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REVIEW 3 major objections 5 minor 65 references

The paper claims that for a real industrial job-shop scheduling problem, current quantum hardware cannot yet handle realistic sizes, while a quantum-inspired classical accelerator already can—and that the decisive factor is the QUBO formula

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 05:29 UTC pith:LE233543

load-bearing objection Solid multi-platform benchmark with a reproducible package; the co-design story holds, but the Fujitsu 'consistently outperforms' claim is measured against a self-built classical heuristic, not a mature solver. the 3 major comments →

arxiv 2607.13325 v1 pith:LE233543 submitted 2026-07-14 cs.AR quant-ph

A Reality Check on Quantum Optimisation: Evidence from an Industrial Case Study

Hila Safi , Karen Wintersperger , Oliver von Sicard , Christoph Niedermeier , Wolfgang Mauerer This is my paper
classification cs.AR quant-ph
keywords quantum optimizationjob-shop schedulingQUBO formulationhardware-software co-designquantum annealingQAOAdigital annealerindustrial case study
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Using a real industrial job-shop scheduling problem, this paper asks whether quantum processors are ready for practical optimization. It finds that current gate-based and annealing quantum hardware only solve toy instances and quickly break down as constraint density grows, while a quantum-inspired digital annealer solves industry-scale instances and consistently beats the paper's classical approximation heuristics on solution quality. The load-bearing reason is not raw qubit count but how the problem is encoded: a compact single-constraint QUBO that splits assignment from sequencing stays tractable across platforms, whereas a more faithful multi-constraint formulation collapses on physical quantum hardware. The implication for practitioners is that hardware-software co-design—choosing which constraints to keep in the hardware-facing model and which to solve classically—is the main lever for near-term usefulness.

Core claim

On the paper's own terms, the central discovery is that hardware performance cannot be separated from modelling choices: the same industrial scheduling instance yields radically different outcomes depending on whether the QUBO encodes one constraint or several. The Single-Constraint Model, with M·J variables, remains solvable across all platforms and produces non-trivial valid solutions even on noisy gate-based devices; the Multi-Constraint Model, with M·J + M·R variables, is too constraint-dense for physical quantum hardware and yields valid solutions only on toy instances. The quantum-inspired digital annealer, which does not require minor embedding, handles both formulations at sizes beyo

What carries the argument

The central mechanism is the two-stage decomposition of the job-shop scheduling problem: job-to-machine assignment is encoded as a QUBO, while job ordering on each machine is solved classically. Two QUBO formulations are compared: a compact Single-Constraint Model that balances total machine durations and minimises rig-change cost with one assignment constraint, and a Multi-Constraint Model that additionally introduces binary rig-configuration variables and three constraint types. The contrast between these formulations exposes the trade-off between modelling fidelity and hardware tractability, and the paper argues that constraint density, rather than problem size alone, is what limits scala

Load-bearing premise

The performance comparison against classical methods rests on a reference standard—the best solution found within one hour by the authors' own MILP and heuristic pipeline—and if that baseline is far from the true optimum, the reported optimality gaps and the digital annealer's advantage are measured against a weak yardstick.

What would settle it

Run the same industrial instances against a state-of-the-art MILP solver with a much longer time limit (e.g., 24 hours) and compare the makespan to the digital annealer's solutions; if the extended classical solver matches or beats the annealer on the large instances, the claim of consistent outperformance would be falsified. Alternatively, test instances where sequencing costs dominate assignment costs to see if the two-stage split's approximation of makespan via duration balancing breaks down.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the paper is right, near-term industrial quantum optimisation should favour compact QUBO formulations that relax secondary constraints over more faithful but denser encodings.
  • The bottleneck for current quantum hardware is constraint density as much as qubit count, so hardware improvements alone will not enable industrial-scale scheduling; formulation redesign must accompany them.
  • Quantum-inspired digital annealers are presently the practical option for assignment-based scheduling at industry-relevant sizes, providing a reference point for what future fault-tolerant quantum annealers could achieve.
  • End-to-end evaluation—feasibility, optimality gap, and runtime—should replace raw objective values when comparing quantum and classical solvers in industrial settings.
  • The assignment/sequencing split means the conclusions apply most directly to problems where sequencing is cheap relative to assignment; harder sequencing problems may require a different decomposition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A testable corollary of the paper's logic is that classical heuristics could be improved by adopting the digital annealer's objective/constraint separation, suggesting a hybrid classical-quantum-inspired design space beyond the platforms studied here.
  • The paper's one-hour time limit for the classical MILP baseline may understate classical performance; if given much longer, a commercial MILP solver might close or reverse the reported gap, so the 'consistently outperforms' claim is conditional on the baseline.
  • For scheduling problems with many machines, the sequencing step will no longer be trivially classically solvable, so the two-stage split may lose its advantage—implying the findings are bounded to assignment-dominated variants of job-shop scheduling.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper presents an industrial case study of quantum, quantum-inspired, and classical methods for a job-shop scheduling variant with sequence-dependent rig-change times. Two QUBO assignment formulations (Single-Constraint and Multi-Constraint) are evaluated on IBM Eagle/Heron gate-based QPUs, the D-Wave Advantage annealer, and the Fujitsu Digital Annealer, and compared against a custom greedy/iterated-greedy heuristic and a time-limited MILP baseline. The authors report outcome shares, optimality gaps, and scaling, and conclude that current quantum hardware is not competitive for large-scale constrained optimisation, while the Fujitsu Digital Annealer supports industry-relevant sizes and consistently outperforms the classical approximation heuristics. They frame hardware-software co-design—particularly compact QUBO formulation and an assignment/sequencing split—as the decisive near-term lever.

