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Applying Grover-mixer quantum alternating operator ansatz algorithm to higher-order unconstrained binary optimization problems

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

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Pith's one-line read For higher-order unconstrained binary optimization problems, the Grover-mixer variant of QAOA outperforms the standard transverse-field variant once the circuit depth crosses a critical value, and its optimal parameters can be pre-optimized

desk verdict Useful numerical comparison on HUBO, but the analytical parameter-selection scheme rests on an incorrect Gaussian average; as written the central derivation doesn't survive. read the letter →

arxiv 2512.23026 v3 pith:2LPGLF7J submitted 2025-12-28 quant-ph

Evgeniy O. Kiktenko , Elizaveta V. Krendeleva , Aleksey K. Fedorov This is my paper · ORCID
classification quant-ph PACS 03.67.Ac
keywords quantumapproximateoptimizationalgorithmGrovermixerHUBOPUBOextremevaluetheoryvariationalparameterpre-optimizationSherrington-KirkpatrickmodelMax-Cutonhypergraphs
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

The paper sets out to show that the Grover-mixer variant of the QAOA variational quantum algorithm is better suited than the standard transverse-field variant for higher-order unconstrained binary optimization (HUBO) problems, where the cost function couples more than two variables at a time. Its numerical study, on random hypergraph Max-Cut and Sherrington-Kirkpatrick models, finds that GM-QAOA's ground-state success probability keeps rising monotonically with circuit depth, while XM-QAOA quickly plateaus; GM-QAOA overtakes XM-QAOA at a critical depth that shrinks as the interaction order grows. A second claim is that the optimal GM-QAOA angles can be approximated classically by modeling the energy landscape as Gaussian and estimating the ground energy with extreme-value theory, yielding a resource-efficient variant that nearly matches fully layerwise-optimized GM-QAOA. If these claims hold, GM-QAOA becomes a practical near-term strategy for problems with genuinely many-body interactions, and the analytical pre-optimization removes much of the hybrid quantum-classical optimization burden.

What carries the argument

The load-bearing object is the Grover mixer unitary, U_G(β) = I + (e^{-2iβ}-1)|sym⟩⟨sym|, a global diffusion gate built from the projector onto the uniform superposition; it replaces the single-qubit X rotations of the standard mixer. The argument is carried by a recursion for the energy-resolved amplitudes, Ψ_k(E) = (e^{-2iβ_k}-1)⟨e^{-iγ_k E}Ψ_{k-1}(E)⟩_E + e^{-iγ_k E}Ψ_{k-1}(E), which, under a Gaussian approximation for the cost spectrum, decomposes into a constant part and an oscillating part, allowing classical computation of near-optimal angles. The minimum-energy target is estimated via extreme-value theory as E_min_est = σ Φ^{-1}(1/2^n), the mode of the Gumbel distribution for the min

What would settle it

Take a fixed HUBO instance family (e.g., n=10, D=4, SK couplings), optimize XM-QAOA globally—optimizing all 2p angles simultaneously with many random restarts—for depths p up to 20, and compare its success probability against layerwise-optimized and globally optimized GM-QAOA at the same depths. If globally optimized XM-QAOA matches or exceeds GM-QAOA, the crossover is an artifact of the layerwise protocol; if GM-QAOA still wins, the claim is robust. The same comparison should be run on a single structured (non-random) instance to test the Gaussian approximation.

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Extended reading notes

Core claim

The central claim is a separation between two mixers: under layerwise variational optimization, XM-QAOA saturates at shallow depth, whereas GM-QAOA improves monotonically and crosses the XM plateau, with the crossover happening earlier for higher Hamiltonian locality. The mechanism is the global Grover diffusion operator, which couples all computational basis states and keeps amplifying low-energy amplitudes as layers are added. The paper also derives an energy-resolved amplitude recursion, closes it under a Gaussian disorder assumption, and uses a Gumbel extreme-value estimate for the minimum energy; maximizing the resulting analytic success probability gives near-optimal parameters. The re

Load-bearing premise

The claimed separation between GM-QAOA and XM-QAOA rests on the layerwise variational optimization protocol being a fair comparison; because that protocol only fits the two new angles per layer, XM-QAOA could be getting stuck in local optima, and the plateau might vanish under full global optimization of all parameters.

