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Extrapolation method to optimize linear-ramp quantum approximate optimization algorithm parameters: Evaluation of runtime scaling

Vanessa Dehn1, Martin Zaefferer2, Gerhard Hellstern3, Karthik Jayadevan1, Florentin Reiter1, and Thomas Wellens1,*

  • *Contact author: thomas.wellens@iaf.fraunhofer.de
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Phys. Rev. A 113, 032413 – Published 9 March, 2026

DOI: https://doi.org/10.1103/l5r4-zcqv

Abstract

The quantum approximate optimization algorithm (QAOA) has been suggested as a promising candidate for the solution of combinatorial optimization problems. Yet, whether—or under what conditions—it may offer an advantage compared with classical algorithms remains to be proven. Using the standard variational form of QAOA requires a high number of circuit parameters that have to be optimized at a sufficiently large depth, which constitutes a bottleneck for achieving a potential scaling advantage. The linear-ramp QAOA has been proposed to address this issue, as it relies on only two parameters that have to be optimized. Based on this, we develop a method to estimate suitable values for those parameters through extrapolation, starting from smaller problem sizes (number of qubits) toward larger problem sizes. We apply this method to several use cases, such as portfolio optimization, feature selection, clustering, and weighted maxcut. From results obtained on a noiseless quantum emulator, we evaluate the quantum runtime scaling for finding the optimal solution and compare it with that of classical methods. In the case of portfolio optimization, we demonstrate superior scaling compared with the classical runtime for the problem sizes of up to 28 qubits that we consider in this work.

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References (41)

