undisclosed scale factor d must be resolved before the paper can be accepted. Given the small instance sizes and the absence of code/data, I recommend major revision rather than rejection; a careful reanalysis with explicit infeasibility handling and hyperparameter disclosure would make the paper acceptable as a proof-of-concept study. The novelty claim of being the first VarQITE application to constrained optimization should also be checked against Ref. [30] and Ref. [31]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a modest, cleanly written empirical study that combines known ingredients—unbalanced QUBO penalties, QUBO-to-Max-Cut reduction, the iHVA ansatz, and VarQITE—into a first benchmark against BFGS-trained VQEs on small MKP instances. The central numerical claim, that QITE+iHVA lands closer to Gurobi optima than the VQE baselines, is plausible, but the specific mean-optimality-gap numbers in Table I are not yet trustworthy because of how the gap is defined and how the rescaled Hamiltonian is tuned.\n\nWhat's good: the pipeline is sensible and reproducible in principle. Using the unbalanced penalty avoids slack qubits; the QUBO-to-Max-Cut conversion is correctly applied from Barahona; the iHVA ansatz is a legitimate choice for this Hamiltonian, and the fact that it converges for 10–13 qubits is reasonable evidence that expressivity is adequate at this scale. The rescaling derivation in Section IV is internally consistent—scaling H by 1/d is equivalent to a larger time step—and the experiments showing d=10 beats d=1 while d=1000 fails are informative. The paper also honestly flags that choosing d is empirical and that scaling does not always converge.\n\nThe soft spots are not fatal but they are load-bearing for the headline. First, Definition 3/(18) defines optimality gap as 1 - C_VQA/C_opt without restricting C_VQA to feasible bit-strings. An infeasible string can have C_VQA > C_opt, giving a negative gap, so a method that outputs infeasible high-value strings scores better by this metric. The text says the mean is over all trials, and Table I shows rescaled QITE+iHVA has 86.7% mean feasibility, meaning 13.3% of trials are infeasible. The paper never states whether infeasible trials are excluded, clipped at zero, or included as negative. Fig 3(c) starts its y-axis at 0, suggesting clipping, but that is not stated. This directly affects every number in Table I.\n\nSecond, the rescaled QITE+iHVA results—the method that wins on all metrics—use an undisclosed value of d. Section IV reports d=10 is best and d=100 cheap on a handful of instances, but the main experiments never say which d was used or how it was chosen per instance. That's a free hyperparameter tuned on the same instances used for evaluation.\n\nThird, the abstract says 'compared to QAOA' but the baselines are ma-QAOA, HEA, and iHVA with BFGS, not QAOA. That overstates the comparison. There is also no published dataset and no significance testing; the 68 instances are small and the spreads in Fig 3 are large.\n\nBottom line: the qualitative direction of the result is plausible and worth a serious referee, but the paper needs revision before the central claim can be accepted. The reviewer should ask for the gap-handling convention to be stated explicitly and the infeasible trials to be reported separately, for d to be disclosed, and for the abstract to match the actual baselines. A reader working on QITE or constrained quantum optimization will find this a useful benchmark to cite once those numbers are cleaned up. It is not a desk reject.","headline":"Plausible but under-specified empirical case for VarQITE on constrained optimization; the headline gap numbers are likely biased by an ambiguous infeasible-solution convention and an undisclosed rescaling factor.","tokens_in":14911,"tokens_out":2533,"would_cite":false,"duration_ms":25219,"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":"Variational quantum imaginary time evolution solves constrained problems like the Multiple Knapsack Problem with lower optimality gaps than QAOA and VQE.","keywords":["variational quantum imaginary time evolution","VarQITE","multiple knapsack problem","QUBO","Max-Cut","imaginary Hamiltonian variational ansatz","constrained combinatorial optimization","Hamiltonian rescaling"],"falsifier":"Run the same QITE+iHVA versus VQE comparison on larger MKP instances (e.g., 20 items) and check whether the mean optimality gap advantage persists; or deliberately pick Max-Cut graphs where the spanning-tree-based iHVA is known to be inexpressive and see whether VarQITE's energy stops following imaginary-time evolution.","tokens_in":13785,"feed_emoji":"🎒","tokens_out":4457,"duration_ms":41845,"temperature":0.7,"pith_summary":"This paper argues that imaginary time evolution, implemented variationally, is a better optimizer for constrained combinatorial problems than conventional variational quantum eigensolver approaches. It tests this on the Multiple Knapsack Problem with up to three knapsacks and four items, converting each instance into an unbalanced QUBO and then a Max-Cut graph to use the imaginary Hamiltonian variational ansatz. Across 68 instances, the QITE+iHVA protocol achieves feasibility rates comparable to VQE methods but with drastically lower mean optimality gaps. Rescaling the Hamiltonian coefficients further narrows the gap and reduces the number of integration steps needed.","feed_headline":"Imaginary-time quantum solver narrows knapsack optimality gap","feed_subtitle":"On 68 instances, variational imaginary time evolution with a Max-Cut ansatz lands much closer to optimal solutions than QAOA or VQE.","key_machinery":"The framework chains three components: the unbalanced penalization QUBO formulation that avoids slack variables, a conversion of the QUBO into a Max-Cut instance by adding one vertex, and the iHVA ansatz whose RZY gates are assembled from