Significance. The paper has several strengths: it uses real industrial instances, covers three distinct hardware paradigms, proposes a sensible two-stage assignment/sequencing decomposition, and ships a Zenodo reproduction package. The qualitative finding that constraint density and formulation choice matter as much as raw qubit count is well supported by the outcome-share data (Figures 1 and 5) and is a useful message for the community. The candid threats-to-validity section (VII) is exemplary. However, the headline quantitative claim—that the Digital Annealer 'consistently outperforms the classical approximation heuristics in solution quality' (Section VIII-B)—is anchored to a self-built heuristic and a time-limited 'best-known' reference. Until the classical baseline is strengthened or the claim is appropriately qualified, this central quantitative assertion is not established. With revision, the paper would be a valuable empirical contribution to quantum/quantum-inspired optimisation benchmarking.

major comments (3)
  1. [§V-A, §VII-C, §VIII-B] The reference standard for large instances is defined in §V-A as 'the best solution found within a one-hour time limit', and the classical comparator is the custom greedy/iterated-greedy heuristic of §III-C. §VII-C concedes that the time-limited MILP 'may not represent the best possible classical approximations'. Since the optimality gaps in Figures 2 and 6 and the conclusion that the Digital Annealer 'consistently outperforms the classical approximation heuristics' are measured against these self-selected baselines, the quantitative outperformance claim is not established against mature classical solvers. I recommend benchmarking against CP-SAT, Gurobi, or a well-established metaheuristic under the same one-hour time limit, or considerably softening the claim to 'outperforms the custom greedy/iterated-greedy heuristic'.
  2. [§VI-D, Figure 8, §VIII-B] Figure 8 shows only four problem instances on the x-axis (labelled 1–4), yet the text states that 'across nearly all instances, both Fujitsu-based models outperform the classical approximation methods' and §VIII-B concludes 'consistently outperforms'. Four instances are insufficient to support the word 'consistently'. Additionally, the figure legend calls the baseline 'optimal solution', but for large instances the baseline is the time-limited best-known solution (per §V-A), not a proven optimum. The number, sizes, and selection of instances in Figure 8 should be reported, and the caption corrected.
  3. [Abstract and §V-A] The abstract claims the approaches are 'benchmark[ed] against an exact classical solver and a MILP formulation', but §V-A states that the exact solver is only used for small tractable instances (largest 6 machines, 5 rigs, 14 jobs) and that larger instances use a one-hour time-limited best solution. This overstates the evaluation. The abstract and introduction should be precise about the distinction between exact small-instance results and time-limited approximations for large instances.
minor comments (5)
  1. [§III-B, Eq. (11)] The text defines 'Matrix r_{ik}' but the equation uses uppercase R_{ik}; unify notation.
  2. [§III-B, Eq. (15)] H_pair is described as 'Minimise rig-change time between the jobs on the same machine', but the expression contains only mean rig-change times and initial rig setup; it does not contain pairwise job-to-job rig-change terms. This is a modeling approximation and should be described as such.
  3. [§V-A and Figure 8] The phrase 'optimal reference as a lower bound' in the Figure 8 caption is misleading for large instances where the reference is a time-limited best-known. Consider using 'best-known' or 'lower bound from time-limited MILP'.
  4. [§V-B2 and §VI-B] For IBM results, the text says the probability of valid/best solutions is 'only slightly above random guessing', but no random baseline is quantified. Providing the expected share under random assignment would strengthen the claim.
  5. [§V-A and Reference [57]] The statement that sequencing is 'efficiently handled classically' cites a sorting-algorithm comparison paper [57]; a scheduling or complexity reference would be more appropriate.

