Pith. sign in

REVIEW 3 major objections 4 minor 67 references

Structural Comparison of Error Mitigation Methods for Ising Machines: Penalty-Spin Model versus Stacked Model

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that how replicas of an Ising model are wired together—through a central penalty-spin layer or directly to neighbors—determines whether parallel search stays robust, and that the neighbor-coupled stacked design wins on con

desk verdict A careful SA comparison with a plausible mechanism, but the 'cooperation collapse' claim is only as strong as the untested shared-schedule premise—worth refereeing, not accepting as-is. read the letter →

arxiv 2601.09462 v1 pith:KGLTS3QE submitted 2026-01-14 cond-mat.stat-mech

Tetsuro Abe , Kanta Hino , Shu Tanaka This is my paper
classification cond-mat.stat-mech PACS 05.10.-a
keywords Isingmachinessimulatedannealingquadraticassignmentproblemreplica-coupledmodelspenalty-spinmodelstackedconstraintsatisfactionerrormitigation
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

Error mitigation for Ising machines is usually discussed as protection against hardware noise; this paper argues it is better understood as a question of how replicas of the problem Hamiltonian are wired together. Using simulated annealing on quadratic assignment problems, it claims that a ferromagnetically coupled stacked model—where each replica talks only to its immediate neighbors—keeps constraints satisfied and improves solution quality over a broad range of penalty and coupling strengths. The penalty-spin model, where all replicas talk through one central auxiliary layer, instead suffers what the authors call cooperation collapse: as the number of replicas grows, averaging in the central layer dilutes the sparse 1-bits of one-hot solutions, so replicas stop coordinating and feasibility itself fails. If this is right, topology, not just coupling strength, is the decisive design variable for parallelized Ising search.

What carries the argument

The comparison rests on three Hamiltonians over P layers: an independent-replica baseline, the penalty-spin (PS) model, in which P−1 problem replicas each couple to one shared auxiliary spin layer with strength J_P, and the stacked model, in which adjacent replicas couple directly in a ring with the same J_P. The diagnostic that carries the argument is the 1–1 correlation ⟨S⟩1, a Jaccard-style measure of how much adjacent replicas agree on which bits are 1; unlike ordinary spin correlations, it is not fooled by agreement on zeros. It is this measure that shows PS-layer averaging washing out sparse solution information as P grows, while local neighbor coupling preserves partial sharing of act

What would settle it

Re-run the PS model at P=30 and |J_P|=2–3 with inverse temperatures rescaled to the model's own coupling scale (or equivalently, normalize J_P by the constraint coefficient µ), and check whether P_Feasible stays near 1; the paper's cooperation-collapse narrative would weaken substantially if feasibility survives a schedule matched to the PS energy scale.

Watch

Extended reading notes

Core claim

The paper's central claim is that the topology of inter-replica couplings decisively influences search robustness in finite-time annealing, independent of hardware noise. On QAP, the stacked model with ferromagnetic inter-replica coupling stably achieves near-perfect feasibility across |J_P| ≤ 3 at fixed penalty µ=5, and its approximation ratio improves monotonically as the number of replicas P and annealing steps grow. In the PS model, the central penalty-spin layer averages configurations from many replicas; because feasible QAP solutions are extremely sparse, averaging drives the layer's mean bit value toward zero and erases the information about where the active bits are. The paper ident

Load-bearing premise

The comparison assumes that applying one shared, per-instance temperature schedule to all three models—whose energy scales differ because PS and stacked models add J_P couplings the independent model lacks—produces a fair test rather than systematically biasing one topology.

Editorial extensions

If this is right

  • If the claim is correct, the ferromagnetically stacked model is the preferred replica-coupling design for constrained problems with one-hot or sparse solutions: it converts extra replicas and longer annealing into better solutions instead of into coordination failure.
  • PS-model users should expect feasibility and cooperation to break down at large P; the paper's guideline is to keep the number of replicas small and pre-check that a usable (µ, J_P) window exists.
  • The AFM-stacked variant offers a bounded benefit: it stabilizes feasibility at low penalty coefficients by frustrating the all-zero state, but it does not improve approximation ratio, so it fits feasibility-first applications only.
  • Benchmark comparisons of error-mitigation methods should control for coupling topology, since the same coupling strength J_P produces opposite scaling behavior in the two architectures.
  • Minimum-energy decoding across all layers is a fair, decoder-agnostic baseline that treats the PS auxiliary layer as a candidate solution, making the reported differences attributable to the search dynamics rather than to readout choice.