Editorial extensions

If this is right

  • If the crossover is real, GM-QAOA gives strictly better asymptotic success probability than XM-QAOA on HUBO instances, with the advantage growing with interaction order D.
  • Classical pre-optimization of angles in GM-QAOA(a) reduces the number of quantum circuit evaluations needed to reach a given success probability, easing near-term hardware requirements.
  • The critical depth grows with system size n but falls with interaction order, suggesting that high-order problems are the natural regime for Grover mixers.
  • The Gaussian/EVT model provides a purely classical prediction of GM-QAOA performance, which can be used to decide in advance whether GM-QAOA will beat XM-QAOA on a given instance family.

Reading between the lines

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

  • The layerwise optimization protocol—optimizing only the two newest angles per added layer—makes the success probability nondecreasing by construction because the new layer can always be set to recover the previous state; without a globally optimized XM-QAOA baseline, the XM plateau and the GM crossover may be partly artifacts of the local optimizer rather than intrinsic mixer properties.
  • The Gaussian-energy assumption treats energy levels as independent random variables; real HUBO instances with structured couplings or degenerate spectra could violate the EVT estimate, and the GM-QAOA(a) angles would then degrade—a testable sensitivity to single-instance statistics.
  • The analytical recursion may extend to other global mixers, weighted diffusion operators, or constrained feasible subspaces, offering a general tool for QAOA parameter scheduling beyond the specific HUBO setting.
  • If GM-QAOA's advantage survives global optimization checks, qudit-based hardware that natively implements multi-controlled phase gates becomes a natural platform to demonstrate the crossover experimentally.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper investigates the Grover-mixer quantum alternating operator ansatz (GM-QAOA) for higher-order unconstrained binary optimization (HUBO/PUBO). It presents numerical comparisons with transverse-field-mixer QAOA (XM-QAOA) under layerwise optimization on random hypergraph Max-Cut and Sherrington-Kirkpatrick (SK) instances, and reports that GM-QAOA outperforms XM-QAOA for high interaction orders and exhibits a monotonic improvement with circuit depth. A second contribution is an analytical model of GM-QAOA dynamics based on a Gaussian energy distribution and an extreme-value-theory estimate of the minimum energy, leading to a resource-efficient parameter pre-optimization scheme called GM-QAOA(a). The paper claims that GM-QAOA(a) nearly matches fully layerwise-optimized GM-QAOA while requiring fewer quantum resources. The central analytical recurrence in Eq. (20) is, however, incorrect for p ≥ 2, which invalidates the GM-QAOA(a) results and the associated resource-efficiency claim as written.

Significance. The problem addressed is relevant: finding practical parameter-setting strategies for QAOA on high-order binary optimization is of active interest. The numerical comparison over 100-instance ensembles is systematic and, if the layerwise caveats are addressed, provides useful evidence about relative performance of the two mixers. The extreme-value-theory minimum-energy estimate is a reasonable heuristic. However, the main advertised theoretical contribution, the analytical recurrence for GM-QAOA amplitudes, is not sound; the papers' central claims about GM-QAOA(a) and its near-optimal resource efficiency rest on an invalid equation. The numerical comparison alone may be salvageable, but the manuscript in its current form does not support its headline conclusions.

