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Evaluating the limits of Quantum Approximate Optimization Algorithm parameter transfer at high rounds on sparse Ising models with geometrically local cubic terms

Elijah Pelofske1,*, Marek M. Rams2,3, Andreas Bärtschi4, Piotr Czarnik2,3, Paolo Braccia5, Lukasz Cincio5, and Stephan Eidenbenz4

  • *Contact author: epelofske@lanl.gov
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Phys. Rev. Research 8, 023023 – Published 7 April, 2026

DOI: https://doi.org/10.1103/p2lg-z4kn

Abstract

The emergent practical applicability of the Quantum Approximate Optimization Algorithm (QAOA) for approximate combinatorial optimization is a subject of considerable interest. One of the primary limitations of QAOA is the task of finding a set of good parameters, which is usually done using a variational optimization loop. Parameter transfer, or parameter concentration, is a phenomenon where QAOA angles trained on problem instances that are self-similar tend to perform well for other problem instances from that similar class. This suggests a potentially highly efficient and scalable nonvariational learning method for QAOA angle finding. In this work, we systematically study QAOA parameter transferability from small problem sizes (16 and 27 decision variables) onto large problem instances (up to 156 qubits) for heavy-hex graph Ising models with geometrically local higher-order terms using the Julia based QAOA simulation tool JuliQAOA to perform classical angle finding for up to 49 QAOA layers (p). Parameter transfer of the fixed angles is validated using a combination of full state vector, projected entangled pair states, matrix product state, and LOWESA numerical simulations. We find that the QAOA parameter transfer from single instances applied to other (unseen) problem instances does not in general provide monotonically improving performance as a function of p—there are many cases where the performance temporarily decreases as a function of p—but despite this the transferred angles have a general trend of improved expectation value as the QAOA depth increases, in many cases converging close to the true ground-state energy of the 100+ qubit instances. We also sample the hardware-compatible Ising models using the ensemble of transfer-learned QAOA parameters on several superconducting qubit IBM quantum processors with 127, 133, and 156 qubits. We find continuous solution quality improvement of the hardware-compatible QAOA circuits run on the IBM noisy intermediate-scale quantum processors up to p=5 on ibm_fez, up to p=9 on ibm_torino, and up to p=10 on ibm_pittsburgh.

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References (106)

  1. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  2. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem, arXiv:1412.6062.
  3. S. Hadfield, Z. Wang, B. O’Gorman, E. Rieffel, D. Venturelli, and R. Biswas, From the quantum approximate optimization algorithm to a quantum alternating operator ansatz, Algorithms 12, 34 (2019).
  4. E. Farhi, D. Gamarnik, and S. Gutmann, The quantum approximate optimization algorithm needs to see the whole graph: A typical case, arXiv:2004.09002.
  5. E. Farhi, D. Gamarnik, and S. Gutmann, The quantum approximate optimization algorithm needs to see the whole graph: Worst case examples, arXiv:2005.08747.
  6. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  7. E. Farhi and A. W. Harrow, Quantum supremacy through the quantum approximate optimization algorithm, arXiv:1602.07674.
  8. R. Shaydulin, et al., Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem, Sci. Adv. 10, eadm6761 (2024).
  9. M. P. Harrigan, et al., Quantum approximate optimization of non-planar graph problems on a planar superconducting processor, Nat. Phys. 17, 332 (2021).
  10. E. Pelofske, A. Bärtschi, J. Golden, and S. Eidenbenz, High-round QAOA for MAX k-SAT on trapped ion NISQ devices, in IEEE International Conference on Quantum Computing and Engineering QCE’23 (IEEE, Bellevue, WA, USA, 2023), pp. 506–517.
  11. R. Shaydulin and M. Pistoia, QAOA with N·p≥200, in IEEE International Conference on Quantum Computing and Engineering QCE’23 (IEEE, Bellevue, WA, USA, 2023), pp. 1074–1077.
