{"id":"aa48ffef-2706-410f-9308-12fc5a233268","arxiv_id":"2505.04700","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Replacing the full QAOA Hamiltonian with an approximate quadratic or SWAP-truncated version yields shallower circuits that can outperform standard QAOA under realistic noise.","lead":"This paper proposes two circuit-simplification strategies for quantum optimization: approximate quadratization of high-order Hamiltonians and truncating the SWAP network for quadratic problems. Simulations and an IBM hardware run show that these shallower circuits can be more noise-robust than standard QAOA, at the cost of lower noiseless solution quality.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The HUBO robustness claim in Fig. 5(a) is partly a selection artifact: a 374-gate circuit retains a >1% noiseless fraction over the plotted depolarizing range, so its flat best-1% curve does not by itself demonstrate noise-robust quadratization.","rationale":"The paper is transparent about the heuristic nature of the HUBO approach and acknowledges the loss of the QAOA convergence guarantee in Section III.A. The QUBO hardware experiment in Section IV provides independent support for the general idea that truncated, shallower Ansätze can improve noisy solution quality. My concern targets a specific piece of evidence rather than the entire construction: Fig. 5(a) uses a best-α metric in a regime where the quadratized circuit has a noiseless survival fraction larger than α, so the flat curve can be explained without any robust encoding of the low-energy HUBO structure. That is a load-bearing weakness because the abstract's central claim is the HUBO noise robustness. The α=0.05 and α=0.1 panels and the QUBO hardware data keep the overall paper valuable, so a conditional verdict remains appropriate; my read does not move the reader's verdict, hence UNCHANGED.","tokens_in":16493,"tokens_out":11315,"duration_ms":121524,"concrete_test":"Re-run the 12-qubit LABS depolarizing simulation while scoring (i) all 10,000 shots (α=1) and (ii) only shots from circuits that experienced at least one depolarizing error, at α=0.01, 0.05, and 0.1. If the quadratized ansatz is no longer less noise-sensitive than standard QAOA under either condition, then the Fig. 5(a) insensitivity is an artifact of the noiseless-shot surplus, and the abstract should be restricted to tail-CVaR statements at α values above the survival fraction (1−λ)^374.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Fig. 5(a) reports the average energy of the best α=1% of 10,000 samples as a function of depolarizing strength λ applied after each two-qubit gate. The quadratized H'_2(θ) circuit has 374 CZ gates, while standard QAOA has 2094 (line) or 600 (all-to-all) CZ gates (Section III.C). Under this error model a shot is noiseless with probability (1−λ)^G. Over the plotted range λ∈[0.001,0.01], the quadratized circuit's noiseless fraction is at least 0.99^374 ≈ 0.023, which exceeds α=0.01. Hence the best 1% of the quadratized ansatz can be populated entirely by noiseless shots at every plotted noise strength: at λ=0.01, roughly 235 of 10,000 shots are expected to be error-free, more than the 100 shots selected. For the standard line ansatz, by contrast, the noiseless fraction at λ=0.01 is 0.99^2094 ≈ 8×10^−10, far below α, so its best 1% must include corrupted shots. The flat purple curve in Fig. 5(a) is therefore consistent with a trivial statement: a 374-gate circuit has a >1% noiseless survival fraction in this noise range. It does not show that the quadratization preserves useful problem structure under noise. The α=0.05 and α=0.1 panels are less vacuous because 0.023 < 0.05, but the abstract's unqualified 'more robust to noise' leans on the α=0.01 'statistically insensitive' statement. The HUBO evidence is also a single 12-qubit LABS instance with no code release, so the headline robustness claim for quadratized high-order Hamiltonians is not yet established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper addresses the excessive circuit depth and gate count of QAOA for high-order (cubic/quartic) cost Hamiltonians and for dense QUBOs. It proposes two approximate quadratizations of HUBOs—a hypergraph clique expansion and a variational fully connected quadratic Hamiltonian H'_2(θ)—that avoid ancilla overhead, and a SWAP-layer truncation method for QUBOs. Noiseless numerics on 12-qubit LABS show that the variational quadratization samples good suboptimal states with much shallower circuits; noisy simulations purport to show improved noise robustness over standard QAOA. Hardware experiments on ibm_fez with three 40-node 3-regular Max-Cut instances show