The reviewed record of science sign in
Pith

arxiv: 2606.24241 · v1 · pith:VHRS6KRU · submitted 2026-06-23 · cond-mat.stat-mech

Evaluating the solution performance of the augmented Lagrangian function on Ising machines

Shunsuke Awai , Takuro Itoh , Keita Takahashi , Kotaro Tanahashi , Shu Tanaka This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-25 22:49 UTCgrok-4.3pith:VHRS6KRUrecord.jsonopen to challenge →

classification cond-mat.stat-mech
keywords Ising machinesaugmented Lagrangian functionquadratic knapsack problemtime-to-epsiloncombinatorial optimizationpenalty functionsolution performance
0
0 comments X

The pith

Augmented Lagrangian formulation reduces time-to-epsilon on Ising machines by roughly an order of magnitude over penalty methods.

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

The paper tests the augmented Lagrangian function as a problem formulation for Ising machines instead of the usual penalty approach. Experiments on the quadratic knapsack problem track how quickly each method reaches solutions within a given accuracy threshold. The augmented version keeps the penalty parameter small yet still produces feasible answers and arrives at high-precision results sooner during the search. These gains appear consistently for representative hyperparameter choices. The findings point to the augmented Lagrangian function as a more effective way to encode constraints when using Ising machines for optimization.

Core claim

The augmented Lagrangian function (ALF) applied to Ising machines for the quadratic knapsack problem reduces TTε by roughly an order of magnitude compared with the penalty function formulation. It achieves this while keeping μ small, obtaining feasible solutions, and reaching high-precision solutions earlier in the search for representative parameter settings.

What carries the argument

The augmented Lagrangian function, which augments the objective with both quadratic penalty terms and linear Lagrange-multiplier terms to enforce constraints during Ising-machine energy minimization.

If this is right

  • The ALF permits smaller μ values while still returning feasible solutions.
  • High-precision solutions appear earlier in the annealing or search trajectory.
  • Overall time-to-epsilon improves by about tenfold for the tested problem class.
  • The formulation extends naturally to other constrained combinatorial problems solved on Ising machines.

Where Pith is reading between the lines

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

  • If the gain generalizes, mapping pipelines for Ising machines could shift from pure penalty methods to augmented Lagrangian encodings as a default.
  • Adaptive schemes that update the Lagrange multipliers on the fly during the machine run might amplify the observed benefit.
  • Hardware designs could incorporate direct support for multiplier variables to reduce the need for extra spins.

Load-bearing premise

Ising machines minimize the augmented Lagrangian formulation with the same fidelity as the penalty formulation, and hyperparameter searches for μ and λ are performed under comparable conditions for both methods.

What would settle it

Execute identical quadratic knapsack instances on physical Ising hardware using both formulations with matched hyperparameter grids and check whether TTε fails to drop by roughly an order of magnitude under the augmented Lagrangian version.

Figures

Figures reproduced from arXiv: 2606.24241 by Keita Takahashi, Kotaro Tanahashi, Shunsuke Awai, Shu Tanaka, Takuro Itoh.

Figure 1
Figure 1. Figure 1: Distribution of TTε (ε = 0.05) in the λ − µ space for the problem instances treated in this study. Panels (a), (b), and (c) correspond to the instances ‘r 100 50 5’ (W = 983), ‘r 200 50 9’ (W = 1105), and ‘r 300 50 4’ (W = 1957), respectively. The blue line corresponds to the penalty function formulation (λ = 0). The computation was performed under identical conditions 50 times each, with µ in steps of 5 o… view at source ↗
Figure 2
Figure 2. Figure 2: Temporal evolution of the incumbent solution within the execution time of 1 s in the Fixstars Amplify AE. ALF denotes the augmented Lagrangian function (ALF), and PF denotes the penalty function (PF). The error bars in the time-axis (horizontal) direction arise from the fact that the timing of data acquisition is not strictly identical across trials. rewritten as X N i=1 wixi + [PITH_FULL_IMAGE:figures/fu… view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of the feasibility in the λ − µ space for the problem instance ‘r 100 50 5’ treated in this study. The feasibility is the empirical probability that the returned item-selection vari￾ables satisfy the original QKP constraint. The weight is W = 983. When the feasibility becomes 1.0 at µ = 0 due to the addition of the Lagrange multiplier term, the obtained solution is the trivial solution in whic… view at source ↗
Figure 4
Figure 4. Figure 4: Distributions in the λ − µ space for the problem instance ‘r 100 50 5’ (W = 983). Panel (a) shows the constraint violation, and panel (b) shows TTε (ε = 0.05). The computation was performed under identical conditions 50 times each, with µ in steps of 1 over the range from 1 to 40 and λ in steps of 5 over the range from 0 to 150. In panel (a), the constraint violation at each point is the average value of t… view at source ↗
read the original abstract