Reading between the lines

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

  • The averaging-collapse mechanism, if generic, transfers beyond QAP: any centralized auxiliary layer used with assignment, matching, or scheduling constraints (all one-hot sparse) should show the same large-P degradation; testing those problems directly is a natural next step.
  • The shared-temperature-schedule protocol may understate PS performance, since PS energy scales include an extra J_P term the independent model lacks; re-running with per-model schedule normalization would isolate structural from energetic effects—this is an inference, not a claim in the paper.
  • For hardware with sparse local connectivity, the stacked model's ring topology maps more naturally onto limited graphs than the PS model's long-range couplings, strengthening the practical case for stacking beyond the noise-free results shown here.
  • A combined design is suggested by the two mechanisms: use AFM coupling early in annealing to avoid the all-zero trap and switch to FM coupling later to coordinate replicas; the present paper does not test this annealed-sign scheme.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper uses simulated annealing as a noise-free testbed to compare three replica-coupled Ising architectures for the quadratic assignment problem: an independent-replica baseline (C model), the penalty-spin (PS) model with a centralized auxiliary layer, and a stacked model with nearest-neighbor ferromagnetic or antiferromagnetic inter-replica couplings. With P total layers, 1000 SA runs per parameter point on QAPLIB instances tai12a-tai20a, the paper reports that the ferromagnetically coupled stacked model maintains feasibility and improves the approximation ratio as P and N_Steps grow, while the PS model loses feasibility at large P (P_Feasible ~ 0 at P=30, |J_P|>=2) and shows vanishing Jaccard-type 1-1 correlation. The proposed mechanism is that many-replica averaging in the PS layer washes out sparse one-hot information, biasing the auxiliary layer toward zero and preventing coordination. The paper also explains low-mu feasibility of the AFM-stacked model by destabilization of the all-zero state. It concludes that inter-replica coupling topology decisively influences search robustness and offers model-selection and tuning guidelines.

Significance. The empirical core is competently executed: the comparison uses an external QAPLIB optimum, a C-model baseline, 1000-run ensembles with standard-deviation error bars, and the infeasible-sample artifact in Fig. 4(d) is explicitly flagged rather than claimed as a performance gain. The bit-level diagnostics (<x>_PS, <S>_1) provide a concrete, falsifiable account of the proposed averaging mechanism. If the central ranking survives a fair-schedule control, the paper would provide a practically useful design principle for replica-coupled annealing: decentralized nearest-neighbor coupling is more robust than centralized auxiliary coupling for sparse one-hot constraints, and AFM coupling is a feasibility aid but not a quality aid. The main limitation is breadth: all numerical evidence comes from one problem family (QAP, L<=20) under one temperature-schedule convention, so the 'decisive topology' conclusion is not yet established at the level of generality claimed.