an optimal number of SWAP layers and QAOA depth, and a CVaR post-selection model reproduces noiseless approximation ratios.","tokens_in":16950,"tokens_out":9594,"duration_ms":92737,"significance":"If established, the core message—that approximating the problem structure in the Ansatz can outperform implementing the full QAOA cost operator under realistic noise—would be practically useful, especially for HUBOs where exact implementation is hopeless on current hardware. The resource-scaling analysis in Table I and Fig. 1 is a useful quantitative contribution, and the hardware study is a genuine benchmark with three graphs and 50,000 samples per point, including a simple analytical noise model for the CVaR post-selection. The paper is also transparent that the variational quadratization is heuristic and that the QAOA convergence guarantee is lost. However, the HUBO noise-robustness evidence is not yet convincing for the reasons given in the major comments; the single-instance, same-cost-function training makes the result a proof of principle rather than a demonstrated method.","major_comments":[{"comment":"The flatness of the purple curve at α=0.01 is quantitatively explained by the gate count alone. With 374 CZ gates, the noiseless survival probability under the depolarizing model is (1−λ)^374 ≥ 0.99^374 ≈ 0.023 for λ∈[0.001,0.01], which is larger than α=0.01. Thus the best 100 of 10,000 samples can be entirely noiseless at every plotted noise strength, so the 'statistically insensitive' behavior does not demonstrate that the quadratization preserves useful problem structure under noise. For the standard line Ansatz (2094 CZ gates), the noiseless fraction at λ=0.01 is 0.99^2094 ≈ 8×10^−10, far below α, so its best 1% necessarily includes corrupted samples; the comparison is therefore biased by circuit size. The authors should repeat the analysis for α above the noiseless fraction, explicitly account for the finite-shot noiseless contribution, or otherwise show that the advantage persists after controlling for gate count. The α=0.05 and α=0.1 panels are less vacuous, but the abstract's unqualified 'more robust to noise' relies on the α=0.01 statement.","section":"§III.C, Fig. 5(a)"},{"comment":"The abstract states that the noise robustness is demonstrated 'through simulations of systems of 8 to 16 qubits with variable noise strengths,' but the noisy simulations in §III.C cover a single 12-variable LABS instance only. The 8–16 qubit data in Fig. 1 and Table I are circuit-resource counts, not solution-quality simulations under noise. This discrepancy overstates the evidence; the claims in the abstract should match the actual experimental scope, or additional noisy-simulation results at other sizes should be provided.","section":"Abstract vs §III.C"},{"comment":"The noiseless MPS reference used for Fig. 6(a) and for the CVaR fit in Fig. 6(c) is computed at bond dimension 20 with no convergence test. For the deepest circuits (40 qubits, k=9, p=3) the MPS truncation error could be non-negligible, and without a χ-dependence study the claims that the noiseless approximation ratio increases monotonically with k and that the fitted α recovers the noiseless value with RMSE=10^−6 are not fully supported. The authors should report at least one convergence check (e.g., χ=10, 20, 40 for one graph) and specify how the RMSE was evaluated. The hardware finding of an optimal (k,p) in Fig. 6(b) is independent of this issue, but the noiseless interpretation is not.","section":"Appendix D and §IV, Fig. 6"}],"minor_comments":[{"comment":"The set 'α∈{0.01,0.05.0.1}' contains a typographical error; it should read 'α∈{0.01,0.05,0.1}'.","section":"§III.C"},{"comment":"There is a missing space in 'Thequadratizationyieldsaninteresting tradeoff'.","section":"§III.B"},{"comment":"The product symbol in Eq. (1) appears as 'pY' in the text; the typesetting should be corrected to a proper product notation.","section":"§II, Eq. (1)"},{"comment":"The authors cite Ref. [74] for the MPS simulator, but the main text does not state which library or implementation was used; a short sentence identifying the software would aid reproducibility.","section":"Appendix D"},{"comment":"The HUBO simulation code and data are not released, while the QUBO code is linked in Ref. [71]; making the HUBO code available would improve reproducibility.