We apply the augmented Lagrangian function (ALF) as a formulation for Ising machines and evaluate its performance by time-to-epsilon (\mathrm{TT\varepsilon}). The ALF has been well studied in continuous optimization for its numerical stability and convergence, and its advantage over the penalty function formulation is demonstrated here through the following results. Using the quadratic knapsack problem as a benchmark, we examine the dependence of \mathrm{TT\varepsilon} on the hyperparameters \mu and \lambda. The augmented Lagrangian formulation reduces \mathrm{TT\varepsilon} by roughly an order of magnitude compared with the penalty function formulation, keeping \mu small while obtaining feasible solutions and, for representative parameter settings, reaching high-precision solutions earlier in the search. These findings indicate that the augmented Lagrangian function is a promising formulation for improving the solution performance of Ising machines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper evaluates the augmented Lagrangian function (ALF) as a formulation for Ising machines on the quadratic knapsack problem benchmark. It reports that ALF reduces time-to-epsilon (TTε) by roughly an order of magnitude relative to the standard penalty formulation, while allowing small μ values, producing feasible solutions, and reaching high-precision solutions earlier for representative parameter settings. The study examines dependence of TTε on hyperparameters μ and λ.

Significance. If the empirical comparison holds under equivalent tuning budgets and machine fidelity, the result would indicate that ALF can improve practical performance of Ising machines on combinatorial problems by leveraging its known stability properties from continuous optimization. This is a targeted, falsifiable benchmark claim rather than a theoretical derivation; its value lies in the quantitative TTε comparison, but verification requires full disclosure of the experimental protocol.

major comments (3)
  1. [Abstract and experimental results section] The central TTε comparison (abstract) rests on hyperparameter search over (μ, λ) for ALF versus μ for the penalty method, yet no protocol is supplied for grid density, ranges, number of trials, or total tuning budget. This leaves open the possibility that the reported ~10× improvement arises from unequal search effort rather than the formulation itself.
  2. [Methods / experimental protocol] The manuscript does not specify the exact Ising machine model, embedding procedure, noise/precision characteristics, or data-exclusion rules used for both formulations. Without these, it is impossible to confirm that the ALF-augmented QUBO is minimized with equivalent fidelity to the penalty QUBO, which is required for the performance claim.
  3. [Results and figures] No error bars, statistical tests, or number of independent runs are reported for the TTε values or the order-of-magnitude reduction. This weakens the quantitative headline result, especially given that TTε is a stochastic performance metric on hardware solvers.
minor comments (2)
  1. [Abstract] Notation for TTε should be defined explicitly on first use, including the precise definition of ε and how feasibility is checked.
  2. [Figures] The dependence of TTε on μ and λ is mentioned but the corresponding plots or tables lack axis labels or legends that clearly separate ALF from penalty curves.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments on our manuscript. We address each of the major comments below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract and experimental results section] The central TTε comparison (abstract) rests on hyperparameter search over (μ, λ) for ALF versus μ for the penalty method, yet no protocol is supplied for grid density, ranges, number of trials, or total tuning budget. This leaves open the possibility that the reported ~10× improvement arises from unequal search effort rather than the formulation itself.