Circularity Check

0 steps flagged

No significant circularity: the empirical benchmark and QUBO derivations are self-contained; the one-hour best-known baseline is an external-validity concern, not a circular reduction.

full rationale

The paper's central comparisons are empirical benchmarks of independently defined QUBO models (Sections III-A and III-B), a classical heuristic/MILP pipeline (Section III-C), and hardware runs (Section IV). The QUBO objectives (Eqs. 4-6 and 12-15) are stated directly from scheduling data (durations and rig-change matrices) and are not defined from any solver output or fitted to benchmark results. The 'best-known' reference for large instances (Section V-A: 'we use the best solution found within a one-hour time limit') is produced by the authors' own pipeline, and Section VII-C concedes that 'MILP results do not necessarily represent the best possible classical approximations'; however, this is a baseline-quality/external-validity limitation, not circularity, because the Digital Annealer's solutions are not constructed to equal that baseline and the reported gaps are computed rather than assumed. Self-citations ([29], [42], [45]) appear for experimental methodology, integration into workflows, and related-work context, but none is load-bearing as a uniqueness theorem, ansatz, or proof of the central claim. No equation-level reduction, fitted-parameter-renamed-as-prediction, or self-citation chain forcing the conclusion was found.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The central claims rest on modeling simplifications and baseline choices, not on new postulated entities. The most consequential inputs are the load-balancing proxy for makespan, the average-initial-rig approximation, the assignment/sequencing split, the time-limited best-known reference, and the DA-as-lower-bound assumption. None of these has independent external support beyond the paper's internal consistency.

free parameters (5)
  • QUBO penalty weights = not reported
    The composite QUBO Hamiltonians H_obj + w_c H_single + w_r H_rig in Sections III-A/B require weighting coefficients; solution shares and gaps depend on them, but no values are given.
  • lambda in classical score = not reported
    Equation (16) mixes machine load and setup cost with lambda; no value or tuning procedure is stated.
  • acceptance temperature T = not reported
    Equation (17) uses exp(delta/T) for uphill acceptance in the iterated greedy heuristic; T is a hand-set parameter with no value reported.
  • D-Wave chain strengths = 0-3000 (varied)
    Chain strength is varied per instance according to D-Wave guidelines and Fig. 3; it is a hardware/tuning parameter, not a derived constant.
  • QAOA depth p = 2, 8, 50 layers tested
    QAOA layer count is a free hyperparameter; results are shown per depth, but no principled selection rule is given.
axioms (5)
  • ad hoc to paper Makespan minimization is approximated by minimizing deviation of machine loads from the mean duration, because the max function cannot be encoded in a QUBO.
    Section III: this substitution changes the objective from the true makespan to a load-balancing proxy and is specific to the paper's decomposition.
  • ad hoc to paper Initial rig-change cost is approximated by an average rig-change time rather than by the actual first job on each machine.
    Sections III-A Eq. (3) and III-B: since the job order is not optimized, the first-job setup is averaged, which can bias load estimates.
  • domain assumption The JSSP is split into a QUBO assignment stage and a classical sequencing stage; the full problem is not encoded in the QUBO.
    Section III: the authors state explicitly that an optimal QUBO solution need not recover the global optimum of the full scheduling problem.
  • domain assumption For larger instances, the best solution found within a one-hour time limit by the paper's own MILP/heuristic pipeline is used as the 'best-known' reference.
    Sections V-A and VII-C: this reference is not proven optimal, so all gap numbers for large instances are relative to a possibly weak baseline.
  • domain assumption The Fujitsu Digital Annealer is treated as a practical lower bound for future fault-tolerant quantum annealing.
    Section VII-B: acknowledged as a threat; the DA has capabilities, such as objective/constraint separation, that current physical quantum annealers do not have.

pith-pipeline@v1.3.0-alltime-deepseek · 16908 in / 13691 out tokens · 130586 ms · 2026-08-02T05:29:19.818930+00:00 · methodology

0 comments
read the original abstract

Quantum Processing Units promise speed-ups for selected computational problems, including combinatorial optimisation, but their industrial utility remains an open challenge. We study an industrial variant of the Job-Shop Scheduling Problem using quantum, quantum-inspired, and classical methods across three platforms: IBM Quantum, the D-Wave Quantum Annealer, and the Fujitsu Digital Annealer. By tailoring formulations to hardware-specific constraints, we show that hardware-software co-design is essential for solution quality and scalability. We benchmark all approaches against an exact classical solver and a MILP formulation, evaluating runtime, solution quality, and scalability. Our results indicate that quantum and quantum-inspired optimisation can support industrial solver selection, integration in classical workflows, modelling decisions, and early proof-of-concept development, while suggesting a potential path towards improved approximations for industrial scheduling.

Figures

Figures reproduced from arXiv: 2607.13325 by Christoph Niedermeier, Hila Safi, Karen Wintersperger, Oliver von Sicard, Wolfgang Mauerer.