major comments (3)
  1. [Sec. IV B, Eq. (20)] The recurrence for A_k is not a consequence of Eq. (14) under the Gaussian assumption. For Gaussian E with variance σ², the characteristic function is ⟨e^{-itE}⟩ = e^{-σ² t² /2}. Substituting Ψ_{k-1}(E)=A_{k-1}+B_{k-1}(E) into Eq. (14) and taking the constant part, the term involving A_{k-2} after two layers is proportional to ⟨e^{-iγ_2 E}e^{-iγ_1 E}⟩ = e^{-σ²(γ_1+γ_2)²/2}. Equation (20) instead uses e^{-σ²(γ_1²+γ_2²)/2}, omitting the cross term 2γ_1γ_2. For k=2 the correct expression is A_2 = (e^{-2iβ_2}-1)[A_1 e^{-σ²γ_2²/2} + A_0 e^{-σ²(γ_1+γ_2)²/2}], not the printed formula. The error propagates to all k≥2. This is not a minor typo: the omitted cross term is generically first-order in γ_1γ_2 and is not negligible for the optimized angles shown in Fig. 4.
  2. [Sec. V, Figs. 4–6] Since Sec. V states that GM-QAOA(a) parameters are obtained by maximizing P(E_est^min) using Eqs. (20)–(21), and Eq. (20) is invalid for p≥2, all reported GM-QAOA(a) results and the conclusion that the analytically pre-optimized variant nearly matches fully layerwise-optimized GM-QAOA are unsupported. Unless the numerical simulations silently used the correct recurrence (Eq. (14)) instead of Eq. (20), which is not stated, the analytical framework does not describe the actual GM-QAOA dynamics. The near-match in Fig. 5 could, with the current text, be an artifact of the incorrect optimization target. This is a load-bearing error for the paper's second central claim.
  3. [Sec. III, Fig. 2 and Sec. II C] The claim that GM-QAOA, unlike XM-QAOA, exhibits monotonic improvement with circuit depth is misleading under the layerwise optimization protocol. Because U_M(0)U_C(0)=I, adding a layer with zero angles leaves the previous state unchanged, so P(Emin) is nondecreasing for both algorithms by construction. The observed contrast is that XM-QAOA saturates while GM-QAOA keeps increasing, not that GM-QAOA is monotone and XM is not. Moreover, no globally optimized XM-QAOA baseline is provided; the XM plateau and the GM crossover could be properties of the local layerwise protocol rather than intrinsic mixer behavior. This weakens the paper's headline comparative claim.
minor comments (4)
  1. [Eq. (21)] The index in the summation appears as 'A_{k-i}' but should be 'A_{k-j}' to match the sum over j.
  2. [Fig. 5 legend] The legend entries read 'GM-QAOA( )' for the constant-angle variant; the 'c' is missing, likely a rendering issue.
  3. [Sec. II B] Typo: 'idienitity' should be 'identity'. Also the section heading 'Layer-wised optimization' should be 'Layer-wise optimization'.
  4. [Sec. IV C] The phrase 'whilecorrectlycapturingtheexponentialgrowth' is missing spaces due to formatting. Additionally, the independence assumption for the energy levels in the EVT estimate is not discussed; for SK spectra this is an approximation that deserves at least a comment.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: analytical parameters are computed from H_C and EVT, not fitted to the target success probabilities; the GM-vs-XM comparison rests on independent numerical simulation.

full rationale

The paper's central analytical pipeline is not circular. The variance sigma^2 is computed from the Hamiltonian via Eq. (18), the minimum-energy target E_est_min is obtained from the Gumbel extreme-value estimate Eq. (25), and the GM-QAOA(a) angles are found by maximizing the model's own proxy P(E_est_min) built from Eqs. (20)-(21). These quantities are not fitted to the reported success probabilities, and the later evaluation against exact layerwise-optimized GM-QAOA and XM-QAOA is an external numerical benchmark. The Gaussian energy assumption is an explicit ansatz motivated by SK normal couplings, and the paper itself limits validation to SK ensembles (Sec. V: "We focus on SK problems because their coupling coefficients are drawn from a normal distribution, which naturally supports the Gaussian approximation..."). Using the same ensemble family for motivation and validation is a methodological self-reference, not a circular derivation, because sigma^2 is still instance-computed and the proxy is not regressed onto the final P(Emin). The GM-XM comparison in Sec. III is direct numerical simulation and does not import conclusions from the analytical model. Two items noted here are correctness/benchmarking concerns rather than circularity. First, under the layerwise protocol of Sec. II C, adding a layer and optimizing only the new parameters makes P(Emin) nondecreasing by construction, since U_M(0)U_C(0)=I in Eq. (3); this explains part of the reported monotonicity but applies equally to XM-QAOA and does not by itself produce the GM-XM crossover. Second, the Gaussian recurrence in Eq. (20) appears to drop cross terms: the characteristic function for a sum of angles is exp(-sigma^2 (gamma1+gamma2)^2/2), not exp(-sigma^2(gamma1^2+gamma2^2)/2), so the analytical recurrence for k>=2 is algebraically questionable. That is a validity risk in the analytical contribution, not a reduction of the paper's claims to their own inputs.