major comments (3)
  1. [Section 3 (Eqs. 16-20) and Section 4.2 (Fig. 4(b))] The load-bearing claim of an intrinsic large-P cooperation collapse in the PS model is made under a temperature schedule derived from 2N single-spin-flip samples of the un-replicated problem Hamiltonian H0, not from the replica-coupled Hamiltonian. In the PS model, flipping one auxiliary spin costs up to 2|J_P|(P-1); at P=30, |J_P|=2 this barrier is about 116 in units of J_P, whereas the stacked model's flip barriers are O(|J_P|) independent of P. The same nominal schedule therefore operates in very different parts of the two models' low-temperature kinetics, and the <x>_PS -> 0 / <S>_1 -> 0 statistics in Section 5.2 may reflect freezing of the auxiliary layer rather than information-theoretic averaging. Please add a control in which the schedule is recomputed from the full PS Hamiltonian (or otherwise matched to each model's acceptance rate) and report whether P_Feasible ~ 0 and the cor
  2. [Section 2.1 (Eqs. 1-3) vs Section 4.2] At fixed P, the PS model contains P-1 problem replicas plus one auxiliary layer without H0, whereas the C and stacked models contain P problem replicas. The paper treats this as an intrinsic structural cost, but it also means the PS model has fewer layers that directly optimize the objective and fewer candidates for minimum-energy decoding. To separate topology from resource allocation, compare the PS model with P+1 total layers (i.e., P problem replicas plus one auxiliary) against the stacked model with P layers. If the PS model still collapses, the topological explanation is supported; otherwise the ranking may reflect the missing problem replica rather than the centralized coupling topology.
  3. [Section 5.2.3 and abstract] The conclusion is generalized to 'assignment, matching, and scheduling' problems based on QAP's one-hot sparsity. The numerical evidence, however, is confined to four QAPLIB instances (tai12a/tai15a/tai17a/tai20a), all from the same random QAP family. The mean-field averaging argument is plausible, but it is not demonstrated for any non-QAP one-hot-constrained problem. Please add at least one independent benchmark (e.g., maximum matching, N-queens, or a scheduling QUBO) or restrict the conclusions to QAP-like sparse assignment problems.
minor comments (4)
  1. [Section 2.1] Typo: 'We preparePlayers in total' should read 'We prepare P layers in total'.
  2. [Figures 3-11] The axis labels and tick labels appear as garbled '/aj17/aj15...' strings in the submitted source. If this is not a rendering artifact, the figures are unreadable and must be replaced with legible axis labels.
  3. [Eq. (17)] The left side 2*beta_base*P_init has dimensions of inverse energy while the right side is dimensionless. Please check the formula or clarify the intended dimensionless convention.
  4. [Section 5.2.1, Fig. 9] The bit configurations shown are representative single samples. Please state how many seeds were inspected and confirm that the displayed patterns are typical, rather than selected outliers.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: model ranking is empirical against external QAPLIB/OpenJij baselines; the explanatory mechanism uses measured bit statistics, and self-citations are not load-bearing.

full rationale

The paper's central comparison is an empirical SA study on QAPLIB instances (tai12a/tai15a/tai17a/tai20a) with known optima taken from QAPLIB, so the performance claims (P_Feasible, R) are checked against external data rather than derived from fitted quantities. The temperature schedule (Section 3, Eqs. 16-20) is computed from single-spin-flip energy-difference samples of the base Hamiltonian and then fixed for all models; while this raises a legitimate fairness/correctness concern (the PS model has extra J_P energy scales not represented in the schedule), it is not circularity in the sense required: the schedule is not fitted to the observed outcomes, and no prediction is equal to an input by construction. The 'cooperation collapse' mechanism in Section 5.2 is an explanatory inference from measured bit statistics (⟨x⟩_PS → 0, ⟨S⟩_1 → 0) and from the explicit coupling structure of Eq. (2); the all-zero bias of the PS layer is an empirical consequence observed in SA runs, not a definitional identity or a fitted parameter. The paper cites several works by the same authors (refs 13-14, 41, 44-46), but these are used for background/motivation (e.g., collective-freezing risk) and not as load-bearing support for the model ranking; no uniqueness theorem or ansatz is imported from them. No equation reduces to a fitted quantity, and the AFM-stacked low-μ feasibility effect is also inferred from measured ⟨x⟩ data. The main caveat is the shared-schedule fairness assumption, which should be handled as a correctness/robustness risk, not as circularity. Under the hard rules, the honest finding is no significant circularity (score 0-2).