breadth-first spanning trees of the Max-Cut graph. Parameter updates come from the McLachlan variational principle, which solves the linear system M θ̇ = V at each imaginary-time Euler step. Scaling the Hamiltonian by 1/d is shown to be equivalent to taking larger time steps, reducing the number of linear-system solves.","core_discovery":"The paper's central claim is that the variational quantum imaginary time evolution (VarQITE) protocol, combined with the Max-Cut-tailored imaginary Hamiltonian variational ansatz (iHVA), finds solutions far closer to the exact optimum than conventional VQE methods. On the 68 tested MKP instances, the median optimality gap for the rescaled QITE+iHVA is roughly 0.2, while HEA and ma-QAOA sit around 2.0-2.5 and iHVA optimized by BFGS near 4.0. The authors attribute this to the McLachlan variational principle, which updates parameters by tracing imaginary-time evolution rather than minimizing a nonconvex expectation landscape.","pith_inferences":["If the iHVA expressivity assumption holds at scale, the advantage over VQE could grow on larger instances where classical local optimizers get trapped in more local minima.","The rescaling trick suggests an adaptive schedule that adjusts d along the evolution could further reduce the number of steps without losing accuracy.","The QUBO-to-Max-Cut conversion may be a key enabler: because iHVA exploits Max-Cut's bit-flip symmetry, a direct QUBO ansatz might not reap the same QITE benefit.","The claimed advantage is measured on 10-13 qubit instances; a natural test is to see whether the mean optimality gap separation persists on 20+ qubit problems."],"forward_implications":["Constrained problems expressible as QUBO can be attacked with VarQITE by converting them to Max-Cut, avoiding the ancilla qubits and multi-controlled Toffoli gates of QAOA+.","Hamiltonian rescaling acts as a cost-accuracy hyperparameter: moderate scaling (e.g., d = 10) improves solution quality, while larger scaling cuts the number of time steps but can miss the optimum.","Because QITE's parameter path bypasses nonconvex gradient landscapes, the method may be less sensitive to random initialization than VQE.","The same pipeline could be applied to other inequality-constrained problems such as bin packing or vehicle routing.","On real hardware, the bottleneck shifts to estimation of the matrix M and vector V, which require circuit evaluations and classical linear algebra at every step."],"supporting_citations":[{"why":"Introduces variational quantum imaginary time evolution and the McLachlan-principle parameter update used throughout.","marker":"[25]"},{"why":"The McLachlan variational principle that VarQITE relies on to approximate imaginary-time evolution.","marker":"[26]"},{"why":"Defines the iHVA ansatz tailored to Max-Cut's bit-flip symmetry, which the paper reuses.","marker":"[31]"},{"why":"Unbalanced penalization QUBO formulation that avoids slack variables for inequality constraints.","marker":"[37]"},{"why":"Algorithm for converting a QUBO instance to a Max-Cut instance, used to build iHVA.","marker":"[39]"},{"why":"Multi-angle QAOA, a baseline method the paper compares against.","marker":"[9]"},{"why":"Gurobi is used to compute exact optima for the 68 MKP instances, defining the optimality gaps.","marker":"[45]"},{"why":"Original non-variational QITE whose gate cost motivates the variational form.","marker":"[24]"}],"fun_headline_variants":["VarQITE narrows knapsack optimality gap vs QAOA","VarQITE outperforms QAOA and VQE on knapsack","Imaginary-time variational solver beats QAOA on knapsack","VarQITE reduces knapsack optimality gap","VarQITE beats QAOA for constrained optimization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The iHVA ansatz must be expressive enough to approximate the true imaginary-time-evolved state at every Euler step; if the ansatz manifold is too small, the reported gap advantage could be an artifact of the 10 to 13 qubit instances selected.","fun_headline_variants_meta":{"raw":{"variants":["VarQITE narrows knapsack optimality gap vs QAOA","VarQITE outperforms QAOA and VQE on knapsack","Imaginary-time variational solver beats QAOA on knapsack","VarQITE reduces knapsack optimality gap","VarQITE beats QAOA for constrained optimization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001464,"raw_usage":{"total_tokens":5911,"prompt_tokens":989,"completion_tokens":4922,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":4836}},"tokens_in":605,"tokens_out":4922,"duration_ms":36132,"temperature":1.0,"reasoning_tokens":4836,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:27:44.077803+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same QITE+iHVA versus VQE comparison on larger MKP instances (e.g., 20 items) and check whether the mean optimality gap advantage persists; or deliberately pick Max-Cut graphs where the spanning-tree-based iHVA is known to be inexpressive and see whether VarQITE's energy stops following imaginary-time evolution.","supporting_citations":[{"cited_title":"Variational ansatz-based quantum simulation of imaginary time evolu- tion,","cited_arxiv_id":null,"evidence_quote":"Introduces variational quantum imaginary time evolution and the McLachlan-principle parameter update used throughout."},{"cited_title":"A variational solution of the time-dependent schrodinger equation,","cited_arxiv_id":null,"evidence_quote":"The McLachlan variational principle that VarQITE relies on to approximate imaginary-time evolution."},{"cited_title":"Experiments in quadratic 0-1 programming,","cited_arxiv_id":null,"evidence_quote":"Algorithm for converting a QUBO instance to a Max-Cut instance, used to build iHVA."},{"cited_title":"Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution,","cited_arxiv_id":null,"evidence_quote":"Original non-variational QITE whose gate cost motivates the variational form."}],"review_version":1}