- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Optimization-free recursive Quantum Approximate Optimization Algorithm for the binary paint shop problem

Gary J Mooney1,*, Jedwin Villanueva1, Bhaskar Roy Bardhan2, Joydip Ghosh2, Charles D Hill1,3, and Lloyd C L Hollenberg1,†

  • *Contact author: mooneyg@unimelb.edu.au
  • †Contact author: lloydch@unimelb.edu.au
PDF

Phys. Rev. Research 8, 013310 – Published 24 March, 2026

DOI: https://doi.org/10.1103/xv79-w3rs

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for near-term quantum advantage, yet its practical impact is hindered by limited performance on symmetric local Hamiltonians and the costly optimization of variational parameters. The recursive-QAOA (RQAOA) introduced by Bravyi et al. [Phys. Rev. Lett. 125, 260505 (2020)] addresses the first limitation while also reducing circuit size, and parameter transfer techniques can be used to bypass the optimization loop effectively. In this work, we combine these two ideas to develop an optimization-free RQAOA and evaluate its performance on the binary paint shop problem (BPSP)—an optimization problem found in manufacturing where a sequence of cars must be painted under constraints while minimizing the number of color changes. The BPSP can be formulated as an Ising ground state problem with a symmetric local Hamiltonian in the form of MAX-CUT and properties well suited for the application of QAOA parameter transfer. We benchmark QAOA and RQAOA with parameter transfer against classical solvers and heuristics, and investigate their resilience to suboptimal parameters. For circuit optimization, we use reverse-causal cones and introduce a method of trimming outer two-qubit gates. To estimate the classical resources needed to simulate these quantum algorithms, we compute entanglement entropy and bond dimensions using matrix product state methods. We also compare circuit sizes and measurement counts across implementations. Our results show that RQAOA is inherently robust to parameter deviations, maintaining near-optimal solutions without noticeable degradation under parameter transfer while substantially reducing quantum resource requirements compared to QAOA. This highlights a viable route toward scalable quantum optimization without the overhead of the classical optimization loop and its challenges with barren plateaus.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (68)