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper is within scope and the hardware portion is a solid benchmark. My main concern is that the HUBO robustness claim is overinterpreted relative to the evidence, particularly in Fig. 5(a), and the abstract's '8 to 16 qubits' statement exceeds what the noisy simulations actually show. I do not see a fundamental correctness error in the methods; the paper can be made publishable by re-analyzing the noise comparison, adding a bond-dimension convergence check, and tightening the claims to match the demonstrated scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does two useful things. First, it trains a quadratic Hamiltonian H'_2(theta) as the QAOA generator for quartic HUBOs, which is genuinely different from ma-QAOA or time-block ansatze and avoids adding ancillas. Second, it gives a clean hardware study of SWAP-layer truncation for Max-Cut on 40-node graphs, showing a real optimum in (k,p) under noise. That second part is the strongest contribution: the ibm_fez data, the MPS-based training, and the simple 1/sqrt(gamma) model for the fitted CVaR quantile all hang together, and the comparison against the earlier Eagle result is informative. The paper is also transparent about losing the QAOA convergence guarantee and about the heuristic nature of the method.\n\nThe soft spots are concentrated in the HUBO robustness claim, and one of them is more serious than the reader flagged. Figure 5(a) compares a 374-CZ quadratized circuit against 2094-CZ and 600-CZ standard QAOA circuits under depolarizing noise after each two-qubit gate. For the quadratized circuit, the noiseless survival fraction is (1-lambda)^374, which stays above the plotted alpha=0.01 for all lambda up to 0.01. So its flat best-1% curve can be populated entirely by noiseless shots, and the 'statistically insensitive to noise' statement at alpha=0.01 is largely a gate-count artifact, not evidence that the quadratization preserves useful structure. The alpha=0.05 and 0.1 panels are less vacuous, but the abstract's unqualified robustness claim leans on the alpha=0.01 panel. The HUBO evidence is also a single 12-variable LABS instance, with no code or data release, and the MPS bond dimension of 20 is not convergence-tested. None of this invalidates the QUBO part, which is well-supported.\n\nWho should read this? Anyone building QAOA-style heuristics for current hardware, especially those interested in SWAP routing or CVaR-based post-selection. The quadratization idea is worth knowing about even if the evidence for it is thin. The paper deserves peer review, but a referee should push for more HUBO instances, a noise model that accounts for gate-count differences properly, and a bond-dimension check.","headline":"A practical, honestly-written QAOA circuit-simplification study whose QUBO hardware results are solid but whose headline HUBO noise-robustness claim is partly an artifact of gate-count differences in the error model.","tokens_in":17487,"tokens_out":614,"would_cite":true,"duration_ms":7187,"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":"The paper argues that deliberately implementing an approximate, shallower version of a combinatorial problem's Hamiltonian in a QAOA Ansatz -- either a trained quadratic projection of a high-order cost function or a SWAP-truncated…","keywords":["approximate quadratization","QAOA","high-order unconstrained binary optimization","LABS problem","Max-Cut","noise-aware Ansatz design","SWAP networks","depolarizing noise"],"falsifier":"Run the same depth-two quadratized-versus-standard QAOA comparison on LABS instances with 16, 20, and 24 variables at depolarizing strengths from 0.001 to 0.01; if for the best 5% of samples the standard Ansatz matches or beats the quadratized Ansatz at the noise strengths where the paper reports a threshold advantage, the central robustness claim is falsified. A second check would execute Max-Cut on more than three 40-node three-regular graphs over all SWAP-layer counts and depths; if the optimal number of SWAP layers is always zero or always maximal, the noise-aware design claim collapses.","tokens_in":16270,"feed_emoji":"⚛️","tokens_out":11365,"duration_ms":98421,"temperature":0.7,"pith_summary":"The paper tries to establish that, on noisy quantum computers, the best Ansatz for a combinatorial optimization problem is not necessarily the one that most faithfully encodes the problem. For high-order cost functions with cubic and quartic terms, it replaces the Hamiltonian used to build the Ansatz (the trial quantum circuit) by a quadratic approximation that adds no qubits; for quadratic problems, it implements only part of the Hamiltonian by limiting SWAP layers. The price is a reduction in noiseless solution quality and the loss of QAOA's asymptotic convergence guarantee. The payoff is a shallower circuit, and in simulations of 8 to 16 qubits and in hardware experiments the approximate Ansatz samples better solutions than the exact one once realistic noise is