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on four load-bearing premises beyond the quoted numerics: noise-free-SA transferability (axiom 2), shared-schedule fairness (axiom 3), the all-zero-state characterization (axiom 5), and the sparse-structure generalization (axiom 6). Free parameters are the data-selected μ and scanned/selected J_P; P and N_Steps are scan axes. No invented entities: the penalty-spin layer and stacked topology are prior constructions (refs 32–33, 43), and ⟨S⟩₁ is a diagnostic metric. The mechanism section is an explanatory narrative consistent with bit statistics rather than a parameter-free derivation, which is why circularity burden is low but soundness is capped.

free parameters (3)
  • μ (constraint-penalty coefficient) = 5 (tai12a), 6 (tai15a/17a), 8 (tai20a)
    Chosen per instance as the smallest value yielding P_Feasible≈1.0 (§4.1, §4.4); the size-scaling and J_P comparisons are performed at these data-selected values, so the ranking is conditional on this choice.
  • J_P (inter-replica coupling) = recommended ≈ −1 at μ=5 (FM); scans use |J_P|=0.6 and 3
    Scanned over [−3,3] and fixed at 0.6/3 for the main comparisons (§4.1–§4.3); the 'optimal' operating point is read off the resulting R curves (§4.2), so it is selected from the data rather than derived.
  • P (number of layers) and N_Steps = scan ranges P=5–30, N_Steps=1000–50000
    Experimental resource axes chosen as scan bounds, not fitted to produce a claim; included for completeness.
assumptions (6)
  • standard math Geman–Geman theorem ⇒ SA converges in the asymptotic limit
    Invoked in §1 (refs 23–26) as the ideal baseline; the finite-time regime is the study's justification, so this is background rather than load-bearing.
  • domain assumption SA is an adequate hardware-noise-free testbed for Ising-machine search dynamics
    Core methodological bet of §1 and §3: conclusions about coupling topology transfer from Metropolis SA to QA/coherent Ising machines; not verified in hardware.
  • domain assumption A single shared temperature schedule (Eqs. 16–20) is fair across models with different energy scales
    Section 3 states the same schedule is used for all models to ensure 'identical search conditions'; if the schedule mismatches the PS model's coupled landscape, the ranking could be partly schedule-driven.
  • standard math QUBO→Ising mapping with penalty μ (Eqs. 10–12) faithfully encodes QAP
    Well-known one-hot encoding with quadratic penalty; the sparsity property (exactly L ones) used by the mechanism arguments follows from this formulation.
  • domain assumption The all-zero state is the typical constraint-violating local minimum for QAP
    Used in §5.3 to explain why AFM coupling aids feasibility; asserted for sparse problems, not characterized across instances.
  • ad hoc to paper QAP's one-hot sparsity represents assignment/matching/scheduling problems generally
    Section 5.2.3 extends the collapse mechanism 'broadly' to these problem classes without experiments; this is the main generalization leap.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Structural Comparison of Error Mitigation Methods for Ising Machines: Penalty-Spin Model versus Stacked Model." pith.science (2026). https://pith.science/paper/KGLTS3QE

@misc{pith2026260109462,
  author       = {Pith},
  title        = {Pith review of: Structural Comparison of Error Mitigation Methods for Ising Machines: Penalty-Spin Model versus Stacked Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KGLTS3QE}},
  note         = {Machine review of arXiv:2601.09462}
}
read the original abstract

Error-mitigation methods for Ising machines are reexamined not merely as noise-suppression techniques but as a structural design problem of replica-coupled Ising models. Using simulated annealing as a hardware-noise-free testbed, we systematically compare the penalty-spin (PS) model, which couples replicas through a centralized auxiliary layer, with the stacked model, which couples adjacent replicas directly. Numerical experiments on the quadratic assignment problem reveal that the ferromagnetically coupled stacked model stably maintains constraint satisfaction and improves solution quality over a broad parameter range, exhibiting favorable scalability with both the number of replicas and problem size. In contrast, the PS model suffers from cooperation collapse at large parallelism: many-replica averaging in the PS layer washes out sparse solution information, preventing effective inter-replica coordination. These findings demonstrate that the topology of inter-replica couplings decisively influences search robustness, and provide practical guidelines for model selection and parameter tuning in constrained optimization.

Figures

Figures reproduced from arXiv: 2601.09462 by the authors.