Figure 1
Figure 1. Figure 1: Scatter plot of best, valid, and invalid solution shares for the Single-Constraint and Multi-Constraint models on Advantage System 6.4. Problem [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Average gap (in %) of all valid solutions relative to the best-known solution (y) for varying instances (x) in form [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Impact of chain strength on the number of valid and best solutions for the Single-Constraint and Multi-Constraint models on Advantage 4.1 and 6.4. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: QAOA outcome shares on the Eagle ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Overview of Fujitsu Digital Annealer results for the ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Execution time on the Fujitsu Digital Annealer as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of normalised makespan across all problem instances for Fujitsu Digital Annealer (Single-Constraint and Multi-Constraint Model), the [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

65 extracted references · 7 canonical work pages

  1. [1]

    Industry quantum computing applications,

    A. and Bayerstadler, G. Becquin, J. Binder, T. Botter, H. Ehm, T. Ehmer, M. Erdmann, N. Gaus, P. Harbach, M. Hess, J. Klepsch, M. Leib, S. Luber, A. Luckow, M. Mansky, W. Mauerer, F. Neukart, C. Niedermeier, L. Palackal, R. Pfeiffer, C. Polenz, J. Sepulveda, T. Sievers, B. Standen, M. Streif, T. Strohm, C. Utschig-Utschig, D. V olz, H. Weiss, and F. Winte...

    doi:10.1140/e 2021

  2. [2]

    T. Yue, W. Mauerer, S. Ali, and D. Taibi,Challenges and Opportunities in Quantum Software Architecture. Springer Nature Switzerland, 2023, p. 1–23. [Online]. Available: http://dx.doi.org/10. 1007/978-3-031-36847-9 1

    2023

  3. [3]

    Gabor, S

    T. Gabor, S. Zielinski, S. Feld, C. Roch, C. Seidel, F. Neukart, I. Galter, W. Mauerer, and C. Linnhoff-Popien,Assessing Solution Quality of 3SAT on a Quantum Annealing Platform. Springer International Publishing, 2019, p. 23–35. [Online]. Available: http: //dx.doi.org/10.1007/978-3-030-14082-3 3

    work page doi:10.1007/978-3-030-14082-3 2019

  4. [4]

    Scheduling problems — an overview,

    A. Muluk, H. Akpolat, and J. Xu, “Scheduling problems — an overview,”Journal of Systems Science and Systems Engineering, vol. 12, pp. 481–492, 01 2003. [Online]. Available: https://doi.org/10. 1007/s11518-006-0149-z

    2003

  5. [5]

    J. E. Mitchell,Integer Programming: Branch-and-Cut Algorithms. Cham: Springer Nature Switzerland, 2025, pp. 1–8. [Online]. Available: https://doi.org/10.1007/978-3-030-54621-2 287-1

    doi:10.1007/978-3-030-54621-2 2025

  6. [6]

    Optimization methods: Choose the right approach,

    Gurobi Optimization, LLC, “Optimization methods: Choose the right approach,” 2026, accessed 2026-04-

    2026

  7. [7]

    1-2-3 reproducibility for quantum software experiments,

    W. Mauerer and S. Scherzinger, “1-2-3 reproducibility for quantum software experiments,” inIEEE SANER, 2022, pp. 1247–1248

    2022

  8. [8]

    The complexity of flowshop and jobshop scheduling,

    M. R. Garey, D. S. Johnson, and R. Sethi, “The complexity of flowshop and jobshop scheduling,”Mathematics of Operations Research, vol. 1, no. 2, pp. 117–129, 1976. [Online]. Available: http://www.jstor.org/stable/3689278

    arXiv 1976

  9. [9]

    Scheduling with sequence- dependent setup times in short-term production planning: A main path analysis-based review,

    K.-C. Ying, P. Pourhejazy, and Z.-R. Lin, “Scheduling with sequence- dependent setup times in short-term production planning: A main path analysis-based review,”Operations Research Perspectives, vol. 14, p. 100340, 2025. [Online]. Available: https://www.sciencedirect.com/ science/article/pii/S2214716025000168

    2025

  10. [10]

    Hybrid mixed integer linear programming for large-scale join order optimisation,

    M. Sch ¨onberger, I. Trummer, and W. Mauerer, “Hybrid mixed integer linear programming for large-scale join order optimisation,” in Proceedings of the VLDB Endowment, 12 2025. [Online]. Available: https://arxiv.org/abs/2510.20308

    arXiv 2025

  11. [11]

    A quantum approximate optimization algorithm,

    E. Farhi, J. Goldstone, and S. Gutmann, “A quantum approximate optimization algorithm,” 2014. [Online]. Available: https://arxiv.org/abs/ 1411.4028

    Pith/arXiv arXiv 2014

  12. [12]

    A review on Quantum Approximate Optimization Algorithm and its variants,

    K. Blekos, D. Brand, A. Ceschini, C.-H. Chou, R.-H. Li, K. Pandya, and A. Summer, “A review on Quantum Approximate Optimization Algorithm and its variants,”Physics Reports, vol. 1068, pp. 1–66, Jun. 2024. [Online]. Available: https://www.sciencedirect.com/science/ article/pii/S0370157324001078

    2024

  13. [13]

    Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices,

    L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, “Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices,”Physical Review X, vol. 10, no. 2, Jun. 2020. [Online]. Available: http://dx.doi.org/10.1103/ physrevx.10.021067

    2020

  14. [14]

    Near-optimal quantum circuit for grover’s unstructured search using a transverse field,