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

No numerical constants are fitted to the reported success probabilities; sigma is computed from the Hamiltonian and the angles are optimization outputs. The Gaussian/EVT choices are modeling assumptions, not fitted parameters. No new physical entities are introduced.

assumptions (3)
  • domain assumption Gaussian energy distribution f(E) with zero mean and variance sigma^2 = sum J^2.
    Used to evaluate the disorder-average recurrence and EVT minimum estimate (Section IV.B); not valid for arbitrary HUBO spectra and ignores correlations between energy levels.
  • domain assumption Energy levels are treated as independent samples for the Fisher-Tippett-Gnedenko extreme value theorem.
    Section IV.C applies i.i.d. EVT to the 2^n energy levels of a single instance; actual eigenenergies are correlated, so the Gumbel minimum estimate is not rigorously justified for finite instances.
  • domain assumption Layerwise optimization faithfully represents each algorithm's achievable performance.
    Section II.C fixes previous layers and optimizes only the newest parameters; because the new layer can be set to identity, success probability is nondecreasing by construction and no global-optimization baseline for XM-QAOA is provided.

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Pith. "Pith review of Applying Grover-mixer quantum alternating operator ansatz algorithm to higher-order unconstrained binary optimization problems." pith.science (2026). https://pith.science/paper/2LPGLF7J

@misc{pith2026251223026,
  author       = {Pith},
  title        = {Pith review of: Applying Grover-mixer quantum alternating operator ansatz algorithm to higher-order unconstrained binary optimization problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2LPGLF7J}},
  note         = {Machine review of arXiv:2512.23026}
}
read the original abstract

The quantum approximate optimization algorithm (QAOA) is among the leading candidates for achieving quantum advantage on near-term processors. While typically implemented with a transverse-field mixer (XM-QAOA), the Grover-mixer variant (GM-QAOA) offers a compelling alternative due to its global search capabilities. This work investigates the application of GM-QAOA to higher-order unconstrained binary optimization (HUBO) problems, also known as polynomial unconstrained binary optimization (PUBO), which form a general class of combinatorial optimization problems involving multivariable interactions. We present a comprehensive numerical study demonstrating that GM-QAOA, unlike XM-QAOA, exhibits monotonic improvement in performance with circuit depth and achieves superior results for HUBO problems within a layerwise optimization framework. An important component of our approach is an analytical framework for modeling GM-QAOA dynamics, which enables a classical approximation of the optimal parameters and helps reduce the optimization overhead. Our resource-efficient parametrized version of GM-QAOA nearly matches the performance of the version optimized using the layerwise approach while being significantly less demanding, making it a highly effective approach for complex optimization tasks. These findings highlight the potential of GM-QAOA and provide a practical pathway for its implementation on current quantum hardware.

Figures

Figures reproduced from arXiv: 2512.23026 by the authors.

Figure 1
Figure 1. (a) General structure of the QAOA circuit. (b) Two implementations of the mixing operator considered [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Performance comparison between GM-QAOA and XM-QAOA for the Max-Cut problem on random [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Changing the critical depth, defined as the minimum circuit depth at which GM-QAOA outperforms [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Comparison of analytically optimized parameters [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Behavior of the success probability P(Emin) as a function of the number of layers for four methods: layer-wise optimized GM-QAOA, XM-QAOA, and two analytical approaches—one with parameter optimization [GM-QAOA(a)] and one without [GM-QAOA(c)] – for various problem size…
Figure 6
Figure 6. Figure 6: Minimum circuit depth at which GM-QAOA(a) outperforms XM-QAOA in terms of the success [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Reference graph

Works this paper leans on

51 extracted references · 9 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke,et al., Reviews of Modern Physics94, 015004 (2022)

    2022

  2. [2]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio,et al., Nature Reviews Physics3, 625 (2021)

    2021

  3. [3]

    Biamonte, P

    J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Nature549, 195 (2017)

    2017

  4. [4]

    McArdle, S

    S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Reviews of Modern Physics92, 015003 (2020)

    2020

  5. [5]

    X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Quantum3, 191 (2019)

    2019

  6. [6]

    X. Xu, J. Sun, S. Endo, Y. Li, S. C. Benjamin, and X. Yuan, Science Bulletin66, 2181 (2021)

    2021

  7. [7]

    A. K. Fedorov, N. Gisin, S. M. Beloussov, and A. I. Lvovsky, arXiv preprint arXiv:2203.17181 (2022)

    arXiv 2022

  8. [8]

    Farhi, J

    E. Farhi, J. Goldstone, and S. Gutmann, arXiv preprint arXiv:1411.4028 (2014). 13

    arXiv 2014

Show all 51 references
  1. [9]

    and has been successfully implemented on various quantum hardware platforms [10–12]. In the literature, the acronym “QAOA” appears in two closely related forms:Quantum Approximate Optimization Algorithm, emphasizing its goal-oriented nature, andQuantum Alternating Operator Ans...