Figure 1
Figure 1. (Color online) Schematic illustrations of replica-coupled models with P = 5 replicas: (a) independent-replica model (C model), (b) penalty￾spin (PS) model, and (c) stacked model. 2.1 Replica-coupled Ising models The basic idea behind error-mitigation methods is to cre￾ate multiple replicas of the original Ising model and introduce inter-replica couplings, thereby enhancing the robustness of the search. In this study… view at source ↗
Figure 2
Figure 2. Schematic of minimum-energy decoding: the replica with the low￾est energy under H0 is selected as the output. We emphasize that the purpose of employing minimum￾energy decoding is not to advocate a specific decoding strat￾egy, but to provide a decoder-agnostic and uniform evaluation rule applicable to all replica-coupled models. By evaluating all layers using the same original problem Hamiltonian H0, this decoding a… view at source ↗
Figure 3
Figure 3. Dependence of PFeasible and R on µ for P = 10. (a) PFeasible at |JP| = 0.6. (b) PFeasible at |JP| = 3. (c) R at |JP| = 0.6. (d) R at |JP| = 3. Error bars indicate standard deviations. Lines connecting data points are guides to the eye. ferromagnetic (FM, JP < 0) and antiferromagnetic (AFM, JP > 0) couplings. Results for the original single-replica set￾ting are also shown as a baseline [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Dependence of PFeasible and R on |JP| at µ = 5. (a) PFeasible for P = 10. (b) PFeasible for P = 30. (c) R for P = 10. (d) R for P = 30. Error bars indicate standard deviations. Lines connecting data points are guides to the eye. We next examine solution quality as meas…
Figure 5
Figure 5. Figure 5: Dependence of R on P and NSteps at µ = 5. (a) P dependence at |JP| = 0.6. (b) P dependence at |JP| = 3. (c) NSteps dependence at |JP| = 0.6 with P = 30. (d) NSteps dependence at |JP| = 3 with P = 30. Error bars indicate standard deviations. Lines connecting data points…
Figure 6
Figure 6. Figure 6: Dependence of R on the QAP size L. Parameters are P = 10, µ = 8, and |JP| = 1. Error bars indicate standard deviations. Lines connecting data points are guides to the eye. 35 10 20 30 P 1.100 1.125 1.150 1.175 1.200 1.225 R (a) 35 10 20 30 P 1.100 1.125 1.150 1.175 1.2…
Figure 7
Figure 7. Figure 7: Dependence of R on P for QAP instances with L = 15, 17, and 20 at |JP| = 0.6. (a) L = 15 with µ = 6. (b) L = 17 with µ = 6. (c) L = 20 with µ = 8. Error bars indicate standard deviations. Lines connecting data points are guides to the eye. size, suggesting that the str…
Figure 8
Figure 8. Figure 8: (c) (|JP| = 0.6), ⟨S ⟩1 decreases with P but does not vanish; it saturates at a finite value (∼0.4–0.5). This can be attributed to the locality of the stacked topology: each replica interacts only with its neighbors, so the character of the in￾teraction does not change…
Figure 9
Figure 9. Figure 9: Bit configurations obtained by SA for QAP (tai12a, L = 12) with |JP| = 0.6 and P = 5. (a) µ = 1. (b) µ = 8. White and black indicate bits 0 and 1, respectively. Replica p = 5 is the PS layer. 1 3 5 7 0.00 0.02 0.04 0.06 0.08 x PS (a) 1 3 5 7 0.00 0.02 0.04 0.06 0.08 (b…
Figure 11
Figure 11. Figure 11: Dependence of the mean bit value ⟨x⟩ over all replicas on µ for P = 10. (a) |JP| = 0.6. (b) |JP| = 3. Dashed lines indicate the theoretical value for feasible solutions. Error bars indicate standard deviations. Lines connecting data points are guides to the eye. consi…
Figure 10
Figure 10. Figure 10: Mean bit value in the PS layer, ⟨x⟩PS. (a) µ dependence at P = 10 and |JP| = 0.6. (b) µ dependence at P = 10 and |JP| = 3. (c) P dependence at µ = 5 and |JP| = 0.6. (d) P dependence at µ = 5 and |JP| = 3. Dashed lines indicate the theoretical value for feasible soluti…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

