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Accuracy and Performance Evaluation of Quantum, Classical and Hybrid Solvers for the Max-Cut Problem

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper reports that on all small Max-Cut instances with known optima, the D-Wave Hybrid solver and two classical simulated-annealing variants reached the global optimum while the fast-annealing QPU usually fell short, and on the…

desk verdict Useful small-instance benchmark and dataset, but the G-set conclusion leans on unverified SBM numbers and a time statement that contradicts the paper's own tables. read the letter →

arxiv 2412.07460 v1 pith:AN2BO3CN submitted 2024-12-10 math.OC quant-ph

Jaka Vodeb , Vid Eržen , Timotej Hrga , Janez Povh This is my paper · ORCID
classification math.OCquant-ph MSC 68Q1290C2790C2690C59
keywords Max-CutQUBOquantumannealingsimulatedbifurcationmachineD-Waveglobaloptimumbenchmark
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish where current quantum, hybrid, and classical solvers actually stand on the NP-hard Max-Cut problem, measuring how close each solver gets to known global optima or best-known solutions. The authors benchmark D-Wave's fast-annealing QPU and hybrid cloud solver against two tuned variants of classical simulated annealing and Toshiba's simulated bifurcation machine across 139 instances with 100 to 10,000 nodes. Their central finding is that on small instances the hybrid solver and both classical annealing variants always reached the global optimum while the QPU usually landed far from it, and on large instances the simulated bifurcation machine and the slower annealing variant produced the best solutions, with the hybrid solver and the faster annealing variant noticeably worse. If this is right, the fast-annealing QPU hardware is not competitive for Max-Cut, and a well-tuned classical annealing with enough runtime is the quality leader alongside the simulated bifurcation machine.

What carries the argument

The central object is the Max-Cut objective written as $\frac{1}{4}x^T L x$ for a cut vector $x\in\{\pm1\}^n$, with $L$ the graph Laplacian; this is the Ising/QUBO Hamiltonian that all solvers approximately minimize. The benchmark machinery is the three-dataset suite: the be and bqp instances with exact optima computed by two exact solvers, and the G-dataset with best-known values from the literature. The two simulated-annealing variants SA1 and SA2 are the same algorithm with different annealing schedules, so they isolate the effect of search time on solution quality. The D-Wave Hybrid solver is a black-box cloud method that combines classical samplers with the QPU through an undisclosed graph decomposition, while the SBM results are taken directly from an external benchmark rather than run by the authors.

What would settle it

If any be or bqp instance is found where the Hybrid solver or either SA variant returns a cut value strictly below the optimum computed by the exact solvers, the paper's claim that these solvers consistently achieve the global optimum is false. For the large-instance ranking, running the simulated bifurcation machine (or an equivalent implementation) on the G-dataset under matched time budgets and hardware, with independently recorded wall-clock times, would confirm or overturn the reported SBM advantage, since the current numbers are taken from the external benchmark.

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Extended reading notes

Core claim

The central discovery is an empirical ranking of four approximate solvers on the Max-Cut problem. On the be and bqp datasets, where global optima are known for instances up to 251 nodes, the Hybrid solver and both simulated-annealing variants recovered the exact optimum on every instance, while the D-Wave fast-annealing QPU returned the optimum on only one of the fifty small instances and otherwise landed thousands of objective units below it. On the G-dataset of 69 instances with 800 to 10,000 nodes, the paper reports that Toshiba's simulated bifurcation machine and the slower simulated-annealing schedule produced the best cut values, the Hybrid solver was noticeably worse in quality but similar in speed, and the faster annealing schedule was the weakest. The conclusion the authors draw is that the new fast-annealing feature of the QPU does not translate into competitive solution quality for Max-Cut, and the leading solvers in this comparison are classical or classical-hybrid.

Load-bearing premise

The large-instance conclusion that the simulated bifurcation machine matches the slower classical annealing depends on SBM solution values and runtimes taken from an external, non-peer-reviewed benchmark that the authors did not run, verify, or compare on equal hardware; if those numbers are inaccurate or not comparable, the G-dataset ranking is not established.