  12. J. Weidenfeller, L. C. Valor, J. Gacon, C. Tornow, L. Bello, S. Woerner, and D. J. Egger, Scaling of the quantum approximate optimization algorithm on superconducting qubit based hardware, Quantum 6, 870 (2022).
  13. J. Golden, A. Bärtschi, S. Eidenbenz, and D. O’Malley, Numerical evidence for exponential speed-up of QAOA over unstructured search for approximate constrained optimization, in IEEE International Conference on Quantum Computing and Engineering QCE’23 (IEEE, Bellevue, WA, USA, 2023), pp. 496–505.
  14. S. Boulebnane and A. Montanaro, Solving Boolean satisfiability problems with the quantum approximate optimization algorithm, PRX Quantum 5, 030348 (2024).
  15. M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
  16. The QAOA parameter similarity property comes by several different terms, including parameter transfer, problem instance independent QAOA parameters, fixed angles, parameter concentration, parameter fixing, size invariant QAOA parameters, fixed control parameters, fixed point QAOA, and transfer learning.
  17. V. Akshay, D. Rabinovich, E. Campos, and J. Biamonte, Parameter concentration in quantum approximate optimization, Phys. Rev. A 104, L010401 (2021).
  18. X. Lee, Y. Saito, D. Cai, and N. Asai, Parameters fixing strategy for quantum approximate optimization algorithm, in IEEE International Conference on Quantum Computing and Engineering QCE’21 (IEEE, Broomfield, CO, USA, 2021), pp. 10–16.
  19. J. Basso, E. Farhi, K. Marwaha, B. Villalonga, and L. Zhou, The quantum approximate optimization algorithm at high depth for MaxCut on large-girth regular graphs and the Sherrington-Kirkpatrick model, in 17th Conference on the Theory of Quantum Computation, Communication and Cryptography TQC’22 (Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2022), Vol. 232, pp. 7:1–7:21.
  20. E. Farhi, J. Goldstone, S. Gutmann, and L. Zhou, The quantum approximate optimization algorithm and the Sherrington-Kirkpatrick model at infinite size, Quantum 6, 759 (2022).
  21. S. Boulebnane and A. Montanaro, Predicting parameters for the quantum approximate optimization algorithm for MAX-CUT from the infinite-size limit, arXiv:2110.10685.
  22. R. Shaydulin, P. C. Lotshaw, J. Larson, J. Ostrowski, and T. S. Humble, Parameter transfer for quantum approximate optimization of weighted MaxCut, ACM Trans. Quantum Comput. 4, 1 (2023).
  23. S. H. Sureshbabu, D. Herman, R. Shaydulin, J. Basso, S. Chakrabarti, Y. Sun, and M. Pistoia, Parameter setting in quantum approximate optimization of weighted problems, Quantum 8, 1231 (2024).
  24. J. Wurtz and D. Lykov, Fixed-angle conjectures for the quantum approximate optimization algorithm on regular MaxCut graphs, Phys. Rev. A 104, 052419 (2021).
  25. J. Wurtz and P. Love, Maxcut quantum approximate optimization algorithm performance guarantees for p>1, Phys. Rev. A 103, 042612 (2021).
  26. M. Streif and M. Leib, Training the quantum approximate optimization algorithm without access to a quantum processing unit, Quantum Sci. Technol. 5, 034008 (2020).
  27. J. A. Montañez-Barrera, D. Willsch, and K. Michielsen, Transfer learning of optimal QAOA parameters in combinatorial optimization, Quantum Inf. Process. 24, 129 (2025).
  28. E. Pelofske, A. Bärtschi, L. Cincio, J. Golden, and S. Eidenbenz, Scaling whole-chip QAOA for higher-order Ising spin glass models on heavy-hex graphs, npj Quantum Inf. 10, 109 (2024).