    Authors: We agree that the hyperparameter tuning protocol requires more detail to ensure the comparison is equitable. In the revised manuscript, we will provide a full description of the grid search procedure, including the ranges and densities for μ and λ, the number of trials conducted, and confirmation that the total tuning budget was equivalent for both the ALF and penalty formulations. This will clarify that the performance improvement is attributable to the formulation rather than search effort. revision: yes

  2. Referee: [Methods / experimental protocol] The manuscript does not specify the exact Ising machine model, embedding procedure, noise/precision characteristics, or data-exclusion rules used for both formulations. Without these, it is impossible to confirm that the ALF-augmented QUBO is minimized with equivalent fidelity to the penalty QUBO, which is required for the performance claim.

    Authors: The referee correctly identifies a gap in the experimental protocol description. We will revise the Methods section to explicitly state the Ising machine model employed, the embedding procedure used, the noise and precision characteristics of the solver, and the rules for data exclusion or handling of invalid solutions. These details will be provided for both formulations to demonstrate equivalent fidelity in the minimization process. revision: yes

  3. Referee: [Results and figures] No error bars, statistical tests, or number of independent runs are reported for the TTε values or the order-of-magnitude reduction. This weakens the quantitative headline result, especially given that TTε is a stochastic performance metric on hardware solvers.

    Authors: We acknowledge that the lack of statistical reporting limits the robustness of the claims. In the revision, we will report the number of independent runs for each TTε measurement, include error bars in the relevant figures, and perform appropriate statistical tests to quantify the significance of the observed order-of-magnitude reduction. This will provide stronger support for the quantitative results. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical benchmark with measured TTε improvements

full rationale

The paper is an empirical benchmark study comparing augmented Lagrangian function (ALF) and penalty formulations on Ising machines for the quadratic knapsack problem, reporting measured reductions in time-to-epsilon (TTε). The central performance claim rests on direct experimental results under varying μ and λ, with no derivation chain, mathematical prediction, or result that reduces by construction to fitted inputs, self-citations, or renamed known quantities. No load-bearing self-citation, self-definitional step, or ansatz smuggling is present in the provided text or abstract. The study is self-contained against external benchmarks via reported TTε values.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The paper relies on the established convergence properties of the augmented Lagrangian function from continuous optimization literature and treats μ and λ as tunable hyperparameters whose values are explored experimentally.

free parameters (2)
  • μ
    Penalty coefficient in the augmented Lagrangian term whose value is varied to study dependence of TTε.
  • λ
    Lagrange multiplier term in the augmented Lagrangian formulation whose value is varied to study dependence of TTε.
axioms (1)
  • domain assumption The augmented Lagrangian function provides numerical stability and convergence advantages over the pure penalty method, as established in continuous optimization.
    Invoked in the abstract as background for the Ising-machine application.

pith-pipeline@v0.9.1-grok · 5678 in / 1349 out tokens · 28661 ms · 2026-06-25T22:49:15.084708+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

61 extracted references · 7 canonical work pages · 3 internal anchors

  1. [1]

    Evaluating the solution performance of the augmented Lagrangian function on Ising machines

    Introduction Combinatorial optimization is a mathematical prob- lem of finding a combination of decision variables that minimizes or maximizes an objective function under given constraints. 1) Many real-world problems, such as logistics optimization, manufacturing scheduling, and communication routing, can be reduced to this form. 2–5) As the number of de...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    First, Subsect

    Input F ormat for the Ising Machine This section presents the form of the energy function that is input to the Ising machine, in order to clar- ify the formulation of the constraint-handling methods described later. First, Subsect. 2.1 describes the Ising model, which is the input format of the Ising machine. Then, Subsect. 2.2 introduces Quadratic Uncons...

  3. [3]

    Method The main comparison in this study is between the penalty function and the augmented Lagrangian func- tion at fixed hyperparameters. The penalty method and the augmented Lagrangian method are described to clar- ify their relation to iterative hyperparameter tuning, and the augmented Lagrangian method is examined as an auxiliary analysis in Sect. 6. ...

  4. [4]

    3.1.3, we conducted two investigations

    Experimental Setup To test the hypothesis described in Subsubsect. 3.1.3, we conducted two investigations. First, we evaluated TTε over theλ−µparameter space, visualized as a heatmap, to identify the parameter region in which TTεis small. 4 J. Phys. Soc. Jpn. FULL P APERS Second, for representative parameter settings selected from this region, we examined...

