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Benchmark study of simulated quantum annealing and asynchronous parallel simulated annealing for quadratic unconstrained binary optimization-based semiprime factorization

Gianbiagio Curato*, Manuel Ponzi, Edoardo Chiatello, and Roberto Giorgetti

  • *Contact author: gianbiagio.curato@intesasanpaolo.com
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APS Open Sci. 1, 000067 – Published 21 July, 2026

DOI: https://doi.org/10.1103/hj46-qw5t

Abstract

Simulated quantum annealing (SQA) has been widely studied as a heuristic for solving hard optimization problems, including integer factorization formulated as a quadratic unconstrained binary optimization (QUBO) problem. In this work, we present a replication-oriented benchmark of annealing-based solvers applied to biprime factorization using the direct QUBO mapping. We perform a controlled comparison of SQA, implemented via path-integral Monte Carlo, simulated annealing (SA), and asynchronous parallel simulated annealing (A-PSA), evaluating their performance in terms of probability of success and time to solution. Here, A-PSA denotes a multistart strategy consisting of multiple independent SA trajectories with no inter-replica interactions. Solver and encoding parameters are systematically optimized using Bayesian optimization to ensure a fair and reproducible comparison. Our results reproduce previously reported performance trends: SQA outperforms conventional SA on the considered QUBO landscapes, while SQA and A-PSA can exhibit closely comparable performance in specific regimes, highlighting the importance of multistart baselines in annealing-based benchmarking. All simulations are performed on commodity multithreaded CPU hardware using open-source software, enabling straightforward reproducibility without reliance on specialized experimental platforms. As a reference benchmark, the largest semiprime successfully factorized in our study is 8219999=32749×251, matching results previously obtained using quantum annealing hardware.

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References (123)

  1. A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography (CRC Press, Boca Raton, FL, 2018).
  2. M. Tesoro, I. Siloi, D. Jaschke, G. Magnifico, and S. Montangero, Integer factorization via tensor-network Schnorr's sieving, Phys. Rev. A 113, 032418 (2026).
  3. W. Stallings, Cryptography and Network Security—Principles and Practice (Pearson Education India, New Delhi, India, 2017), 7th ed.
  4. D. J. Bernstein and T. Lange, Post-quantum cryptography, Nature (London) 549, 188 (2017).
  5. K. Rabah, Review of methods for integer factorization applied to cryptography, J. Appl. Sci. 6, 458 (2006).
  6. H. Riesel, J. Oesterlé, and A. Weinstein, Prime Numbers and Computer Methods for Factorization (Springer, Boston, MA, 1994), Vol. 2.
  7. F. K. Wilhelm and R. Steinwandt, Status of quantum computer development, Technical Report 477, 2024, https://www.bsi.bund.de/SharedDocs/Downloads/DE/BSI/Publikationen/Studien/Quantencomputer/Entwicklungstand_QC_V_2_1.html.
  8. R. Mengoni, D. Ottaviani, and P. Iorio, Breaking RSA security with a low noise D-Wave 2000Q quantum annealer: Computational times, limitations and prospects, arXiv:2005.02268.
  9. P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, in Proceedings 35th Annual Symposium on Foundations of Computer Science (IEEE, New York, 1994), p. 124.
  10. R. Babbush, R. King, S. Boixo, W. Huggins, T. Khattar, G. H. Low, J. R. McClean, T. O’Brien, and N. C. Rubin, Grand challenge of quantum applications, PRX Quantum 7, 020101 (2026).
  11. U. Skosana and M. Tame, Demonstration of Shor’s factoring algorithm for N = 21 on IBM quantum processors, Sci. Rep. 11, 16599 (2021).
  12. J. Palsberg, J. Cong, Y. Ding, B. Fefferman, M. Qureshi, G. S. Ravi, K. N. Smith, H. Wang, X. Wu, and H. Yuen, Computer science challenges in quantum computing: Early fault-tolerance and beyond, arXiv:2601.20247.