  29. F. G. S. L. Brandao, M. Broughton, E. Farhi, S. Gutmann, and H. Neven, For fixed control parameters the quantum approximate optimization algorithm's objective function value concentrates for typical instances, arXiv:1812.04170.
  30. R. Sakai, H. Matsuyama, W.-H. Tam, Y. Yamashiro, and K. Fujii, Linearly simplified QAOA parameters and transferability, arXiv:2405.00655.
  31. V. Katial, K. Smith-Miles, C. Hill, and L. Hollenberg, On the instance dependence of optimal parameters for the quantum approximate optimisation algorithm: Insights via instance space analysis, INFORMS J. Comput. 37, 146 (2025).
  32. J. Sud, S. Hadfield, E. Rieffel, N. Tubman, and T. Hogg, Parameter-setting heuristic for the quantum alternating operator ansatz, Phys. Rev. Res. 6, 023171 (2024).
  33. A. Y. Chernyavskiy, D. A. Kulikov, B. I. Bantysh, Y. I. Bogdanov, A. K. Fedorov, and E. O. Kiktenko, Improving QAOA to find approximate QUBO solutions in O(1) shots, arXiv:2509.19035.
  34. A. Galda, E. Gupta, J. Falla, X. Liu, D. Lykov, Y. Alexeev, and I. Safro, Similarity-based parameter transferability in the quantum approximate optimization algorithm, Front. Quantum Sci. Technol. 2, 1200975 (2023).
  35. A. Galda, X. Liu, D. Lykov, Y. Alexeev, and I. Safro, Transferability of optimal QAOA parameters between random graphs, arXiv:2106.07531.
  36. I. Lyngfelt and L. García-Álvarez, Symmetry-informed transferability of optimal parameters in the quantum approximate optimization algorithm, Phys. Rev. A 111, 022418 (2025).
  37. A. Ozaeta, W. van Dam, and P. L. McMahon, Expectation values from the single-layer quantum approximate optimization algorithm on Ising problems, Quantum Sci. Technol. 7, 045036 (2022).
  38. Z. He, R. Shaydulin, D. Herman, C. Li, R. Raymond, S. H. Sureshbabu, and M. Pistoia, Parameter setting heuristics make the quantum approximate optimization algorithm suitable for the early fault-tolerant era, in 43rd IEEE/ACM International Conference on Computer-Aided Design ICCAD’24 (Association for Computing Machinery, New York, NY, USA, 2024), pp. 63:1–36:7.
  39. S. H. Sack and M. Serbyn, Quantum annealing initialization of the quantum approximate optimization algorithm, Quantum 5, 491 (2021).
  40. J. A. Montañez-Barrera and K. Michielsen, Toward a linear-ramp QAOA protocol: Evidence of a scaling advantage in solving some combinatorial optimization problems, npj Quantum Inf. 11, 131 (2025).
  41. E. Pelofske, A. Bärtschi, and S. Eidenbenz, Short-depth QAOA circuits and quantum annealing on higher-order Ising models, npj Quantum Inf. 10, 30 (2024).
  42. E. Pelofske, A. Bärtschi, and S. Eidenbenz, Quantum annealing vs. QAOA: 127 qubit higher-order Ising problems on NISQ computers, in International Conference on High Performance Computing ISC HPC’23 (Springer, Cham, 2023), pp. 240–258.
  43. S. V. Romero, A.-M. Visuri, A. G. Cadavid, E. Solano, and N. N. Hegade, Bias-field digitized counterdiabatic quantum algorithm for higher-order binary optimization, Commun. Phys. 8, 348 (2025).
  44. S. H. Sack and D. J. Egger, Large-scale quantum approximate optimization on nonplanar graphs with machine learning noise mitigation, Phys. Rev. Res. 6, 013223 (2024).
  45. The QAOA parameter p can be found under several different terms in the literature, including layers, rounds, depth (importantly not the same quantity as circuit depth), and levels; in this study, we will typically refer to this parameter as either rounds or depth.