    Z. Jiang, E. G. Rieffel, and Z. Wang, “Near-optimal quantum circuit for grover’s unstructured search using a transverse field,” Phys. Rev. A, vol. 95, Jun 2017. [Online]. Available: https: //link.aps.org/doi/10.1103/PhysRevA.95.062317

    work page doi:10.1103/physreva.95.062317 2017

  15. [15]

    Xy mixers: Analytical and numerical results for the quantum alternating operator ansatz,

    Z. Wang, N. C. Rubin, J. M. Dominy, and E. G. Rieffel, “Xy mixers: Analytical and numerical results for the quantum alternating operator ansatz,”Physical Review A, vol. 101, no. 1, Jan. 2020. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.101.012320

    doi:10.1103/physreva.101.012320 2020

  16. [16]

    Grover mixers for qaoa: Shifting complexity from mixer design to state preparation,

    A. B ¨artschi and S. Eidenbenz, “Grover mixers for qaoa: Shifting complexity from mixer design to state preparation,” in2020 IEEE International Conference on Quantum Computing and Engineering (QCE), 2020, pp. 72–82

    2020

  17. [17]

    Obstacles to variational quantum optimization from symmetry protection,

    S. Bravyi, A. Kliesch, R. Koenig, and E. Tang, “Obstacles to variational quantum optimization from symmetry protection,” inPhysical Review Letters, vol. 125, no. 26. APS, 2020, p. 260505

    2020

  18. [18]

    Warm-starting quantum optimization,

    D. J. Egger, J. Mare ˇcek, and S. Woerner, “Warm-starting quantum optimization,”Quantum, vol. 5, no. 479, p. 479, 2021

    2021

  19. [19]

    Quantum walks with iterated open quantum dynamics,

    A. Awasthi, A. B ¨artschi, S. Eidenbenz, E. Le, and S. Zhu, “Quantum walks with iterated open quantum dynamics,”Communications Physics, vol. 7, no. 1, Dec. 2023. [Online]. Available: http: //dx.doi.org/10.1038/s42005-023-01453-0

    work page doi:10.1038/s42005-023-01453-0 2023

  20. [20]

    Available: https://www.gurobi.com/resources/blog/ optimization-methods-choosing-the-right-approach-with-gurobi

    [Online]. Available: https://www.gurobi.com/resources/blog/ optimization-methods-choosing-the-right-approach-with-gurobi

  21. [21]

    Bridging classical and quantum with sdp initialized warm- starts for qaoa,

    R. Tate, M. Moondra, B. Gard, M. Mohseni, S. E. Economou, and E. Barnes, “Bridging classical and quantum with sdp initialized warm- starts for qaoa,”Communications Physics, vol. 7, no. 1, Nov. 2023. [Online]. Available: http://dx.doi.org/10.1038/s42005-023-01439-y

    work page doi:10.1038/s42005-023-01439-y 2023

  22. [22]

    Large scale quantum approximate optimization algorithm on non-planar graphs with machine learning noise mitigation,

    V . Vijendran, R. S. Das, L. C. L. Hollenberg, C. D. Hill, and J. Thompson, “Large scale quantum approximate optimization algorithm on non-planar graphs with machine learning noise mitigation,”

  23. [23]

    Towards a universal qaoa protocol: Evidence of quantum advantage in solving combinatorial optimization problems,

    B. Montanez-Barrera, K. Kottmann, R. Or ´us, V . Dunjko, and M. Mauri, “Towards a universal qaoa protocol: Evidence of quantum advantage in solving combinatorial optimization problems,” 2024. [Online]. Available: https://arxiv.org/abs/2405.09169

    Pith/arXiv arXiv 2024

  24. [24]

    Qaoa-in-qaoa: solving large-scale maxcut problems on small quantum machines,

    J. Sud, M. P. Harrigan, N. Rubin, N. M. Tubman, D. Lidar, E. Knill, and R. Babbush, “Qaoa-in-qaoa: solving large-scale maxcut problems on small quantum machines,” 2024. [Online]. Available: https://arxiv.org/abs/2405.05087

    Pith/arXiv arXiv 2024

  25. [25]

    Training the quantum approximate optimization algorithm without access to a quantum processing unit,

    M. Streif and M. Leib, “Training the quantum approximate optimization algorithm without access to a quantum processing unit,”Quantum Science and Technology, vol. 5, no. 3, p. 034008, 2020

    2020

  26. [26]

    A parity quantum approximate optimization algorithm,

    C. Fingar, T. M ¨uller, B. Zanger, and W. Lechner, “A parity quantum approximate optimization algorithm,” 2024. [Online]. Available: https://arxiv.org/abs/2406.20084

    Pith/arXiv arXiv 2024

  27. [27]

    Path matters: Industrial data meet quantum optimization,

    L. Schmidbauer, C. A. Riofr ´ıo, F. Heinrich, V . Junk, U. Schwenk, T. Husslein, and W. Mauerer, “Path matters: Industrial data meet quantum optimization,” inProceedings of the IEEE International Conference on Quantum Computing and Engineering, 5 2025. [Online]. Available: https://arxiv.org/abs/2504.16607

    arXiv 2025

  28. [28]