  13. J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  14. D. Willsch, P. Hanussek, G. Hoever, M. Willsch, F. Jin, H. De Raedt, and K. Michielsen, The state of factoring on quantum computers, NIC Symposium Proc. 52, 423 (2025).
  15. B. Wang, X. Yang, and D. Zhang, Research on quantum annealing integer factorization based on different columns, Front. Phys. 10, 914578 (2022).
  16. Š. Grigaliūnas and R. Brūzgienė, Towards a unified quantum risk assessment, Electronics 14, 3338 (2025).
  17. R. Martoňák, G. E. Santoro, and E. Tosatti, Quantum annealing by the path-integral Monte Carlo method: The two-dimensional random Ising model, Phys. Rev. B 66, 094203 (2002).
  18. T. Albash and D. A. Lidar, Adiabatic quantum computation and quantum annealing: Theory and practice, Rev. Mod. Phys. 90, 015002 (2018).
  19. V. Bapst and G. Semerjian, Thermal, quantum and simulated quantum annealing: Analytical comparisons for simple models, J. Phys.: Conf. Ser., 473, 012011 (2013).
  20. E. Crosson and A. W. Harrow, Simulated quantum annealing can be exponentially faster than classical simulated annealing, in Proceedings of the 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Piscataway, NJ, 2016), pp. 714–723.
  21. C. Gomes, J. P. Fernandes, G. Falcao, S. Kar, and S. Tayur, A systematic mapping study on quantum and quantum-inspired algorithms in operations research, ACM Comput. Surv. 57, 1 (2025).
  22. H. M. Waidyasooriya and M. Hariyama, Temporal and spatial parallel processing of simulated quantum annealing on a multicore CPU, J. Supercomput. 78, 8733 (2022).
  23. H. M. Waidyasooriya and M. Hariyama, A GPU-based quantum annealing simulator for fully-connected Ising models utilizing spatial and temporal parallelism, IEEE Access 8, 67929 (2020).
  24. B. Tasseff, T. Albash, Z. Morrell, M. Vuffray, A. Y. Lokhov, S. Misra, and C. Coffrin, On the emerging potential of quantum annealing hardware for combinatorial optimization, J. Heuristics 30, 325 (2024).
  25. H. Oshiyama and M. Ohzeki, Benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary optimization, Sci. Rep. 12, 2146 (2022).
  26. G. Schaller and R. Schützhold, The role of symmetries in adiabatic quantum algorithms, Quantum Inf. Comput. 10, 109 (2010).
  27. S. Kirkpatrick, C. Gelatt, and M. Vecchi, Optimization by simulated annealing, Science 220, 671 (1983).
  28. E. Onbaşoğlu and L. Özdamar, Parallel simulated annealing algorithms in global optimization, J. Global Optim. 19, 27 (2001).
  29. A. M. Ferreiro, J. García, J. G. López-Salas, and C. Vázquez, An efficient implementation of parallel simulated annealing algorithm in GPUs, J. Global Optim. 57, 863 (2013).
  30. A. L. B. Almeida, J. de C. Lima, and M. A. M. Carvalho, Systematic literature review on parallel trajectory-based metaheuristics, ACM Comput. Surv. 55, 1 (2023).
  31. P. Siarry, Metaheuristics (Springer, Cham, 2016), p. 19.
  32. C. Baldassi, C. Borgs, J. T. Chayes, A. Ingrosso, C. Lucibello, L. Saglietti, and R. Zecchina, Unreasonable effectiveness of learning neural networks: From accessible states and robust ensembles to basic algorithmic schemes, Proc. Natl. Acad. Sci. USA 113, E7655 (2016).
  33. C. Baldassi and R. Zecchina, Efficiency of quantum vs. classical annealing in nonconvex learning problems, Proc. Natl. Acad. Sci. USA 115, 1457 (2018).
  34. V. Gripon, M. Löwe, and F. Vermet, Some remarks on replicated simulated annealing, J. Stat. Phys. 182, 51 (2021).
  35. G. E. Santoro, R. Martonák, E. Tosatti, and R. Car, Theory of quantum annealing of an Ising spin glass, Science 295, 2427 (2002).