  5. [5]

    Hobj =− X 1≤i≤j≤N pi,jxixj, x i ∈ {0,1},(14) wherex i is a binary variable that takes the value 1 when itemiis included in the knapsack and 0 when it is not

    p N,N   .(13) Since the original problem maximizes the total value, its objective function is expressed as the following mini- mization problem by negating the total value. Hobj =− X 1≤i≤j≤N pi,jxixj, x i ∈ {0,1},(14) wherex i is a binary variable that takes the value 1 when itemiis included in the knapsack and 0 when it is not. As the constraint, an u...

    1957

  6. [6]

    Subsection 5.1 presents the hyperparameter dependence of TTε

    Results This section presents the results obtained in this study. Subsection 5.1 presents the hyperparameter dependence of TTε. The following Subsect. 5.2 presents the compar- ison of the evolution of the incumbent solution under specific hyperparameter settings, based on the findings obtained from the heatmaps. 5.1TTε(ε= 0.05) This subsection describes t...

  7. [7]

    Discussion This section discusses the results presented in Sect. 5. Subsection 6.1 examines, from the structural change in the formulation, the mechanism by which the augmented Lagrangian function reduces TTε, and Subsect. 6.2 ex- amines, as an auxiliary analysis, whether the augmented Lagrangian method can reach the favorable parameter region without a p...

    1957

  8. [8]

    The initial valueλ (0) was commonly set toλ (0) = 0

    In addition, for the hyperparameter update trajec- tories of the augmented Lagrangian method, the initial valueµ (0) and the growth factorαwere set under multi- ple conditions, and the computation was performed. The initial valueλ (0) was commonly set toλ (0) = 0. At each step, the hyperparameters were retained until at least one feasible solution, in the...

  9. [9]

    Promoting the application of advanced quantum technology platforms to social issues

    Conclusion In this study, the augmented Lagrangian function was evaluated as a formulation for the Ising machine, us- ing TTεas the evaluation metric. For the QKP, which is an inequality-constrained combinatorial optimization problem, the dependence of TTεon the hyperparame- tersµandλwas investigated using the benchmark in- stances listed in Table II. The...

  10. [10]

    R. M. Karp, Reducibility among combinatorial problems, 50 Years of Integer Programming 1958-2008: from the Early Years to the State-of-the-Art, pp. 219–241. Springer, 2009

    1958

  11. [11]

    B.Dantzig and J

    G. B.Dantzig and J. H.Ramser: Management Science6(1959) 80

    1959

  12. [12]

    Laporte: Transportation Science43(2009) 408

    G. Laporte: Transportation Science43(2009) 408

    2009

  13. [13]

    T. L. Magnanti and R. T. Wong: Transportation Science18 (1984) 1

    1984

  14. [14]

    Alonso-Mora, S

    J. Alonso-Mora, S. Samaranayake, A. Wallar, E. Frazzoli, and D. Rus: Proceedings of the National Academy of Sciences114 (2017) 462

    2017

  15. [15]

    Mohseni, P

    N. Mohseni, P. L. McMahon, and T. Byrnes: Nature Reviews Physics4(2022) 363

    2022

  16. [16]

    Rosenberg, P

    G. Rosenberg, P. Haghnegahdar, P. Goddard, P. Carr, K. Wu, and M. L. De Prado: Proceedings of the 8th workshop on high performance computational finance, 2015, pp. 1–7

    2015

  17. [17]

    Tatsumura, R

    K. Tatsumura, R. Hidaka, J. Nakayama, T. Kashimata, and M. Yamasaki: IEEE Access11(2023) 120023

    2023

  18. [18]

    Takahashi, T

    K. Takahashi, T. Abe, Y. Nakamura, R. Hidaka, S. Kikuchi, and S. Tanaka: arXiv preprint arXiv:2510.23310 (2025)

    work page arXiv 2025

  19. [19]

    Tanahashi, S

    K. Tanahashi, S. Takayanagi, T. Motohashi, and S. Tanaka: Journal of the Physical Society of Japan88(2019) 061010

    2019

  20. [20]