Editorial extensions

If this is right

  • For Max-Cut instances with known optima up to 251 nodes, the D-Wave Hybrid solver and both simulated-annealing schedules are equivalent in solution quality, so any claim of quantum advantage for these cases must beat a classical baseline that never misses the optimum.
  • The fast-annealing QPU, as configured out of the box, cannot be recommended as a near-optimal solver for Max-Cut: it produced the optimum on only one of the fifty small instances and could not handle instances above 151 nodes.
  • On large G-instances, solution quality for simulated annealing is bought with runtime: the slow SA2 schedule beats the fast SA1 schedule, so reported runtimes for SA are meaningful only together with the annealing schedule.
  • The Hybrid solver's overall quality on the G-dataset sits between the fast and slow classical schedules, with speed comparable to the simulated bifurcation machine, making it a fast but not the best-quality option.
  • SBM and SA2 are the two leading solvers by solution quality on the G-dataset, with SA2 orders of magnitude slower, so the practical choice depends on whether runtime or best cut value matters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's negative QPU result is for generic Max-Cut instances; earlier work cited in the paper suggests tunneling-friendly planted problems can favor quantum annealing, so the finding should not be extrapolated to such structures.
  • A fair time-equality comparison is missing: SA2 runs for roughly 600 to 1000 seconds while SBM is credited with about 10 seconds, so it remains an open question how SA2 would perform if capped at SBM's time budget.
  • The Hybrid solver's black-box decomposition prevents isolating why it loses quality on large instances; testing it on sparse versus dense subgraphs or varying decomposition parameters would localize the bottleneck.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents a benchmark of four approximate solvers for the Max-Cut problem on 139 instances drawn from three datasets (be, bqp, and G). The solvers are D-Wave's fast-annealing QPU, D-Wave's Hybrid solver, two variants of a simulated-annealing algorithm (SA1, SA2) from Myklebust, and Toshiba's Simulated Bifurcation Machine (SBM), with the SBM results transcribed from a non-peer-reviewed Medium post. For instances with at most 251 vertices, the paper reports certified global optima obtained with the BiqBin and MADAM exact solvers; for larger instances it uses best-known values from the literature. The main reported findings are that Hybrid, SA1, and SA2 always match the global optimum on the small instances while the QPU does not, and that on the G-set SBM and SA2 deliver the best solution values while Hybrid and SA1 perform noticeably worse. The paper also reports computation times and argues that Hybrid and SBM are fast while SA2 trades time for quality. The underlying datasets are made publicly available.

Significance. If the small-instance results hold, they provide a credible additional negative data point for the fast-annealing D-Wave QPU on Max-Cut, and the open datasets with certified optima are a useful community resource. The comparison on the G-set is potentially important because it suggests that a classical SA tuned for Max-Cut can match or beat a commercial hybrid quantum-classical solver when given enough runtime. However, the large-instance conclusion rests on SBM values and times taken from a Medium post that the authors did not run or independently verify, and several entries have a suspicious constant 10.00 s runtime that looks like a fixed time budget rather than a measured time-to-solution. The paper honestly discloses that [4] is not peer-reviewed, but that disclosure does not cure the reliability problem. With that caveat, the paper's own SA1/SA2/Hybrid comparisons on the G-set are internally consistent and are a valid contribution; the SBM-based efficiency claims are not.