  36. T. F. Rønnow, Z. Wang, J. Job, S. Boixo, S. V. Isakov, D. Wecker, J. M. Martinis, D. A. Lidar, and M. Troyer, Defining and detecting quantum speedup, Science 345, 420 (2014).
  37. S. V. Isakov, G. Mazzola, V. N. Smelyanskiy, Z. Jiang, S. Boixo, H. Neven, and M. Troyer, Understanding quantum tunneling through quantum Monte Carlo simulations, Phys. Rev. Lett. 117, 180402 (2016).
  38. E. Raimondo, E. Garzón, Y. Shao, A. Grimaldi, S. Chiappini, R. Tomasello, N. Davila-Melendez, J. A. Katine, M. Carpentieri, M. Chiappini, et al., High-performance and reliable probabilistic Ising machine based on simulated quantum annealing, Phys. Rev. X 15, 041001 (2025).
  39. S. Chowdhury, N. A. Aadit, A. Grimaldi, E. Raimondo, A. Raut, P. A. Lott, J. H. Mentink, M. M. Rams, F. Ricci-Tersenghi, M. Chiappini, et al., Pushing the boundary of quantum advantage in hard combinatorial optimization with probabilistic computers, Nat. Commun. 16, 9193 (2025).
  40. T. Okuyama, M. Hayashi, and M. Yamaoka, An Ising computer based on simulated quantum annealing by path integral Monte Carlo method, in 2017 IEEE International Conference on Rebooting Computing (ICRC) (IEEE, Washington, DC, 2017), p. 1.
  41. L. T. Brady and W. van Dam, Quantum Monte Carlo simulations of tunneling in quantum adiabatic optimization, Phys. Rev. A 93, 032304 (2016).
  42. L. Hormozi, E. W. Brown, G. Carleo, and M. Troyer, Nonstoquastic Hamiltonians and quantum annealing of an Ising spin glass, Phys. Rev. B 95, 184416 (2017).
  43. C. J. Burges, Factoring as optimization, Technical Report MSR-TR-2002-83, 2002, https://www.microsoft.com/en-us/research/publication/factoring-as-optimization/.
  44. A. Grimaldi, K. Selcuk, N. A. Aadit, K. Kobayashi, Q. Cao, S. Chowdhury, G. Finocchio, S. Kanai, H. Ohno, S. Fukami, and K. Y. Camsari, Experimental evaluation of simulated quantum annealing with MTJ-augmented p-bits, in Proceedings of the 2022 IEEE International Electron Devices Meeting (IEDM) (IEEE, Piscataway, NJ, 2022), pp. 22.4.1–22.4.4.
  45. D. Bernal, Qubovert: A Python library for QUBO conversion and manipulation (2022), https://qubovert.readthedocs.io/en/latest/.
  46. K. Nishimura, Y. Sakamoto, T. Shimizu, K. Suzuki, and Y. Yamashiro, OpenJij: An open-source framework for Ising and QUBO optimization (2021), https://jij-inc.github.io/OpenJij-Reference-Page/.
  47. D-Wave, Dwave-neal: D-Wave systems Python library for classical sampling methods (2022), https://docs.ocean.dwavesys.com/projects/neal/en/latest/.
  48. Scikit-optimize: Sequential model-based optimization in Python (2016), https://scikit-optimize.github.io/stable/.
  49. T. Akiba, S. Sano, T. Yanase, T. Ohta, and M. Koyama, Optuna: A next-generation hyperparameter optimization framework, in Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (ACM, New York, NY, 2019), pp. 2623–2631.
  50. D. Dobrynin, A. Renaudineau, M. Hizzani, D. Strukov, M. Mohseni, and J. P. Strachan, Energy landscapes of combinatorial optimization in Ising machines, Phys. Rev. E 110, 045308 (2024).
  51. National Institute of Standards and Technology, Digital Signature Standard (DSS), Federal Information Processing Standards Publication FIPS 186-5 (U.S. Department of Commerce, Washington, DC, 2023).