  46. J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev. 59, 65 (2017).
  47. J. Golden, A. Baertschi, D. O’Malley, E. Pelofske, and S. Eidenbenz, JuliQAOA: Fast, flexible QAOA simulation, in Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis SC-W’23 (Association for Computing Machinery, New York, NY, USA, 2023), pp. 1454–1459.
  48. J. Golden, A. Bärtschi, D. O’Malley, and S. Eidenbenz, Threshold-based quantum optimization, in IEEE International Conference on Quantum Computing and Engineering QCE’21 (IEEE, Broomfield, CO, USA, 2021), pp. 137–147.
  49. J. Golden, A. Bärtschi, D. O’Malley, and S. Eidenbenz, The quantum alternating operator ansatz for satisfiability problems, in IEEE International Conference on Quantum Computing and Engineering QCE’23 (IEEE, Bellevue, WA, USA, 2023), pp. 307–312.
  50. G. Pagano, A. Bapat, P. Becker, K. S. Collins, A. De, P. W. Hess, H. B. Kaplan, A. Kyprianidis, W. L. Tan, C. Baldwin, L. T. Brady, A. Deshpande, F. Liu, S. Jordan, A. V. Gorshkov, and C. Monroe, Quantum approximate optimization of the long-range Ising model with a trapped-ion quantum simulator, Proc. Natl. Acad. Sci. USA 117, 25396 (2020).
  51. S. H. Sack, R. A. Medina, R. Kueng, and M. Serbyn, Recursive greedy initialization of the quantum approximate optimization algorithm with guaranteed improvement, Phys. Rev. A 107, 062404 (2023).
  52. A. Apte, S. H. Sureshbabu, R. Shaydulin, S. Boulebnane, Z. He, D. Herman, J. Sud, and M. Pistoia, Iterative interpolation schedules for quantum approximate optimization algorithm, arXiv:2504.01694.
  53. X. Lee, N. Xie, Y. Saito, D. Cai, and N. Asai, A depth-progressive initialization strategy for quantum approximate optimization algorithm, arXiv:2209.11348.
  54. C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, Topological and subsystem codes on low-degree graphs with flag qubits, Phys. Rev. X 10, 011022 (2020).
  55. C. Benito, E. López, B. Peropadre, and A. Bermudez, Comparative study of quantum error correction strategies for the heavy-hexagonal lattice, Quantum 9, 1623 (2025).
  56. B. Hetényi and J. R. Wootton, Creating entangled logical qubits in the heavy-hex lattice with topological codes, PRX Quantum 5, 040334 (2024).
  57. S. V. Barron, D. J. Egger, E. Pelofske, A. Bärtschi, S. Eidenbenz, M. Lehmkuehler, and S. Woerner, Provable bounds for noise-free expectation values computed from noisy samples, Nat. Comput. Sci. 4, 865 (2024).
  58. T. O’Leary, P. Czarnik, E. Pelofske, A. T. Sornborger, M. McKerns, and L. Cincio, Efficient online quantum circuit learning with no upfront training, Commun. Phys. 8, 514 (2025).
  59. E. Pelofske, A. Bärtschi, and S. Eidenbenz, Classical combinatorial optimization scaling for random Ising models on 2D heavy-hex graphs, arXiv:2412.15572.
  60. A. A. Hagberg, D. A. Schult, and P. J. Swart, Exploring network structure, dynamics, and function using NetworkX, Python in Science Conference (2008).
  61. A. Kotil, E. Pelofske, S. Riedmüller, D. J. Egger, S. Eidenbenz, T. Koch, and S. Woerner, Quantum approximate multi-objective optimization, Nat. Comput. Sci. 5, 1168 (2025).
  62. J. M. Chow, A. D. Córcoles, J. M. Gambetta, C. Rigetti, B. R. Johnson, J. A. Smolin, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, Simple all-microwave entangling gate for fixed-frequency superconducting qubits, Phys. Rev. Lett. 107, 080502 (2011).