included. If this is right, a concrete design principle follows: choose Ansatz depth and problem fidelity together under the device's error budget, rather than insisting on exact problem encoding.","feed_headline":"Shallower quantum circuits beat exact QAOA under noise","feed_subtitle":"Replacing hard four-body terms with a trained two-body Ansatz cuts circuit depth and, under noise, samples better solutions.","key_machinery":"The machinery is approximate quadratization without ancilla qubits. One variant, the hypergraph clique expansion, replaces each k-body Pauli term by a clique of pairwise ZZ edges, setting edge weights to the weighted mean that minimizes the squared deviation from the original hyperedge weights; for LABS this yields the closed form $w_{ij} = 2 - \\frac{I_{ij}}{2N_{ij} + I_{ij}}$, where $I_{ij}$ marks whether the pair appears in the quadratic part and $N_{ij}$ counts hyperedges containing the pair. The variant behind the main robustness result is the variational projection $H'_2(\\theta) = \\sum_{i>j} \\theta_{ij} Z_i Z_j + \\sum_i \\theta_i Z_i$, whose weights are optimized together with the QAOA angles to minimize the energy of the original quartic Hamiltonian $H_C$. For quadratic problems, the analogous mechanism is SWAP-layer truncation: only $k$ of the $n-2$ SWAP layers needed for full connectivity are applied, so the circuit evolves a sub-Hamiltonian $H_C(k)$ that is a subgraph of the full cost Hamiltonian, and $k$ becomes a tunable noise-versus-fidelity knob.","core_discovery":"The central discovery is that approximate quadratization -- projecting a quartic Hamiltonian onto a fully connected quadratic Hamiltonian whose ZZ coefficients are optimized as variational parameters -- yields a QAOA Ansatz that is markedly less sensitive to depolarizing noise than the standard depth-two QAOA Ansatz for the 12-variable LABS problem. In the paper's noisy simulations, the best one percent of samples from the quadratized Ansatz barely degrade as noise grows, while standard QAOA energies worsen noticeably; at the ten-percent best-samples level the quadratized Ansatz is 25% and 36% less noise-sensitive than the line-transpiled and all-to-all QAOA Ansatze. The paper also shows, on 40-node Max-Cut circuits, that the measured approximation ratio rises with the number of SWAP layers only up to an optimal point, after which extra layers or extra QAOA depth reduce solution quality. Both findings support the paper's thesis that a noisy device can deliver better solutions from an approximate implementation of the full problem structure than from the exact one.","pith_inferences":["A natural extension the paper leaves implicit is to use the trained quadratic Hamiltonian as a warm start for QAOA parameters of the exact problem, potentially improving convergence of the full algorithm.","The clique expansion's failure on LABS suggests a testable design rule for approximate quadratization: the quadratic projection should preserve the ground-state Hamming-weight structure of the original Hamiltonian, since converting weight-1 or weight-3 minima into weight-2 minima destroyed the solution quality.","The SWAP-truncation idea generalizes to any connectivity-limited compilation: instead of a fixed $k$, one could order Hamiltonian terms by their SWAP cost or estimated error and include them in decreasing fidelity order; the paper's $k$-layer scheme is the special case where all included terms share one cost class.","Because the HUBO robustness result rests on a single 12-variable instance, the practical scope remains open; a noisy simulation across 16 to 24 variables would show whether the quadratization advantage grows, shrinks, or disappears with system size."],"forward_implications":["For dense quartic problems such as a fully-connected four-local Hamiltonian, the quadratized Ansatz reduces the two-qubit gate count from $O(n^4)$ to $O(n^2)$; for LABS the reduction is $O(n)$.","For the 12-variable LABS instance studied, the quadratized Ansatz samples sub-optimal but high-quality solutions, for example a 97.1% approximation ratio with 95.7% probability at depth two, while using fewer two-qubit gates than the standard Ansatz.","Because the quadratized Ansatz is less noise-sensitive, there is a threshold noise strength above which it outperforms standard QAOA; the paper identifies such thresholds for the best 1% and 5% of samples.","On hardware, for 40-node three-regular Max-Cut instances, the measured approximation ratio has an interior optimum in the number of SWAP layers and QAOA depth, so adding problem structure beyond that point lowers quality.","The sampling overhead needed to recover noiseless approximation ratios