67 extracted references · 6 linked inside Pith

  1. [1]

    wrong”) Hamiltonian, severely deteriorating the scaling of search performance. This “wrong-Hamiltonian

    Introduction Ising machines address combinatorial optimization prob- lems by encoding them into an Ising Hamiltonian and search- ing for its ground state through physical or physics-inspired dynamics.1–4) Using Ising machines, a wide variety of com- binatorial optimization problems have been investigated in recent years.5–14) Representative annealing-base...

    arXiv 2026

  2. [2]

    # !"# !"# !

    Models and Problem In this section, we define the replica-coupled Ising models evaluated in this work and describe the target optimization problem. We first review the structures of the penalty-spin (PS) model and the stacked model, which were originally pro- posed for error mitigation, and clarify the framework for com- parison. We then present an Ising ...

  3. [3]

    Simulation Methods To isolate the effect of inter-replica coupling topology on search dynamics, we employ simulated annealing (SA) as a hardware-noise-free testbed. We useSASamplerfrom the open-source Python library OpenJij (version 0.10.6),54) which implements a Markov-chain Monte Carlo (MCMC) algorithm based on the Metropolis rule.55, 56) The same updat...

  4. [4]

    2.1: the independent-replica model (C model), the PS model, and the stacked model

    Results: Performance, Robustness, and Size Scaling on QAP In this section, we report SA-based numerical results for the three replica-coupled models defined in Sect. 2.1: the independent-replica model (C model), the PS model, and the stacked model. Throughout the numerical comparisons be- low, the parameterPdenotes the total number of layers. In the PS mo...

  5. [5]

    Mechanism: Replica Correlations and Bit Configura- tions In the previous section, we compared optimization perfor- mance and robustness primarily through the feasibility rate PFeasible and the approximation ratioR. We found that the stacked model (especially with ferromagnetic coupling) sta- bly satisfies constraints and improves solution quality over a w...

  6. [6]

    stronger cou- pling is always better

    Practical Guidelines Based on the findings of this study, we summarize practi- cal guidelines for selecting a replica-coupled model and tun- ing its parameters when applying Ising machines (or equiva- lent annealing-type solvers) to real-world problems. Because our analysis isolates structural effects using noise-free SA, the guidelines below should be vi...

  7. [7]

    Promoting the applica- tion of advanced quantum technology platforms to social is- sues

    Conclusion In this study, we reframed error-mitigation methods for Ising machines as a structural design problem of replica- coupled Ising models, rather than merely as noise-suppression techniques. We systematically compared the penalty-spin (PS) model and the stacked model, analyzing how their struc- tural differences affect optimization performance and...

  8. [8]

    Lucas: Front

    A. Lucas: Front. Phys.2(2014) 5

    2014

Show all 67 references
  1. [9]

    Mohseni, P

    N. Mohseni, P. L. McMahon, and T. Byrnes: Nat. Rev. Phys.4(2022) 363

    2022

  2. [10]

    Tanaka, R

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

    2017

  3. [11]

    Tanahashi, S

    K. Tanahashi, S. Takayanagi, T. Motohashi, and S. Tanaka: J. Phys. Soc. Jpn.88(2019) 061010

    2019

  4. [12]

    Neukart, G

    F. Neukart, G. Compostella, C. Seidel, D. V on Dollen, S. Yarkoni, and B. Parney: Front. ICT4(2017) 29

    2017

  5. [13]

    Ohzeki, A

    M. Ohzeki, A. Miki, M. J. Miyama, and M. Terabe: Front. Comput. Sci. 1(2019) 9

    2019

  6. [14]

    Kitai, J

    K. Kitai, J. Guo, S. Ju, S. Tanaka, K. Tsuda, J. Shiomi, and R. Tamura: Phys. Rev. Res.2(2020) 013319

    2020

  7. [15]

    Tamura, T

    K. Tamura, T. Shirai, H. Katsura, S. Tanaka, and N. Togawa: IEEE Ac- cess9(2021) 81032

    2021

  8. [16]