  1. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  2. 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).
  3. C. Grange, M. Poss, and E. Bourreau, An introduction to variational quantum algorithms for combinatorial optimization problems, Ann. Oper. Res. 21, 363 (2023).
  4. K. Blekos, D. Brand, A. Ceschini, C.-H. Chou, R.-H. Li, K. Pandya, and A. Summer, A review on quantum approximate optimization algorithm and its variants, Phys. Rep. 1068, 1 (2024).
  5. R. Shaydulin and Y. Alexeev, Evaluating quantum approximate optimization algorithm: A case study, in 2019 Tenth International Green and Sustainable Computing Conference (IGSC) (IEEE, Alexandria, VA, 2019), pp. 1–6.
  6. 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).
  7. E. Farhi, S. Gutmann, D. Ranard, and B. Villalonga, Lower bounding the maxcut of high girth 3-regular graphs using the QAOA, arXiv:2503.12789.
  8. V. Kremenetski, T. Hogg, S. Hadfield, S. J. Cotton, and N. M. Tubman, Quantum alternating operator ansatz (QAOA) phase diagrams and applications for quantum chemistry, arXiv:2108.13056.
  9. 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).
  10. 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.
  11. L. Zhou, S.-T. Wang, S. Choi, H. Pichler, and M. D. Lukin, Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices, Phys. Rev. X 10, 021067 (2020).
  12. S. H. Sack and M. Serbyn, Quantum annealing initialization of the quantum approximate optimization algorithm, Quantum 5, 491 (2021).
  13. V. Kremenetski, A. Apte, T. Hogg, S. Hadfield, and N. M. Tubman, Quantum alternating operator ansatz (QAOA) beyond low depth with gradually changing unitaries, arXiv:2305.04455.
  14. R. Sakai, H. Matsuyama, W.-H. Tam, Y. Yamashiro, and K. Fujii, Linearly simplified QAOA parameters and transferability, arXiv:2405.00655.
  15. R. Shaydulin, S. Hadfield, T. Hogg, and I. Safro, Classical symmetries and the quantum approximate optimization algorithm, Quantum Inf. Process. 20, 359 (2021).
  16. G. B. Mbeng, R. Fazio, and G. Santoro, Quantum annealing: A journey through digitalization, control, and hybrid quantum variational schemes, arXiv:1906.08948.
  17. G. E. Crooks, Performance of the quantum approximate optimization algorithm on the maximum cut problem, arXiv:1811.08419.
  18. J. A. Montanez-Barrera and K. Michielsen, Towards a universal QAOA protocol: Evidence of a scaling advantage in solving some combinatorial optimization problems, npj Quantum Inf. 11, 131 (2025).
  19. S. Boulebnane and A. Montanaro, Solving Boolean satisfiability problems with the quantum approximate optimization algorithm, PRX Quantum 5, 030348 (2024).
  20. R. Shaydulin, C. Li, S. Chakrabarti, M. DeCross, D. Herman, N. Kumar, J. Larson, D. Lykov, P. Minssen, Y. Sun, Y. Alexeev, J. M. Dreiling, J. P. Gaebler, T. M. Gatterman, J. A. Gerber, K. Gilmore, D. Gresh, N. Hewitt, C. V. Horst, S. Hu, et al., Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem, Sci. Adv. 10, eadm6761 (2024).
  21. F. Glover, G. Kochenberger, and Y. Du, A tutorial on formulating and using QUBO models, arXiv:1811.11538.
  22. H. Markowitz, Portfolio selection, J. Finance 7, 77 (1952).
  23. S. Brandhofer, D. Braun, V. Dehn, G. Hellstern, M. Hüls, Y. Ji, I. Polian, A. S. Bhatia, and T. Wellens, Benchmarking the performance of portfolio optimization with QAOA, Quantum Inf. Process. 22, 25 (2022).
  24. G. Hellstern, V. Dehn, and M. Zaefferer, Quantum computer based feature selection in machine learning, IET Quantum Commun. 5, 232 (2024).
  25. H. Hofmann, Statlog (German credit data), UCI Machine Learning Repository (1994), https://doi.org/10.24432/C5NC77.
  26. S. Lloyd, Least squares quantization in pcm, IEEE Trans. Inf. Theory 28, 129 (1982).
  27. R. Sibson, SLINK: an optimally efficient algorithm for the single-link cluster method, Comput. J. 16, 30 (1973).
  28. M. Ester, H.-P. Kriegel, J. Sander, X. Xu, et al., A density- based algorithm for discovering clusters in large spatial databases with noise, in KDD'96: Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (AAAI Press, Portland, OR, 1996), Vol. 96, pp. 226–231.
  29. J. Saiphet, S. Suwanna, T. Chotibut, and A. Chantasri, Quantum approximate optimization and k-means algorithms for data clustering, J. Phys.: Conf. Ser. 1719, 012100 (2021).
  30. J. S. Otterbach, R. Manenti, N. Alidoust, A. Bestwick, M. Block, B. Bloom, S. Caldwell, N. Didier, E. S. Fried, S. Hong, P. Karalekas, C. B. Osborn, A. Papageorge, E. C. Peterson, G. Prawiroatmodjo, N. Rubin, C. A. Ryan, D. Scarabelli, M. Scheer, E. A. Sete, et al., Unsupervised machine learning on a hybrid quantum computer, arXiv:1712.05771.
  31. F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, Scikit-learn: Machine learning in Python, J. Mach. Learn. Res. 12, 2825 (2011).
  32. IBM ILOG CPLEX, V12. 1: User's manual for CPLEX, IBM ILOG CPLEX V12.1 (2009), https://public.dhe.ibm.com/software/websphere/ilog/docs/optimization/cplex/ps_usrmancplex.pdf.
  33. Gurobi Optimization, LLC, Gurobi optimizer reference manual (2023), https://www.gurobi.com/.
  34. I. Dunning, S. Gupta, and J. Silberholz, What works best when? a systematic evaluation of heuristics for max-cut and QUBO, INFORMS J. Comput. 30, 608 (2018).
  35. M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42, 1115 (1995).
  36. Qiskit contributors, Qiskit: An open-source framework for quantum computing, Zenodo (2023), https://doi.org/10.5281/zenodo.8190968.
  37. Z. Zhu, A. J. Ochoa, and H. G. Katzgraber, Fair sampling of ground-state configurations of binary optimization problems, Phys. Rev. E 99, 063314 (2019).
  38. M. Born and V. Fock, Beweis des Adiabatensatzes, Z. Phys. 51, 165 (1928).
  39. A. Dranov, J. Kellendonk, and R. Seiler, Discrete time adiabatic theorems for quantum mechanical systems, J. Math. Phys. 39, 1340 (1998).
  40. 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).
  41. T. Wellens, V. Dehn, G. Hellstern, K. Jayadevan, F. Reiter, and T. Wellens, Extrapolation method to optimize linear-ramp quantum approximate optimization algorithm parameters: dataset [Data set], Zenodo (2026), https://doi.org/10.5281/zenodo.18302993.

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Phys. Rev. A 113, 032413– Published 9 March, 2026
  • Received 17 April 2025
  • Accepted 15 January 2026
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