    Mukasa, T

    Y. Mukasa, T. Wakaizumi, S. Tanaka, and N. Togawa: IEICE TRANSACTIONS on Information and Systems104(2021) 1592

    2021

  21. [21]

    S. J. Weinberg, F. Sanches, T. Ide, K. Kamiya, and R. Correll: Scientific Reports13(2023) 4770

    2023

  22. [22]

    Kanai, M

    H. Kanai, M. Yamashita, K. Tanahashi, and S. Tanaka: IEEE Access12(2024) 157669

    2024

  23. [23]

    Scientific Reports14(1), 23053 (2024)

    E. Kawase, S. Kikuchi, H. Tamai, and S. Tanaka: Scientific Re- ports (2026). Advance online publication, doi:10.1038/s41598- 026-57443-z

    work page doi:10.1038/s41598- 2026

  24. [24]

    Harris, Y

    R. Harris, Y. Sato, A. J. Berkley, M. Reis, F. Altomare, M. H. Amin, K. Boothby, P. Bunyk, C. Deng, C. Enderud, et al.: Science361(2018) 162

    2018

  25. [25]

    A. D. King, J. Carrasquilla, J. Raymond, I. Ozfidan, E. An- driyash, A. Berkley, M. Reis, T. Lanting, R. Harris, F. Al- tomare, et al.: Nature560(2018) 456

    2018

  26. [26]

    Kitai, J

    K. Kitai, J. Guo, S. Ju, S. Tanaka, K. Tsuda, J. Shiomi, and R. Tamura: Physical Review Research2(2020) 013319

    2020

  27. [27]

    Utimula, T

    K. Utimula, T. Ichibha, G. I. Prayogo, K. Hongo, K. Nakano, and R. Maezono: Scientific Reports11(2021) 7261

    2021

  28. [28]

    Sampei, K

    H. Sampei, K. Saegusa, K. Chishima, T. Higo, S. Tanaka, Y. Yayama, M. Nakamura, K. Kimura, and Y. Sekine: JACS Au3(2023) 991

    2023

  29. [29]

    Couzini´ e, Y

    Y. Couzini´ e, Y. Seki, Y. Nishiya, H. Nishi, T. Kosugi, S. Tanaka, and Y.-i. Matsushita: Journal of the Physical So- ciety of Japan94(2025) 044802

    2025

  30. [30]

    K. Endo, Y. Matsuda, S. Tanaka, and M. Muramatsu: Scien- tific Reports12(2022) 10794

    2022

  31. [31]

    Honda, K

    R. Honda, K. Endo, T. Kaji, Y. Suzuki, Y. Matsuda, S. Tanaka, and M. Muramatsu: Scientific Reports14(2024) 13872

    2024

  32. [32]

    Kondo, T

    T. Kondo, T. Kohira, and Y. Minamoto: Transactions of So- ciety of Automotive Engineers of Japan56(2025)

    2025

  33. [33]

    Perdomo-Ortiz, N

    A. Perdomo-Ortiz, N. Dickson, M. Drew-Brook, G. Rose, and A. Aspuru-Guzik: Scientific Reports2(2012) 571

    2012

  34. [34]

    Irb¨ ack, L

    A. Irb¨ ack, L. Knuthson, S. Mohanty, and C. Peterson: Physi- cal Review Research4(2022) 043013

    2022

  35. [35]

    Kikuchi and S

    S. Kikuchi and S. Tanaka: 2026 International Conference on Quantum Communications, Networking, and Computing (QCNC), 2026, pp. 924–929

    2026

  36. [36]

    Kikuchi and S

    S. Kikuchi and S. Tanaka: Scientific Reports (2026). Advance online publication, doi:10.1038/s41598-026-50891-7

    work page doi:10.1038/s41598-026-50891-7 2026

  37. [37]

    Kirkpatrick, C

    S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi: Science220 (1983) 671

    1983

  38. [38]

    D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon: 10 J. Phys. Soc. Jpn. FULL P APERS 0 50 100 150 10 20 30 40 (0) = 1, = 1.1 (0) = 5, = 1.1 (0) = 1, = 1.5 (0) = 1, = 2.0 0 10 20 30 40 50 Constraint violation (a) 0 50 100 150 10 20 30 40 (0) = 1, = 1.1 (0) = 5, = 1.1 (0) = 1, = 1.5 (0) = 1, = 2.0 103 104 105 TT [ms] (b) Fig. 4.Distributions in th...