  63. Qiskit contributors, Qiskit: An open-source framework for quantum computing (2023), https://zenodo.org/record/2573505.
  64. A. Javadi-Abhari, M. Treinish, K. Krsulich, C. J. Wood, J. Lishman, J. Gacon, S. Martiel, P. D. Nation, L. S. Bishop, A. W. Cross, B. R. Johnson, and J. M. Gambetta, Quantum computing with Qiskit, arXiv:2405.08810.
  65. IBM ILOG CPLEX, V12.10.0: User's Manual for CPLEX, (International Business Machines Corporation, 2019), Vol. 46, p. 157, https://public.dhe.ibm.com/software/websphere/ilog/docs/optimization/cplex/ps_usrmancplex.pdf.
  66. Gurobi Optimization, LLC, Gurobi optimizer reference manual (2024), https://www.gurobi.com/.
  67. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  68. Y.-Y. Shi, L.-M. Duan, and G. Vidal, Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev. A 74, 022320 (2006).
  69. R. Jozsa, On the simulation of quantum circuits, arXiv:quant-ph/0603163.
  70. R. Alkabetz and I. Arad, Tensor networks contraction and the belief propagation algorithm, Phys. Rev. Res. 3, 023073 (2021).
  71. J. Tindall and M. Fishman, Gauging tensor networks with belief propagation, SciPost Phys. 15, 222 (2023).
  72. H. C. Jiang, Z. Y. Weng, and T. Xiang, Accurate determination of tensor network state of quantum lattice models in two dimensions, Phys. Rev. Lett. 101, 090603 (2008).
  73. J. Tindall, M. Fishman, E. M. Stoudenmire, and D. Sels, Efficient tensor network simulation of IBM's Eagle kicked Ising experiment, PRX Quantum 5, 010308 (2024).
  74. T. Begušić, J. Gray, and G. K.-L. Chan, Fast and converged classical simulations of evidence for the utility of quantum computing before fault tolerance, Sci. Adv. 10, eadk4321 (2024).
  75. S. Patra, S. S. Jahromi, S. Singh, and R. Orús, Efficient tensor network simulation of IBM's largest quantum processors, Phys. Rev. Res. 6, 013326 (2024).
  76. M. S. Rudolph and J. Tindall, Simulating and sampling from quantum circuits with 2D tensor networks, arXiv:2507.11424.
  77. M. M. Rams, G. Wójtowicz, A. Sinha, and J. Hasik, YASTN: Yet another symmetric tensor networks; a Python library for Abelian symmetric tensor network calculations, SciPost Phys. Codebases, 52 (2025).
  78. M. Fishman, S. R. White, and E. M. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Phys. Codebases, 4 (2022).
  79. M. Fishman, S. R. White, and E. M. Stoudenmire, Codebase release 0.3 for ITensor, SciPost Phys. Codebases, 4-r0.3 (2022).
  80. M. Rams, PEPS heavy-hex QAOA simulation Github, Github (2025), https://github.com/marekrams/hex.
  81. E. Fontana, M. S. Rudolph, R. Duncan, I. Rungger, and C. Cîrstoiu, Classical simulations of noisy variational quantum circuits, npj Quantum Inf. 11, 84 (2025).
  82. M. S. Rudolph, E. Fontana, Z. Holmes, and L. Cincio, Classical surrogate simulation of quantum systems with LOWESA, arXiv:2308.09109.
  83. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  84. G. E. Crooks, Performance of the quantum approximate optimization algorithm on the maximum cut problem, arXiv:1811.08419.
  85. T. Müller, A. Singh, F. K. Wilhelm, and T. Bode, Limitations of quantum approximate optimization in solving generic higher-order constraint-satisfaction problems, Phys. Rev. Res. 7, 023165 (2025).
  86. P. Díez-Valle, F. J. Gómez-Ruiz, D. Porras, and J. J. García-Ripoll, Universal resources for quantum approximate optimization algorithm and quantum annealing, Phys. Rev. Res. 8, 013211 (2026).