major comments (3)
  1. [Section 4.2, Tables 3-4] The central large-instance conclusion that 'SBM and SA2 are very competitive' while Hybrid and SA1 are worse depends entirely on the SBM column transcribed from the non-peer-reviewed Medium post [4]. Many tSBM entries are exactly 10.00 s (e.g., G35-G39, G55, G57, G60-G63, G70, G72, and also G14, G23, G53, G54), which strongly suggests that these are fixed time budgets rather than measured computation times. If [4] reports a stopping limit rather than time-to-solution, the paper's claims about SBM's computational efficiency are unsupported. The authors should either run the SBM code (or a comparable simulated-bifurcation implementation) themselves, or clearly label the SBM column as externally reported values and avoid time-based efficiency claims that depend on unknown stopping rules.
  2. [Section 5, Discussion vs. Tables 3-4] The statement that SA2 takes 'factor 10000' more time than SBM is internally inconsistent with the reported tables. For most G-set instances with tSBM capped at 10.00 s, the ratio tSA2/tSBM is about 60-100 (e.g., G35: 683.91/10.00 ≈ 68, G67: 630.33/10.01 ≈ 63), and a factor of 10000 holds only for a few instances with tSBM ≈ 0.02 s such as G1 and G6. This discrepancy indicates that the externally sourced time data were not checked against the other columns; the text should report actual ratios or remove the factor.
  3. [Section 3 and Section 4.2] The QPU benchmark is not reproducible as reported. The paper does not give the annealing time, the number of reads, the embedding parameters, or whether any post-processing (e.g., majority vote or multiple restarts) was applied, and no QPU computation times are reported in Table 1 even though computational efficiency is part of the paper's stated objectives. These parameters are essential for interpreting the QPU's solution quality and for comparing it with the other solvers; please add them or state explicitly that they are unavailable.
minor comments (5)
  1. [Section 1.1] The sentence 'for problem sizes below 500, these are global optima calculated using exact solvers BiqBin and MADAM' is imprecise: the bqp500 instances have n = 501 and are later described as having only best-known values without a certificate of optimality (Section 4.2).
  2. [Tables 3 and 4] The column header 'tSBAM(s)' contains a typo; it should be 'tSBM(s)' to match the rest of the text.
  3. [Table 1] The objective values in Table 1 are negative (because the D-Wave input was formulated as a minimization problem), but the text consistently discusses Max-Cut maximization; an explicit note explaining the sign convention would prevent confusion.
  4. [Section 4.2] The sentence that SA1 'often slightly more than SBM' is not supported by the tables: for the largest G-set instances SA1 times are typically 1.5-3 s while many SBM times are 10.00 s, so SA1 is often faster; please rephrase.
  5. [Section 4.2] The claim that the bold values in Tables 3-4 'are also the best-known solutions so far' is stronger than what the authors can support from the cited literature alone; suggest 'best among the compared solvers' or 'best-known to the authors'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: empirical benchmark with independently measured solvers; the only external data (SBM) is openly disclosed as unverified provenance, not a circular derivation.

full rationale

This is an empirical benchmarking study, not a derivation chain. The core comparisons are measured solver outputs (QPU, Hybrid, SA1, SA2) against known or best-known Max-Cut values. The self-citations to BiqBin [5] and MADAM [6] are exact solvers that produce certified global optima, so they constitute independent evidence rather than a circular premise. The only external results are the SBM values transcribed from [4], a non-peer-reviewed Medium post; the paper explicitly states that the authors had no access to SBM and relied entirely on that source, which is a data-provenance and comparability limitation, not a circular reduction of the authors' own equations. The large-instance best-known values are taken from the literature, and the tables show the authors' own SA2 and Hybrid runs frequently disagree with the transcribed SBM values (e.g., G35: SBM 7685 vs SA2 7686; G58: SBM 19257 vs SA2 19118), so the SBM conclusion is not forced by construction. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is imported from the authors' prior work. Hence no significant circularity is present.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The free parameters are the annealing schedules and cloud-solver settings that determine the measured outcomes but are not derived from the data. The axioms are the trust placed in external exact solvers and in the non-peer-reviewed SBM numbers, plus standard Max-Cut and QUBO equivalences. No invented entities are introduced.