    Out of the Loop: Structural Approximation of Optimisation Landscapes and non-Iterative Quantum Optimisation,

    T. Kr ¨uger and W. Mauerer, “Out of the Loop: Structural Approximation of Optimisation Landscapes and non-Iterative Quantum Optimisation,” Quantum, vol. 9, p. 1903, Nov. 2025. [Online]. Available: https: //doi.org/10.22331/q-2025-11-06-1903

    doi:10.22331/q-2025-11-06-1903 1903

  29. [29]

    QEF: Reproducible and Exploratory Quantum Software Experiments,

    V . Gierisch and W. Mauerer, “QEF: Reproducible and Exploratory Quantum Software Experiments,” inService-Oriented Computing – ICSOC 2025 Workshops, ser. Lecture Notes in Computer Science, vol. 16441. Springer Singapore, 2026, forthcoming. [Online]. Available: https://arxiv.org/abs/2511.04563

    arXiv 2025

  30. [30]

    Systematic benchmarking of quantum computers: status and recommendations,

    J. M. Lorenz, T. Monz, J. Eisert, D. Reitzner, F. Schopfer, F. Barbaresco, K. Kurowski, W. van der Schoot, T. Strohm, J. Senellart, C. M. Perrault, M. Knufinke, Z. Amodjee, and M. Giardini, “Systematic benchmarking of quantum computers: status and recommendations,”

  31. [31]

    Next-generation topology of d-wave quantum processors,

    K. Boothby, P. Bunyk, J. Raymond, and A. Roy, “Next-generation topology of d-wave quantum processors,” 2020. [Online]. Available: https://arxiv.org/abs/2003.00133

    Pith/arXiv arXiv 2020

  32. [32]

    Quantum Annealing-Based Software Components: An Experimental Case Study with SAT Solving,

    T. Kr ¨uger and W. Mauerer, “Quantum Annealing-Based Software Components: An Experimental Case Study with SAT Solving,” in Proceedings of the IEEE/ACM 42nd International Conference on Software Engineering Workshops, ser. ICSEW’20. New York, NY , USA: Association for Computing Machinery, 2020, pp. 445–450. [Online]. Available: https://doi.org/10.1145/33879...

    arXiv 2020

  33. [33]

    Quantum annealing with manufactured spins,

    M. W. e. a. Johnson, “Quantum annealing with manufactured spins,” Nature, vol. 473, pp. 194–198, 2011

    2011

  34. [34]

    It’s Quick to be Square: Fast Quadratisation for Quantum Toolchains,

    L. Schmidbauer, E. Lobe, I. Schaefer, and W. Mauerer, “It’s Quick to be Square: Fast Quadratisation for Quantum Toolchains,”ACM Transactions on Quantum Computing, vol. 7, no. 3, pp. 1–46, 2026. [Online]. Available: https://doi.org/10.1145/3800943

    doi:10.1145/3800943 2026

  35. [35]

    Approximate approximation on a quantum annealer,

    I. Sax, S. Feld, S. Zielinski, T. Gabor, C. Linnhoff-Popien, and W. Mauerer, “Approximate approximation on a quantum annealer,” inProceedings of the 17th ACM International Conference on Computing Frontiers, ser. CF ’20. New York, NY , USA: Association for Computing Machinery, 2020, p. 108–117. [Online]. Available: https://doi.org/10.1145/3387902.3392635

    arXiv 2020

  36. [36]

    SAT Strikes Back: Parameter and Path Relations in Quantum Toolchains,

    L. Schmidbauer and W. Mauerer, “SAT Strikes Back: Parameter and Path Relations in Quantum Toolchains,” in2025 IEEE International Conference on Quantum Software (QSW). Los Alamitos, CA, USA: IEEE Computer Society, Jul. 2025, pp. 1–12. [Online]. Available: https://doi.org/10.1109/QSW67625.2025.00021

    arXiv 2025

  37. [37]

    Fujitsu digital annealer: Architecture and applications,

    F. Laboratories, “Fujitsu digital annealer: Architecture and applications,” Fujitsu Technical Whitepaper, 2020

    2020

  38. [38]

    Quantum annealing implementation of job-shop scheduling,

    D. Venturelli, D. J. J. Marchand, and G. Rojo, “Quantum annealing implementation of job-shop scheduling,” 2016. [Online]. Available: https://arxiv.org/abs/1506.08479

    Pith/arXiv arXiv 2016

  39. [39]

    Physics-inspired optimization for quadratic unconstrained problems using a digital annealer,

    M. Aramon, G. Rosenberg, E. Valiante, T. Miyazawa, H. Tamura, and H. Katzgraber, “Physics-inspired optimization for quadratic unconstrained problems using a digital annealer,”Frontiers in Physics, vol. 7, p. 48, 04 2019. [Online]. Available: http://dx.doi.org/10.3389/ fphy.2019.00048

    arXiv 2019

  40. [40]

    Quantum annealing to solve the unrelated parallel machine scheduling problem,

    F. Orts, A. M. Puertas, E. M. Garz ´on, and G. Ortega, “Quantum annealing to solve the unrelated parallel machine scheduling problem,” inParallel Processing and Applied Mathematics: 14th International Conference, PPAM 2022, Gdansk, Poland, September 11–14, 2022, Revised Selected Papers, Part II. Berlin, Heidelberg: Springer- Verlag, 2022, p. 165–176. [Onl...