    2025 arXiv

  2. [10]

    Farhi and A

    E. Farhi and A. W. Harrow, arXiv preprint arXiv:1602.07674 (2016)

    2016 arXiv

  3. [11]

    Pagano, A

    G. Pagano, A. Bapat, P. Becker, K. S. Collins, A. De, P. W. Hess, H. B. Kaplan, A. Kyprianidis, W. L. Tan, C. Baldwin,et al., Proceedings of the National Academy of Sciences117, 25396 (2020)

    2020

  4. [12]

    M. P. Harrigan, K. J. Sung, M. Neeley, K. J. Satzinger, F. Arute, K. Arya, J. Atalaya, J. C. Bardin, R. Barends, S. Boixo,et al., Nature Physics17, 332 (2021)

    2021

  5. [13]

    Zhou, S.-T

    L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Physical Review X10, 021067 (2020)

    2020

  6. [14]

    L. K. Grover, inProceedings of the twenty-eighth annual ACM symposium on Theory of computing(1996) pp. 212–219

    1996

  7. [15]

    Bärtschi and S

    A. Bärtschi and S. Eidenbenz, in2020 IEEE International Conference on Quantum Computing and Engineering (QCE)(IEEE, 2020) pp. 72–82

    2020

  8. [16]

    G. A. Bridi and F. d. L. Marquezino, Physical Review A110, 052409 (2024)

    2024

  9. [17]

    N. Xie, J. Xu, T. Chen, X. Lee, Y. Saito, N. Asai, and D. Cai, Physical Review A111, 012401 (2025)

    2025

  10. [18]

    Golden, A

    J. Golden, A. Bärtschi, D. O’Malley, and S. Eidenbenz, in2021 IEEE International Conference on Quantum Com- puting and Engineering (QCE)(IEEE, 2021) pp. 137–147

    2021

  11. [19]

    Benchasattabuse, A

    N. Benchasattabuse, A. Bärtschi, L. P. García-Pintos, J. Golden, N. Lemons, and S. Eidenbenz, arXiv preprint arXiv:2308.15442 (2023)

    2023 arXiv

  12. [20]

    Pelofske, Physical Review E111, 054103 (2025)

    E. Pelofske, Physical Review E111, 054103 (2025)

    2025

  13. [21]

    T. Y. Ng, J. M. Koh, and D. E. Koh, arXiv preprint arXiv:2411.09745 (2024)

    2024 arXiv

  14. [22]

    Zhukov, A

    A. Zhukov, A. Lebedev, and W. Pogosov, Computer Physics Communications , 109627 (2025)

    2025

  15. [23]

    Tsvelikhovskiy, M

    B. Tsvelikhovskiy, M. Nuyten, and B. N. Bakalov, arXiv preprint arXiv:2509.10424 (2025)

    2025

  16. [24]

    E. O. Kiktenko, A. S. Nikolaeva, P. Xu, G. V. Shlyapnikov, and A. K. Fedorov, Physical Review A101, 022304 (2020)

    2020

  17. [25]

    A. S. Nikolaeva, E. O. Kiktenko, and A. K. Fedorov, Physical review A105, 032621 (2022)

    2022

  18. [26]

    E. O. Kiktenko, A. S. Nikolaeva, and A. K. Fedorov, Reviews of Modern Physics97, 021003 (2025)

    2025

  19. [27]

    A. S. Nikolaeva, E. O. Kiktenko, and A. K. Fedorov, EPJ Quantum Technology11, 1 (2024)

    2024

  20. [28]

    A. S. Nikolaeva, I. V. Zalivako, A. S. Borisenko, N. V. Semenin, K. P. Galstyan, A. E. Korolkov, E. O. Kiktenko, K. Y. Khabarova, I. A. Semerikov, A. K. Fedorov,et al., Physical Review Letters135, 060601 (2025)