    Inoue, Y

    T. Inoue, Y . Seki, S. Tanaka, N. Togawa, K. Ishizaki, and S. Noda: Opt. Express30(2022) 43503

    2022

  9. [17]

    Yarkoni, E

    S. Yarkoni, E. Raponi, T. B ¨ack, and S. Schmitt: Rep. Prog. Phys.85 (2022) 104001

    2022

  10. [18]

    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

  11. [19]

    Terui, Y

    Y . Terui, Y . Inoue, Y . Hamakawa, K. Tatsumura, and K. Kudo: Front. Big Data8(2025) 1599704

    2025

  12. [20]

    Ikeuchi, Y

    K. Ikeuchi, Y . Matsuda, and S. Tanaka: arXiv preprint arXiv:2510.24237 (2025)

    2025 arXiv

  13. [21]

    Kikuchi and S

    S. Kikuchi and S. Tanaka: arXiv preprint arXiv:2601.01860 (2026)

    2026 arXiv

  14. [22]

    Kadowaki and H

    T. Kadowaki and H. Nishimori: Phys. Rev. E58(1998) 5355

    1998

  15. [23]

    Das and B

    A. Das and B. K. Chakrabarti:Quantum annealing and related opti- mization methods(Springer Science & Business Media, 2005), V ol. 679

    2005

  16. [24]

    Das and B

    A. Das and B. K. Chakrabarti: Rev. Mod. Phys.80(2008) 1061

    2008

  17. [25]

    Hauke, H

    P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver: Rep. Prog. Phys.83(2020) 054401

    2020

  18. [26]

    B. K. Chakrabarti, H. Leschke, P. Ray, T. Shirai, and S. Tanaka: Philos. Trans. R. Soc. A381(2023) 20210419

    2023

  19. [27]

    Kirkpatrick, C

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

    1983

  20. [28]

    D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon: Oper. Res.37(1989) 865

    1989

  21. [29]

    D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon: Oper. Res.39(1991) 378

    1991

  22. [30]

    T. Kato: J. Phys. Soc. Jpn.5(1950) 435

    1950

  23. [31]

    Morita and H

    S. Morita and H. Nishimori: J. Phys. Soc. Jpn.76(2007) 064002

    2007

  24. [32]

    Morita and H

    S. Morita and H. Nishimori: J. Math. Phys.49(2008) 125210

    2008

  25. [33]

    Geman and D

    S. Geman and D. Geman: IEEE Trans. Pattern Anal. Mach. Intell. (1984) 721. 13 J. Phys. Soc. Jpn. FULL PAPERS

    1984

  26. [34]

    Nifle and H

    M. Nifle and H. J. Hilhorst: Phys. Rev. Lett.68(1992) 2992

    1992

  27. [35]

    Nifle: Phys

    M. Nifle: Phys. Rev. B57(1998) 492

    1998

  28. [36]

    H. G. Katzgraber and F. Krza ¸kała: Phys. Rev. Lett.98(2007) 017201

    2007

  29. [37]

    Albash, V

    T. Albash, V . Martin-Mayor, and I. Hen: Quantum Sci. Technol.4 (2019) 02LT03

    2019

  30. [38]

    Pearson, A

    A. Pearson, A. Mishra, I. Hen, and D. A. Lidar: npj Quantum Inf.5 (2019) 107

    2019

  31. [39]

    K. C. Young, R. Blume-Kohout, and D. A. Lidar: Phys. Rev. A88(2013) 062314

    2013

  32. [40]

    K. L. Pudenz, T. Albash, and D. A. Lidar: Nat. Commun.5(2014) 3243

    2014

  33. [41]

    K. L. Pudenz, T. Albash, and D. A. Lidar: Phys. Rev. A91(2015) 042302

    2015

  34. [42]

    Vinci, T

    W. Vinci, T. Albash, G. Paz-Silva, I. Hen, and D. A. Lidar: Phys. Rev. A92(2015) 042310

    2015

  35. [43]