  1. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  2. S. Brandhofer, D. Braun, V. Dehn, G. Hellstern, M. Hüls, Y. Ji, I. Polian, A. Singh Bhatia, and T. Wellens, Benchmarking the performance of portfolio optimization with QAOA, Quantum Inf. Proc. 22, 25 (2022).
  3. M. T. Khumalo, H. A. Chieza, K. Prag, and M. Woolway, An investigation of IBM quantum computing device performance on combinatorial optimisation problems, Neural Computing Appl. 37, 611 (2025).
  4. M. Streif, S. Yarkoni, A. Skolik, F. Neukart, and M. Leib, Beating classical heuristics for the binary paint shop problem with the quantum approximate optimization algorithm, Phys. Rev. A 104, 012403 (2021).
  5. P. Kl. Barkoutsos, G. Nannicini, A. Robert, I. Tavernelli, and S. Woerner, Improving variational quantum optimization using CVaR, Quantum 4, 256 (2020).
  6. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  7. 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).
  8. A. Abbas, A. Ambainis, B. Augustino, A. Bärtschi, H. Buhrman, C. Coffrin, G. Cortiana, V. Dunjko, D. J. Egger, B. G. Elmegreen, et al., Challenges and opportunities in quantum optimization, Nat. Rev. Phys. 6, 718 (2024).
  9. P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, in Proceedings of 35th Annual Symposium on Foundations of Computer Science (IEEE, Santa Fe, NM, 1994), pp. 124–134.
  10. L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing (ACM, Pennsylvania, Philadelphia, 1996), pp. 212–219.
  11. D. S. Johnson, G. Gutin, L. A. McGeoch, A. Yeo, W. Zhang, and A. Zverovitch, Experimental analysis of heuristics for the ATSP, The Traveling Salesman Problem and its Variations, edited by G. Gutin and A. P. Punnen, Combinatorial Optimization, Vol. 12 (Springer, Boston, MA, 2007), pp. 445–487.
  12. D. Applegate, W. Cook, and A. Rohe, Chained Lin-Kernighan for large traveling salesman problems, INFORMS J. Comput. 15, 82 (2003).
  13. K. Helsgaun, An effective implementation of the Lin-Kernighan traveling salesman heuristic, Eur. J. Oper. Res. 126, 106 (2000).
  14. S. Arora, Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, J. ACM 45, 753 (1998).
  15. D. S. Johnson and L. A. Mcgeoch, The traveling salesman problem: A case study, in Local Search in Combinatorial Optimization, edited by E. H. L. Aarts and J. K. Lenstra (John Wiley and Sons, London, 1997), pp. 215–310.
  16. M. L. Fredman, D. S. Johnson, L. A. McGeoch, and G. Ostheimer, Data structures for traveling salesmen, J. Algorithms 18, 432 (1995).
  17. S. Bravyi, A. Kliesch, R. Koenig, and E. Tang, Obstacles to variational quantum optimization from symmetry protection, Phys. Rev. Lett. 125, 260505 (2020).
  18. Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Connecting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum 3, 010313 (2022).
  19. S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Noise-induced barren plateaus in variational quantum algorithms, Nat. Commun. 12, 6961 (2021).
  20. J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Barren plateaus in quantum neural network training landscapes, Nat. Commun. 9, 4812 (2018).
  21. R. Zhang, Environment-aware production scheduling for paint shops in automobile manufacturing: A multi-objective optimization approach, Int. J. Environ. Res. Public Healths 15, 32 (2018).
  22. P. Bonsma, T. Epping, and W. Hochstättler, Complexity results on restricted instances of a paint shop problem for words, Discrete Appl. Math. 154, 1335 (2006).
  23. F. Meunier and A. Sebő, Paintshop, odd cycles and necklace splitting, Discrete Appl. Math. 157, 780 (2009).
  24. M. R. Garey, D. S. Johnson, and L. Stockmeyer, Some simplified NP-complete problems, in Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, STOC '74 (Association for Computing Machinery, Seattle, Washington, 1974), pp. 47–63.
  25. J. Wurtz and D. Lykov, Fixed-angle conjectures for the quantum approximate optimization algorithm on regular MaxCut graphs, Phys. Rev. A 104, 052419 (2021).
  26. 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).
  27. M. Streif and M. Leib, Training the quantum approximate optimization algorithm without access to a quantum processing unit, Quantum Sci. Technol. 5, 034008 (2020).
  28. R. Shaydulin, K. Marwaha, J. Wurtz, and P. C. Lotshaw, QAOAKit: A toolkit for reproducible study, application, and verification of the QAOA, in 2021 IEEE/ACM Second International Workshop on Quantum Computing Software (QCS) (IEEE, St. Louis, MO, 2021), pp. 64–71.
  29. 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).
  30. V. Vijendran, D. E. Koh, P. K. Lam, and S. M. Assad, Classical and quantum heuristics for the binary paint shop problem, arXiv:2509.15294.