  52. Z. Luo, Y. Yuan, Z. Wang, M. Lai, and P. Lu, General integer factorization algorithm based on Ising machine, EPJ Quantum Technol. 13, 1 (2026).
  53. D. Zhang, H. Wang, S. Li, and B. Wang, Progress in the prime factorization of large numbers, J. Supercomput. 80, 11382 (2024).
  54. S. Jiang, K. A. Britt, A. J. McCaskey, T. S. Humble, and S. Kais, Quantum annealing for prime factorization, Sci. Rep. 8, 17667 (2018).
  55. K. Jun and H. Lee, HUBO and QUBO models for prime factorization, Sci. Rep. 13, 10080 (2023).
  56. F. Glover, G. Kochenberger, and Y. Du, Quantum Bridge Analytics I: A tutorial on formulating and using QUBO models, Ann. Oper. Res. 314, 141 (2022).
  57. Q.-G. Zeng, X.-P. Cui, B. Liu, Y. Wang, P. Mosharev, and M.-H. Yung, Performance of quantum annealing inspired algorithms for combinatorial optimization problems, Commun. Phys. 7, 249 (2024).
  58. M. Aramon, G. Rosenberg, E. Valiante, T. Miyazawa, H. Tamura, and H. G. Katzgraber, Physics-inspired optimization for quadratic unconstrained problems using a digital annealer, Front. Phys. 7, 48 (2019).
  59. M. Sobhani, Y. Chai, T. Hartung, and K. Jansen, Variational quantum eigensolver approach to prime factorization on IBM’s noisy intermediate scale quantum computer, Phys. Rev. A 111, 042413 (2025).
  60. S. McGarraghy and M. Venkova, Quantum-inspired heuristics, in Handbook of Heuristics, edited by R. Martí, P. M. Pardalos, and M. G. Resende (Springer Nature Switzerland, Cham, 2025), p. 259.
  61. C. Hong, Z. Pei, Q. Wang, S. Yang, J. Yu, and C. Wang, Quantum attack on RSA by D-Wave Advantage: A first break of 80-bit RSA, Sci. China Inf. Sci. 68, 129501 (2025).
  62. J. Ding, G. Spallitta, and R. Sebastiani, Effective prime factorization via quantum annealing by modular locally-structured embedding, Sci. Rep. 14, 3518 (2024).
  63. T. Kadowaki and H. Nishimori, Quantum annealing in the transverse Ising model, Phys. Rev. E 58, 5355 (1998).
  64. S. Morita and H. Nishimori, Mathematical foundation of quantum annealing, J. Math. Phys. 49, 125210 (2008).
  65. H. Nishimori and Y. Nonomura, Quantum effects in neural networks, J. Phys. Soc. Jpn. 65, 3780 (1996).
  66. A. Das and B. K. Chakrabarti, Quantum Annealing And Related Optimization Methods (Springer, Berlin, 2005), Vol. 679.
  67. A. Rajak, S. Suzuki, A. Dutta, and B. K. Chakrabarti, Quantum annealing: An overview, Philos. Trans. R. Soc. A 381, 20210417 (2023).
  68. V. Mukherjee and B. K. Chakrabarti, Multivariable optimization: Quantum annealing and computation, Eur. Phys. J.: Spec. Top. 224, 17 (2015).
  69. S. Tanaka, R. Tamura, and B. K. Chakrabarti, Quantum Spin Glasses, Annealing and Computation (Cambridge University Press, Cambridge, 2017), 1st ed.
  70. B. K. Chakrabarti, A. Dutta, and P. Sen, Quantum Ising Phases and Transitions in Transverse Ising Models (Springer, Berlin, 2008), Chap. 3, pp. 50–57.
  71. Y. Okabe and M. Kikuchi, Quantum Monte Carlo Methods in Equilibrium and Nonequilibrium Systems, edited by M. Suzuki (Springer, Berlin, 1987), Chap. 3, pp. 34–40.