    1989

  39. [39]

    D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon: Operations Research39(1991) 378

    1991

  40. [40]

    Kadowaki and H

    T. Kadowaki and H. Nishimori: Physical Review E58(1998) 5355

    1998

  41. [41]

    Takehara, D

    K. Takehara, D. Oku, Y. Matsuda, S. Tanaka, and N. To- gawa: 2019 IEEE 9th International Conference on Consumer Electronics (ICCE-Berlin), 2019, pp. 64–69

    2019

  42. [42]

    Ayodele: European Conference on Evolutionary Computa- tion in Combinatorial Optimization (Part of EvoStar), 2022, pp

    M. Ayodele: European Conference on Evolutionary Computa- tion in Combinatorial Optimization (Part of EvoStar), 2022, pp. 159–174

    2022

  43. [43]

    F. Yin, H. Tamura, Y. Furue, and Y. Watanabe: IEEE Access 12(2024) 168303

    2024

  44. [44]

    S. Ide, S. Kikuchi, and S. Tanaka: arXiv preprint arXiv:2509.19280 (2025)

    work page arXiv 2025

  45. [45]

    Yu, and X.-B.Wang: Quantum Science and Tech- nology11(2026) 015039

    J.-Q.Qin, Y. Yu, and X.-B.Wang: Quantum Science and Tech- nology11(2026) 015039

    2026

  46. [46]

    Unfair Sampling of Quantum Annealing in Weighted Graph Bipartitioning Problems

    S. Ide and S. Tanaka: arXiv preprint arXiv:2604.11449 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  47. [47]

    M. R. Hestenes: Journal of Optimization Theory and Appli- cations4(1969) 303

    1969

  48. [48]

    H. N. Djidjev: Advanced Quantum Technologies6(2023) 2300104

    2023

  49. [49]

    Cellini, A

    L. Cellini, A. Macaluso, and M. Lombardi: Scientific Reports 14(2024) 5142

    2024

  50. [50]

    W. Hong, W. Xu, and F. Teng: arXiv preprint arXiv:2502.15917 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  51. [51]

    Tanahashi and S

    K. Tanahashi and S. Tanaka. Augmented Lagrangian Method for Constrained Optimization Problems in Quantum Anneal- ing, 2021. in Adiabatic Quantum Compututing Conference

    2021

  52. [52]

    H. N. Djidjev: Quantum Science and Technology8(2023) 035013

    2023

  53. [53]

    Ising: Zeitschrift f¨ ur Physik31(1925) 253

    E. Ising: Zeitschrift f¨ ur Physik31(1925) 253

    1925

  54. [54]

    T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis, D. A. Lidar, and M. Troyer: Science 345(2014) 420

    2014

  55. [55]

    Munoz-Bauza and D

    H. Munoz-Bauza and D. A. Lidar: Physical Review Letters 134(2025) 160601

    2025

  56. [56]

    Lucas: Frontiers in Physics2(2014) 5

    A. Lucas: Frontiers in Physics2(2014) 5

    2014

  57. [57]

    Tanaka, R

    S. Tanaka, R. Tamura, and B. K. Chakrabarti:Quantum spin glasses, annealing and computation(Cambridge University Press, 2017)

    2017

  58. [58]

    Zaman, K

    M. Zaman, K. Tanahashi, and S. Tanaka: IEEE Transactions on Computers71(2021) 838

    2021

  59. [59]

    Fixstars Amplify Engine Performance

    Fixstars Corporation. Fixstars Amplify Engine Performance. https://amplify.fixstars.com/ja/engine#performance, 2026

    2026

  60. [60]

    E. Soutif. Instances of the 0-1 Quadratic Knapsack Problem (QKP).https://cedric.cnam.fr/ ~soutif/QKP/, April 2005

    2005

  61. [61]

    Parizy: Ph

    M. Parizy: Ph. D thesis, Waseda University (2022). 11

    2022