  87. S. Boulebnane, J. Sud, R. Shaydulin, and M. Pistoia, Equivalence of quantum approximate optimization algorithm and linear-time quantum annealing for the Sherrington-Kirkpatrick model, arXiv:2503.09563.
  88. L. K. Kovalsky, F. A. Calderon-Vargas, M. D. Grace, A. B. Magann, J. B. Larsen, A. D. Baczewski, and M. Sarovar, Self-healing of Trotter error in digital adiabatic state preparation, Phys. Rev. Lett. 131, 060602 (2023).
  89. G. B. Mbeng, R. Fazio, and G. Santoro, Quantum annealing: A journey through digitalization, control, and hybrid quantum variational schemes, arXiv:1906.08948.
  90. S. Dasgupta and T. S. Humble, Characterizing the stability of NISQ devices, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Denver, CO, USA, 2020), pp. 419–429.
  91. S. Dasgupta and T. S. Humble, Stability of noisy quantum computing devices, arXiv:2105.09472.
  92. J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature (London) 453, 1031 (2008).
  93. E. Magesan, J. M. Gambetta, and J. Emerson, Characterizing quantum gates via randomized benchmarking, Phys. Rev. A 85, 042311 (2012).
  94. R. Harper, I. Hincks, C. Ferrie, S. T. Flammia, and J. J. Wallman, Statistical analysis of randomized benchmarking, Phys. Rev. A 99, 052350 (2019).
  95. D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficient Z gates for quantum computing, Phys. Rev. A 96, 022330 (2017).
  96. A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta, Validating quantum computers using randomized model circuits, Phys. Rev. A 100, 032328 (2019).
  97. C. H. Baldwin, K. Mayer, N. C. Brown, C. Ryan-Anderson, and D. Hayes, Re-examining the quantum volume test: Ideal distributions, compiler optimizations, confidence intervals, and scalable resource estimations, Quantum 6, 707 (2022).
  98. E. Pelofske, A. Bärtschi, and S. Eidenbenz, Quantum volume in practice: What users can expect from NISQ devices, IEEE Trans. Quantum Eng. 3, 1 (2022).
  99. P. Jurcevic, et al., Demonstration of quantum volume 64 on a superconducting quantum computing system, Quantum Sci. Technol. 6, 025020 (2021).
  100. M. Dupont, N. Didier, M. J. Hodson, J. E. Moore, and M. J. Reagor, Entanglement perspective on the quantum approximate optimization algorithm, Phys. Rev. A 106, 022423 (2022).
  101. E. Pelofske, L. Cincio, M. Rams, A. Bärtschi, P. Czarnik, P. Braccia, and S. Eidenbenz, Dataset for the paper titled “Evaluating the limits of Quantum Approximate Optimization Algorithm parameter transfer at high rounds on sparse Ising models with geometrically local cubic terms”, Zenodo (2025), http://doi.org/10.5281/zenodo.17110170.
  102. M. S. Rudolph, T. Jones, Y. Teng, A. Angrisani, and Z. Holmes, Pauli propagation: A computational framework for simulating quantum systems, arXiv:2505.21606.
  103. T. Nishino and K. Okunishi, A density matrix algorithm for 3D classical models, J. Phys. Soc. Jpn. 67, 3066 (1998).
  104. R. Orús and G. Vidal, Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction, Phys. Rev. B 80, 094403 (2009).
  105. P. Corboz, S. R. White, G. Vidal, and M. Troyer, Stripes in the two-dimensional t−J model with infinite projected entangled-pair states, Phys. Rev. B 84, 041108(R) (2011).
  106. F. Verstraete, V. Murg, and J. I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).

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Phys. Rev. Research 8, 023023– Published 7 April, 2026
  • Received 20 September 2025
  • Accepted 2 March 2026
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