through CVaR post-selection follows the estimate $1/\\sqrt{\\gamma}$ based on two-qubit gate fidelities, so it can be predicted before execution."],"supporting_citations":[{"why":"Defines the QAOA Ansatz and its p-to-infinity convergence guarantee, which the approximate method explicitly sacrifices.","marker":"[18]"},{"why":"Supplies the LABS Hamiltonian and evidence of QAOA's potential scaling advantage, motivating LABS as the test problem.","marker":"[37]"},{"why":"Introduces the time-block Ansatz design for QUBOs, the closest prior approach to the paper's SWAP-truncation method.","marker":"[41]"},{"why":"Provides the SAT-based qubit-mapping method used to minimize SWAP layers in the Max-Cut experiments.","marker":"[53]"},{"why":"Shows that exact quadratization of each four-body term costs two ancilla qubits, the overhead the approximate method avoids.","marker":"[54]"},{"why":"Defines the hypergraph clique-expansion quadratization and its squared-difference weight objective, which the paper adapts.","marker":"[55]"},{"why":"Reports the earlier hardware QAOA approximation ratio used as the baseline the new Max-Cut measurements beat.","marker":"[59]"},{"why":"Introduces CVaR aggregation, used to post-select samples and connect noisy approximation ratios to noiseless ones.","marker":"[62]"},{"why":"Supplies the MPS simulation method used to evaluate noiseless energies for 40-qubit circuits with bond dimension 20.","marker":"[72]"}],"fun_headline_variants":["Under noise, approximate QAOA surpasses its exact counterpart","Quadratized Hamiltonians make QAOA noise-proof","Noise-aware ansatz design: shallow circuits win","Approximate terms, better noisy quantum optimization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that a trained quadratic Hamiltonian captures enough of the low-energy structure of the original high-order problem that sampling from it yields good solutions once noise is added, and this assumption is currently demonstrated on only a single 12-variable LABS instance, with the paper explicitly noting that the usual QAOA convergence guarantee is lost.","fun_headline_variants_meta":{"raw":{"variants":["Under noise, approximate QAOA surpasses its exact counterpart","Quadratized Hamiltonians make QAOA noise-proof","Noise-aware ansatz design: shallow circuits win","Approximate terms, better noisy quantum optimization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000438,"raw_usage":{"total_tokens":2236,"prompt_tokens":968,"completion_tokens":1268,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":584,"completion_tokens_details":{"reasoning_tokens":1206}},"tokens_in":584,"tokens_out":1268,"duration_ms":10228,"temperature":1.0,"reasoning_tokens":1206,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:23:03.777037+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same depth-two quadratized-versus-standard QAOA comparison on LABS instances with 16, 20, and 24 variables at depolarizing strengths from 0.001 to 0.01; if for the best 5% of samples the standard Ansatz matches or beats the quadratized Ansatz at the noise strengths where the paper reports a threshold advantage, the central robustness claim is falsified. A second check would execute Max-Cut on more than three 40-node three-regular graphs over all SWAP-layer counts and depths; if the optimal number of SWAP layers is always zero or always maximal, the noise-aware design claim collapses.","supporting_citations":[{"cited_title":"Shaydulin, C","cited_arxiv_id":null,"evidence_quote":"Supplies the LABS Hamiltonian and evidence of QAOA's potential scaling advantage, motivating LABS as the test problem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the time-block Ansatz design for QUBOs, the closest prior approach to the paper's SWAP-truncation method."},{"cited_title":"Matsuo, S","cited_arxiv_id":null,"evidence_quote":"Provides the SAT-based qubit-mapping method used to minimize SWAP layers in the Max-Cut experiments."},{"cited_title":"Mandal, A","cited_arxiv_id":null,"evidence_quote":"Shows that exact quadratization of each four-body term costs two ancilla qubits, the overhead the approximate method avoids."},{"cited_title":"Suppakitpaisarn and J.-K","cited_arxiv_id":null,"evidence_quote":"Defines the hypergraph clique-expansion quadratization and its squared-difference weight objective, which the paper adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the earlier hardware QAOA approximation ratio used as the baseline the new Max-Cut measurements beat."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the MPS simulation method used to evaluate noiseless energies for 40-qubit circuits with bond dimension 20."}],"review_version":1}