    2022

  41. [41]

    Application of quantum approximate optimization algorithm to job shop scheduling problem,

    K. Kurowski, T. Pecyna, M. Slysz, R. R ´o˙zycki, G. Walig ´ora, and J. Weglarz, “Application of quantum approximate optimization algorithm to job shop scheduling problem,”European Journal of Operational Research, vol. 310, no. 2, pp. 518–528, None 2023. [Online]. Available: https://ideas.repec.org/a/eee/ejores/v310y2023i2p518-528.html

    2023

  42. [42]

    Solving flexible job shop scheduling problems in manufacturing with quantum annealing,

    P. Schworm, X. Wu, M. Glatt, and J. C. Aurich, “Solving flexible job shop scheduling problems in manufacturing with quantum annealing,” Production Engineering, no. 17, pp. 105 – 115, 2024. [Online]. Available: https://nbn-resolving.de/urn:nbn:de:hbz:386-kluedo-79093

    2024

  43. [43]

    An r —— cmax quantum scheduling algorithm,

    F. Lu and D. C. Marinescu, “An r —— cmax quantum scheduling algorithm,”Quantum Information Processing, vol. 6, no. 3, p. 159–178, Jun. 2007. [Online]. Available: https://doi.org/10.1007/ s11128-006-0048-8

    2007

  44. [44]

    Load balancing for high performance computing using quantum annealing,

    O. Rathore, A. Basden, N. Chancellor, and H. Kusumaatmaja, “Load balancing for high performance computing using quantum annealing,” Phys. Rev. Res., vol. 7, p. 013067, 01 2025. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevResearch.7.013067

    work page doi:10.1103/physrevresearch.7.013067 2025

  45. [45]

    Investigating the variational quantum eigensolver to solve scheduling for identical parallel machines with sequence-dependent setup-times,

    T. Schwenzow, K. Wintersperger, H. Safi, O. Sicard, C. Niedermeier, C. Liebrecht, J. Franke, and S. Reitelsh¨ofer, “Investigating the variational quantum eigensolver to solve scheduling for identical parallel machines with sequence-dependent setup-times,” 08 2024

    2024

  46. [46]

    Worst-case error bounds for parallel machine scheduling problems with bounded sequence-dependent setup times,

    I. M. Ovacik and R. Uzhoy, “Worst-case error bounds for parallel machine scheduling problems with bounded sequence-dependent setup times,”Operations Research Letters, vol. 14, no. 5, pp. 251–256, 1993. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ 016763779390089Y

    1993

  47. [47]

    Heuristic scheduling of parallel machines with sequence-dependent set-up times,

    M. E. Kurz and R. G. Askin, “Heuristic scheduling of parallel machines with sequence-dependent set-up times,”International Journal of Production Research, vol. 39, no. 16, pp. 3747–3769, 2001. [Online]. Available: https://doi.org/10.1080/00207540110064938

    work page doi:10.1080/00207540110064938 2001

  48. [48]

    Influence of hw-sw-co- design on quantum computing scalability,

    H. Safi, K. Wintersperger, and W. Mauerer, “Influence of hw-sw-co- design on quantum computing scalability,” in2023 IEEE International Conference on Quantum Software (QSW), 2023, pp. 104–115. [Online]. Available: https://www.lfdr.de/Publications/2023/SaWiMa23.pdf

    2023

  49. [49]

    Parallel machine scheduling with earliness and tardiness penalties,

    F. Sivrikaya-S ¸erifoˇglu and G. Ulusoy, “Parallel machine scheduling with earliness and tardiness penalties,”Computers & Operations Research, vol. 26, no. 8, pp. 773–787, 1999. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0305054898000902

    1999

  50. [50]

    Simulated annealing for parallel machine scheduling with earliness-tardiness penalties and sequence-dependent set-up times,

    S. Radhakrishnan and J. A. Ventura, “Simulated annealing for parallel machine scheduling with earliness-tardiness penalties and sequence-dependent set-up times,”International Journal of Production Research, vol. 38, no. 10, pp. 2233–2252, 2000. [Online]. Available: https://doi.org/10.1080/00207540050028070

    work page doi:10.1080/00207540050028070 2000

  51. [51]

    A tabu search heuristic for the multiprocessor scheduling problem with sequence dependent setup times,

    P. M. Franc ¸a, M. Gendreau, G. Laporte, and F. M. M ¨uller, “A tabu search heuristic for the multiprocessor scheduling problem with sequence dependent setup times,”International Journal of Production Economics, vol. 43, no. 2, pp. 79–89, 1996. [Online]. Available: https://www.sciencedirect.com/science/article/pii/092552739600031X

    arXiv 1996

  52. [52]