free parameters (5)
  • SA1 annealing schedule = T0=10000, decrement step 2e-4
    Chosen to keep SA1 runtimes comparable to D-Wave; directly affects solution quality and runtime.
  • SA2 annealing schedule = T0=40000, decrement step 2e-6
    Adopted from Myklebust [3] with a smaller decrement to improve quality; this parameter choice determines SA2's superior performance on the G-set.
  • QPU annealing parameters = not reported (annealing time, number of reads, chain strength)
    The paper does not report these cloud-job parameters, yet they determine QPU solution quality.
  • Hybrid solver configuration = out-of-the-box defaults (unspecified)
    LeapHybridSampler() defaults were used; internal decomposition and reconstruction details are unknown.
  • SBM time limit = 10 s for many instances
    From [4]; many tSBM entries equal 10.00 s, indicating a fixed time budget rather than time-to-solution.
assumptions (4)
  • domain assumption BiqBin and MADAM provide correct global optima for all instances with n <= 251.
    These are the authors' own exact solvers using branch-and-bound and certificate methods, but no certificates are shown for every n=251 instance.
  • domain assumption The best-known values cited for the G-set are correct and up-to-date.
    The paper asserts computed best values are also the best-known, but does not provide a systematic provenance table; partly relies on [3] and [4].
  • ad hoc to paper The SBM results reported in [4] are accurate and directly comparable despite different hardware.
    The paper uses SBM numbers from a non-peer-reviewed Medium post without access to the solver; timing comparison is acknowledged as non-ideal.
  • standard math Max-Cut, QUBO, and Ising formulations are equivalent for these instances.
    Standard equivalence used to encode problems for D-Wave; not a source of concern.

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Cite this review

Pith. "Pith review of Accuracy and Performance Evaluation of Quantum, Classical and Hybrid Solvers for the Max-Cut Problem." pith.science (2026). https://pith.science/paper/AN2BO3CN

@misc{pith2026241207460,
  author       = {Pith},
  title        = {Pith review of: Accuracy and Performance Evaluation of Quantum, Classical and Hybrid Solvers for the Max-Cut Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AN2BO3CN}},
  note         = {Machine review of arXiv:2412.07460}
}
read the original abstract

This paper investigates the performance of quantum, classical, and hybrid solvers on the NP-hard Max-Cut and QUBO problems, examining their solution quality relative to the global optima and their computational efficiency. We benchmark the new fast annealing D-Wave quantum processing unit (QPU) and D-Wave Hybrid solver against the state-of-the-art classical simulated annealing algorithm (SA) and Toshiba's simulated bifurcation machine (SBM). Our study leverages three datasets encompassing 139 instances of the Max-Cut problem with sizes ranging from 100 to 10,000 nodes. For instances below 251 nodes, global optima are known and reported, while for larger instances, we utilize the best-known solutions from the literature. Our findings reveal that for the smaller instances where the global optimum is known, the Hybrid solver and SA algorithm consistently achieve the global optimum, outperforming the QPU. For larger instances where global optima are unknown, we observe that the SBM and the slower variant of SA deliver competitive solution quality, while the Hybrid solver and the faster variant of SA performed noticeably worse. Although computing time varies due to differing underlying hardware, the Hybrid solver and the SBM demonstrate both efficient computation times, while for SA reduction in computation time can be achieved at the expense of solution quality.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Cosm: Collective Switched Motion for Fast and Accurate Sparse Ising Optimization

    cs.CE 2026-04 accept novelty 7.5 of 10

    Cosm finds certified optimal cuts on Gset G72/G77/G81 and reduces best-known times-to-target on G61/G70 from hundreds of hours to 36–303 s via switched circular dynamics.

  2. Limitations of tensor network approaches for optimization and sampling: A comparison to quantum and classical Ising machines

    cond-mat.dis-nn 2024-11 accept novelty 6.0 of 10

    A tensor-network branch-and-bound solver is slower and slightly less accurate than Ising machines on large random Pegasus and Zephyr spin glasses, but beats them on planted-instance energy.

  3. Exact Spin Elimination in Ising Hamiltonians and Energy-Based Machine Learning

    quant-ph 2025-05 reject novelty 4.0 of 10

    The exact spin-elimination idea via Walsh-Hadamard expansion is sound, but the paper's explicit two-spin, three-spin, and other gadget formulas contain sign errors that break the claimed ground-state preservation.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.