    2025

  21. [29]

    J. Chu, X. He, Y. Zhou, J. Yuan, L. Zhang, Q. Guo, Y. Hai, Z. Han, C.-K. Hu, W. Huang,et al., Nature physics 19, 126 (2023)

    2023

  22. [30]

    D. A. Chermoshentsev, A. O. Malyshev, M. Esencan, E. S. Tiunov, D. Mendoza, A. Aspuru-Guzik, A. K. Fedorov, and A. I. Lvovsky, arXiv preprint arXiv:2106.13167 (2021)

    2021 arXiv

  23. [31]

    I. G. Rosenberg, Cahiers du Centre d’Études de Recherche Opérationnelle17, 71 (1975)

    1975

  24. [32]

    Boros and A

    E. Boros and A. Gruber, arXiv preprint arXiv:1404.6538 (2014)

    2014 arXiv

  25. [33]

    Semenov, S

    A. Semenov, S. Usmanov, and A. Fedorov, Problems of Information Transmission61, 110 (2025)

    2025

  26. [34]

    Dattani, arXiv preprint arXiv:1901.04405 (2019)

    N. Dattani, arXiv preprint arXiv:1901.04405 (2019)

    1901 arXiv

  27. [35]

    T. J. Sejnowskiet al., inAIP Conference Proceedings, Vol. 151 (American Institute of Physics, 1986) pp. 398–403

    1986

  28. [36]

    T. G. Kolda and B. W. Bader, SIAM review51, 455 (2009)

    2009

  29. [37]

    Chang and C.-J

    C.-C. Chang and C.-J. Lin, ACM transactions on intelligent systems and technology (TIST)2, 1 (2011)

    2011

  30. [38]

    W.-H. Wei, G. Hemani, and C. S. Haley, Nature Reviews Genetics15, 722 (2014)

    2014

  31. [39]

    M. L. Klein and W. Shinoda, science321, 798 (2008)

    2008

  32. [40]

    C. C. Ribeiro, D. Aloise, T. F. Noronha, C. Rocha, and S. Urrutia, European Journal of Operational Research191, 981 (2008)

    2008

  33. [41]

    Bierwirth and D

    C. Bierwirth and D. C. Mattfeld, Evolutionary computation7, 1 (1999)

    1999

  34. [42]

    Shaydulin, C

    R. Shaydulin, C. Li, S. Chakrabarti, M. DeCross, D. Herman, N. Kumar, J. Larson, D. Lykov, P. Minssen, Y. Sun, et al., Science Advances10, eadm6761 (2024)

    2024

  35. [43]

    Kempe, A

    J. Kempe, A. Kitaev, and O. Regev, Siam journal on computing35, 1070 (2006)

    2006

  36. [44]

    R. A. Fisher and L. H. C. Tippett, inMathematical proceedings of the Cambridge philosophical society, Vol. 24 (Cambridge University Press, 1928) pp. 180–190

    1928

  37. [45]

    Gnedenko, Annals of mathematics44, 423 (1943)

    B. Gnedenko, Annals of mathematics44, 423 (1943)

    1943

  38. [46]

    A. F. Jenkinson, Quarterly Journal of the Royal meteorological society81, 158 (1955)

    1955

  39. [47]

    Pickands III, the Annals of Statistics , 119 (1975)

    J. Pickands III, the Annals of Statistics , 119 (1975)

    1975

  40. [48]

    Coles, J

    S. Coles, J. Bawa, L. Trenner, and P. Dorazio,An introduction to statistical modeling of extreme values, Vol. 208 (Springer, 2001)

    2001

  41. [49]

    Beirlant, Y

    J. Beirlant, Y. Goegebeur, J. Segers, and J. L. Teugels,Statistics of extremes: theory and applications(John Wiley & Sons, 2006)

    2006

  42. [50]

    Haan and A

    L. Haan and A. Ferreira,Extreme value theory: an introduction, Vol. 3 (Springer, 2006)

    2006

  43. [51]

    A. Y. Chernyavskiy, D. Kulikov, B. Bantysh, Y. I. Bogdanov, A. Fedorov, and E. Kiktenko, arXiv preprint arXiv:2509.19035 (2025)

    2025

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