    Superconducting qubits: Current state of play,

    M. Kjaergaard, M. E. Schwartz, J. Braum ¨uller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, “Superconducting qubits: Current state of play,”Annual Review of Condensed Matter Physics, vol. 11, pp. 369– 395, 2020

    2020

  53. [53]

    A quantum engineer’s guide to superconducting qubits,

    P. Krantz, M. Kjaergaard, F. Yan, T. Orlando, S. Gustavsson, and W. Oliver, “A quantum engineer’s guide to superconducting qubits,” Applied Physics Reviews, vol. 6, p. 021318, 06 2019. [Online]. Available: http://dx.doi.org/10.1063/1.5089550

    doi:10.1063/1.5089550 2019

  54. [54]

    The distributionally robust machine scheduling problem with job selection and sequence-dependent setup times,

    M. Bruni, S. Khodaparasti, and E. Demeulemeester, “The distributionally robust machine scheduling problem with job selection and sequence-dependent setup times,”Computers & Operations Research, vol. 123, p. 105017, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0305054820301349

    2020

  55. [55]

    Ibm heron processor technology overview,

    I. Quantum, “Ibm heron processor technology overview,” https://quantum.cloud.ibm.com/docs/de/guides/processor-types, 2025, accessed: 2025-07-01

    2025

  56. [56]

    Predict and Conquer: Navigating Algorithm Trade-Offs with Quantum Design Automation,

    S. Thelen and W. Mauerer, “Predict and Conquer: Navigating Algorithm Trade-Offs with Quantum Design Automation,” in2025 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 1. Los Alamitos, CA, USA: IEEE Computer Society, 2025, pp. 591–602. [Online]. Available: https://doi.org/10.1109/QCE65121. 2025.00071

    arXiv 2025

  57. [57]

    Topological and subsystem codes on low-degree graphs with flag qubits,

    C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, “Topological and subsystem codes on low-degree graphs with flag qubits,”Phys. Rev. X, vol. 10, p. 011022, Jan 2020. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevX.10.011022

    doi:10.1103/physrevx.10.011022 2020

  58. [58]

    Programming the D-Wave QPU: Setting the chain strength,

    D-Wave, “Programming the D-Wave QPU: Setting the chain strength,” D-Wave Systems Inc., Burnaby, BC, Canada, Tech. Rep., April 2020, white Paper. [Online]. Available: https://www.dwavequantum. com/media/vsufwv1d/14-1041a-a setting the chain strength.pdf

    2020

  59. [59]

    Effects of imperfections on quantum algorithms: A software engineering perspective,

    F. Greiwe, T. Kr ¨uger, and W. Mauerer, “Effects of imperfections on quantum algorithms: A software engineering perspective,” inIEEE QSW, 2023, pp. 31–42

    2023

  60. [60]

    Comparative analysis of five sorting algorithms on the basis of best case, average case, and worst case,

    M. Khan, S. Shaheen, and F. Qureshi, “Comparative analysis of five sorting algorithms on the basis of best case, average case, and worst case,”International Journal of Information Technology and Electrical Engineering 2306-708X, vol. 3, pp. 1–10, 05 2014. [Online]. Available: https://api.semanticscholar.org/CorpusID:63715675

    2014

  61. [61]

    Evidence tetris in the pixelated world of validity threats,

    M. Wyrich and S. Apel, “Evidence tetris in the pixelated world of validity threats,” inProceedings of the 1st IEEE/ACM International Workshop on Methodological Issues with Empirical Studies in Software Engineering, ser. WSESE ’24. New York, NY , USA: Association for Computing Machinery, 2024, p. 13–16. [Online]. Available: https://doi.org/10.1145/3643664.3648203

    arXiv 2024

  62. [62]

    Threats to validity in controlled experiments in software engineering: what the experts say and why this is relevant,

    E. Teixeira, L. Fonseca, and S. Soares, “Threats to validity in controlled experiments in software engineering: what the experts say and why this is relevant,” inProceedings of the XXXII Brazilian Symposium on Software Engineering, ser. SBES ’18. New York, NY , USA: Association for Computing Machinery, 2018, p. 52–61. [Online]. Available: https://doi.org/...

    arXiv 2018

  63. [63]

    Approximating under the influence of quantum noise and compute power,

    S. Thelen, H. Safi, and W. Mauerer, “Approximating under the influence of quantum noise and compute power,” inIEEE QCE, vol. 02, 2024, pp. 274–279

    2024

  64. [2024]

    Available: https://arxiv.org/abs/2408.02342

    [Online]. Available: https://arxiv.org/abs/2408.02342

    Pith/arXiv arXiv

  65. [2025]

    Available: https://arxiv.org/abs/2503.04905

    [Online]. Available: https://arxiv.org/abs/2503.04905

    Pith/arXiv arXiv