  72. J. E. Gubernatis, N. Kawashima, and P. Werner, Quantum Monte Carlo Methods: Algorithms for Lattice Models (Cambridge University Press, Cambridge, 2016), Chap. 5, pp. 84–118.
  73. D. R. Jones, A taxonomy of global optimization methods based on response surfaces, J. Global Optim. 21, 345 (2001).
  74. See Supplemental Material at http://link.aps.org/supplemental/10.1103/hj46-qw5t for details on the QUBO construction, solver benchmarks, and convergence analysis of the path-integral Monte Carlo simulations.
  75. G. E. Santoro and E. Tosatti, Optimization using quantum mechanics: Quantum annealing through adiabatic evolution, J. Phys. A: Math. Gen. 39, R393 (2006).
  76. T. Albash and J. Marshall, Comparing relaxation mechanisms in quantum and classical transverse-field annealing, Phys. Rev. Appl. 15, 014029 (2021).
  77. M. Bernaschi, I. González-Adalid Pemartín, V. Martín-Mayor, and G. Parisi, The quantum transition of the two-dimensional Ising spin glass, Nature (London) 631, 749 (2024).
  78. B. Heim, T. F. Rønnow, S. V. Isakov, and M. Troyer, Quantum versus classical annealing of Ising spin glasses, Science 348, 215 (2015).
  79. C. R. McLeod and M. Sasdelli, Benchmarking D-Wave quantum annealers: Spectral gap scaling of maximum cardinality matching problems, in Proceedings of the International Conference on Computational Science (Springer, Cham, 2022), p. 150.
  80. D. Volpe, G. A. Cirillo, M. Zamboni, and G. Turvani, Integration of simulated quantum annealing in parallel tempering and population annealing for heterogeneous-profile QUBO exploration, IEEE Access 11, 30390 (2023).
  81. E. Crosson and D. Lidar, Prospects for quantum enhancement with diabatic quantum annealing, Nat. Rev. Phys. 3, 466 (2021).
  82. D. De Santis, S. Tirone, S. Marmi, and V. Giovannetti, Optimized QUBO formulation methods for quantum computing, Quantum Sci. Technol. 11, 015056 (2026).
  83. D. Volpe, G. Orlandi, and G. Turvani, Improving the solving of optimization problems: A comprehensive review of quantum approaches, Quantum Rep. 7, 3 (2025).
  84. F. R. K. Chung, Spectral Graph Theory (American Mathematical Society, Providence, 1997).
  85. D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett. 35, 1792 (1975).
  86. L. Petrov, Lectures on random matrices (Spring 2025), Lecture 2: Wigner semicircle law, course notes (2025), https://lpetrov.cc/rmt25/rmt25-notes/rmt2025-l02.pdf.
  87. D. Sherrington and S. Kirkpatrick, 50 years of spin glass theory, Nat. Rev. Phys. 7, 528 (2025).
  88. J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
  89. S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F: Met. Phys. 5, 965 (1975).
  90. S. F. Edwards and P. W. Anderson, Theory of spin glasses. II, J. Phys. F: Met. Phys. 6, 1927 (1976).
  91. R. Moessner and A. P. Ramirez, Geometrical frustration, Phys. Today 59(2), 24 (2006).
  92. C. Nisoli, R. Moessner, and P. Schiffer, Colloquium: Artificial spin ice: Designing and imaging magnetic frustration, Rev. Mod. Phys. 85, 1473 (2013).
  93. B. B. Bhattacharya and S. Mukherjee, Inference in Ising models, Bernoulli 24, 493 (2018).
  94. S. Chatterjee, Features of a spin glass in the random field Ising model, Commun. Math. Phys. 405, 93 (2024).
  95. A. R. Feier, Methods of proof in random matrix theory, Master’s thesis, Harvard University, 2012.
  96. L. F. Cugliandolo, Advanced statistical physics: Frustration, Lecture Notes (2017), https://www.lpthe.jussieu.fr/~leticia/TEACHING/Master2017/frustration.pdf.