    Vinci, T

    W. Vinci, T. Albash, and D. A. Lidar: npj Quantum Inf.2(2016) 1

    2016

  36. [44]

    Matsuura, H

    S. Matsuura, H. Nishimori, T. Albash, and D. A. Lidar: Phys. Rev. Lett. 116(2016) 220501

    2016

  37. [45]

    Matsuura, H

    S. Matsuura, H. Nishimori, W. Vinci, T. Albash, and D. A. Lidar: Phys. Rev. A95(2017) 022308

    2017

  38. [46]

    Vinci and D

    W. Vinci and D. A. Lidar: Phys. Rev. A97(2018) 022308

    2018

  39. [47]

    Matsuura, H

    S. Matsuura, H. Nishimori, W. Vinci, and D. A. Lidar: Phys. Rev. A99 (2019) 062307

    2019

  40. [48]

    Hattori and S

    T. Hattori and S. Tanaka: arXiv preprint arXiv:2509.11217 (2025)

    2025 arXiv

  41. [49]

    Kanamaru, K

    S. Kanamaru, K. Kawamura, S. Tanaka, Y . Tomita, and N. Togawa: IEICE Trans. Inf. & Syst.104(2021) 226

    2021

  42. [50]

    Bennett, A

    J. Bennett, A. Callison, T. O’Leary, M. West, N. Chancellor, and V . Kendon: Quantum Sci. Technol.8(2023) 035031

    2023

  43. [51]

    Tanaka and S

    S. Tanaka and S. Miyashita: J. Phys. Soc. Jpn.78(2009) 084002

    2009

  44. [52]

    Kikuchi, N

    S. Kikuchi, N. Togawa, and S. Tanaka: IEEE Access11(2023) 95493

    2023

  45. [53]

    T. Abe, K. Hino, and S. Tanaka: 2025 IEEE International Conference on Quantum Computing and Engineering (QCE), V ol. 2, 2025, pp. 532– 533

    2025

  46. [54]

    T. C. Koopmans and M. Beckmann: Econometrica: Journal of the Econometric Society25(1957) 53

    1957

  47. [55]

    Hukushima and K

    K. Hukushima and K. Nemoto: J. Phys. Soc. Jpn.65(1996) 1604

    1996

  48. [56]

    U. H. Hansmann: Chem. Phys. Lett.281(1997) 140

    1997

  49. [57]

    D. J. Earl and M. W. Deem: Phys. Chem. Chem. Phys.7(2005) 3910

    2005

  50. [58]

    Sahni and T

    S. Sahni and T. Gonzalez: J. ACM23(1976) 555

    1976

  51. [59]

    Kikuchi, K

    S. Kikuchi, K. Takahashi, and S. Tanaka: arXiv preprint arXiv:2410.11198 (2024)

    2024 arXiv

  52. [60]

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

    2025

  53. [61]

    Nishimura, Y

    K. Nishimura, Y . Sakamoto, T. Shimizu, K. Suzuki, and Y . Yamashiro. OpenJij. DOI: 10.5281/zenodo.15790495

    doi

  54. [62]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller: J. Chem. Phys.21(1953) 1087

    1953

  55. [63]

    W. K. Hastings: Biometrika57(1970) 97

    1970

  56. [64]

    R. E. Burkard, S. E. Karisch, and F. Rendl. QAPLIB - A Quadratic Assignment Problem Library.https://coral.ise.lehigh.edu/ data-sets/qaplib/. accessed Jan. 13, 2026

    2026

  57. [65]

    A. D. King, T. Lanting, and R. Harris: arXiv preprint arXiv:1502.02098 (2015)

    2015 arXiv

  58. [66]

    C. Roch, D. Ratke, J. N ¨ußlein, T. Gabor, and S. Feld: ACM Trans. Quantum Comput.4(2023) 1

    2023

  59. [67]

    Mirkarimi, I

    P. Mirkarimi, I. Shukla, D. C. Hoyle, R. Williams, and N. Chancellor: Phys. Rev. Res.6(2024) 043241. 14

    2024

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.