  97. F. Wegner, Inverse participation ratio in 2+ɛ dimensions, Z. Phys. B 36, 209 (1980).
  98. B. Kramer and A. MacKinnon, Localization: Theory and experiment, Rep. Prog. Phys. 56, 1469 (1993).
  99. F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  100. F. Haake, Quantum Signatures of Chaos (Springer, Berlin, Heidelberg, 2010), 3rd ed.
  101. I. J. Farkas, I. Derényi, A.-L. Barabási, and T. Vicsek, Spectra of “real-world” graphs: Beyond the semicircle law, Phys. Rev. E 64, 026704 (2001).
  102. A. J. Bray and M. A. Moore, Replica theory of quantum spin glasses, J. Phys. C 14, 2629 (1981).
  103. G. Parisi, A sequence of approximated solutions to the S-K model for spin glasses, J. Phys. A: Math. Gen. 13, L115 (1980).
  104. M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
  105. K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801 (1986).
  106. D. J. Amit, H. Gutfreund, and H. Sompolinsky, Storing infinite numbers of patterns in a spin-glass model of neural networks, Phys. Rev. Lett. 55, 1530 (1985).
  107. D. J. Amit, H. Gutfreund, and H. Sompolinsky, Statistical mechanics of neural networks near saturation, Ann. Phys. 173, 30 (1987).
  108. J. Lin, Divergence measures based on the Shannon entropy, IEEE Trans. Inf. Theory 37, 145 (2002).
  109. F. J. Massey, Jr., The Kolmogorov-Smirnov test for goodness of fit, J. Am. Stat. Assoc. 46, 68 (1951).
  110. R. B. Cattell, The scree test for the number of factors, Multivar. Behav. Res. 1, 245 (1966).
  111. Z. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices (Springer, New York, 2010).
  112. A. J. Bray and S. A. Roberts, Replica theory of the spherical model, J. Phys. C 13, 5405 (1980).
  113. D. Herr, E. Brown, B. Heim, M. Könz, G. Mazzola, and M. Troyer, Optimizing schedules for quantum annealing, arXiv:1705.00420.
  114. G. Quiroz, Robust quantum control for adiabatic quantum computation, Phys. Rev. A 99, 062306 (2019).
  115. W. Vinci, T. Albash, G. Paz-Silva, I. Hen, and D. A. Lidar, Quantum annealing correction with minor embedding, Phys. Rev. A 92, 042310 (2015).
  116. E. Grant, T. S. Humble, and B. Stump, Benchmarking quantum annealing controls with portfolio optimization, Phys. Rev. Appl. 15, 014012 (2021).
  117. J. Marshall, E. G. Rieffel, and I. Hen, Thermalization, freeze-out, and noise: Deciphering experimental quantum annealers, Phys. Rev. Appl. 8, 064025 (2017).
  118. J. Marshall, D. Venturelli, I. Hen, and E. G. Rieffel, Power of pausing: Advancing understanding of thermalization in experimental quantum annealers, Phys. Rev. Appl. 11, 044083 (2019).
  119. Y. Susa, Y. Yamashiro, M. Yamamoto, I. Hen, D. A. Lidar, and H. Nishimori, Quantum annealing of the p-spin model under inhomogeneous transverse field driving, Phys. Rev. A 98, 042326 (2018).
  120. J. I. Adame and P. L. McMahon, Inhomogeneous driving in quantum annealers can result in orders-of-magnitude improvements in performance, Quantum Sci. Technol. 5, 035011 (2020).
  121. V. Mehta, H. De Raedt, K. Michielsen, and F. Jin, Unraveling reverse annealing: A study of D-Wave quantum annealers, Phys. Rev. A 112, 012414 (2025).
  122. D. Venturelli and A. Kondratyev, Reverse quantum annealing approach to portfolio optimization problems, Quantum Mach. Intell. 1, 17 (2019).
  123. B. Yan and N. A. Sinitsyn, Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian, Nat. Commun. 13, 2212 (2022).

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APS Open Sci. 1, 000067– Published 21 July, 2026
Vol. 1
  • Received 3 March 2026
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