  31. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  32. IBM ILOG CPLEX, V12.1: User's manual for CPLEX, Int. Bus. Mach. Corp. 46, 157 (2009).
  33. M. Karpinski, Approximability of the minimum bisection problem: An algorithmic challenge, in Mathematical Foundations of Computer Science (Springer, Berlin, Heidelberg, 2002), pp. 59–67.
  34. T. Epping, W. Hochstättler, and P. Oertel, Complexity results on a paint shop problem, Discrete Appl. Math. 136, 217 (2004).
  35. S. Yarkoni, A. Alekseyenko, M. Streif, D. Von Dollen, F. Neukart, and T. Bäck, Multi-car paint shop optimization with quantum annealing, in 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Broomfield, CO, 2021), pp. 35–411.
  36. Y.-H. Han, C. Zhou, B. Bras, L. McGinnis, C. Carmichael, and P. J. Newcomb, Paint line color change reduction in automobile assembly through simulation, in Winter Simulation Conference (2003), Vol. 2, p. 1204.
  37. E. Farhi, D. Gamarnik, and S. Gutmann, The quantum approximate optimization algorithm needs to see the whole graph: A typical case, arXiv:2004.09002.
  38. E. Farhi, D. Gamarnik, and S. Gutmann, The quantum approximate optimization algorithm needs to see the whole graph: Worst case examples, arXiv:2005.08747.
  39. J. Basso, D. Gamarnik, S. Mei, and L. Zhou, Performance and limitations of the QAOA at constant levels on large sparse hypergraphs and spin glass models, in 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, Denver, CO, 2022), pp. 335–343.
  40. M. B. Hastings, Classical and quantum bounded depth approximation algorithms, arXiv:1905.07047 .
  41. E. Bae and S. Lee, Recursive QAOA outperforms the original QAOA for the MAX-CUT problem on complete graphs, Quantum Inf. Process., 23, 78 (2024).
  42. Y. J. Patel, S. Jerbi, T. Bäck, and V. Dunjko, Reinforcement learning assisted recursive QAOA, EPJ Quantum Technol. 11, 6 (2024).
  43. J. R. Finžgar, A. Kerschbaumer, M. J. A. Schuetz, C. B. Mendl, and H. G. Katzgraber, Quantum-informed recursive optimization algorithms, PRX Quantum 5, 020327 (2024).
  44. R. Kondo, Y. Sato, R. Raymond, and N. Yamamoto, Recursive quantum relaxation for combinatorial optimization problems, Quantum 9, 1594 (2025).
  45. B. Gulbahar, Majority voting with recursive QAOA and cost-restricted uniform sampling for maximum-likelihood detection in massive mimo, IEEE Trans. Wireless Commun. 24, 2620 (2025).
  46. J. Cook, S. Eidenbenz, and A. Bärtschi, The quantum alternating operator ansatz on maximum k-vertex cover, in 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Denver, CO, 2020), pp. 83–92.
  47. M. Alam, A. Ash-Saki, and S. Ghosh, Accelerating quantum approximate optimization algorithm using machine learning, in 2020 Design, Automation & Test in Europe Conference & Exhibition (DATE) (IEEE, Grenoble, France, 2020), pp. 686–689.
  48. M. M. Wauters, E. Panizon, G. B. Mbeng, and G. E. Santoro, Reinforcement-learning-assisted quantum optimization, Phys. Rev. Res. 2, 033446 (2020).
  49. O. Amosy, T. Danzig, O. Lev, E. Porat, G. Chechik, and A. Makmal, Iteration-free quantum approximate optimization algorithm using neural networks, Quantum Mach. Intell. 6, 38 (2024).
  50. A. Galda, X. Liu, D. Lykov, Y. Alexeev, and I. Safro, Transferability of optimal QAOA parameters between random graphs, in 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Broomfield, CO, 2021), pp. 171–180.
  51. P. C. Lotshaw, T. S. Humble, R. Herrman, J. Ostrowski, and G. Siopsis, Empirical performance bounds for quantum approximate optimization, Quantum Inf. Proc. 20, 403 (2021).
  52. 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).
  53. J. Wurtz and P. Love, MaxCut quantum approximate optimization algorithm performance guarantees for p > 1, Phys. Rev. A 103, 042612 (2021).
  54. M. X. Goemans and D. P. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. ACM 42, 1115 (1995).
  55. 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.
  56. Z. Wang, S. Hadfield, Z. Jiang, and E. G. Rieffel, Quantum approximate optimization algorithm for MaxCut: A fermionic view, Phys. Rev. A 97, 022304 (2018).
  57. S. A. Hadfield, Quantum Algorithms for Scientific Computing and Approximate Optimization (American Physical Society, 2018).
  58. 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).
  59. S. Khairy, R. Shaydulin, L. Cincio, Y. Alexeev, and P. Balaprakash, Learning to optimize variational quantum circuits to solve combinatorial problems, AAAI 34, 2367 (2020).
  60. I. Lyngfelt and L. García-Álvarez, Symmetry-informed transferability of optimal parameters in the quantum approximate optimization algorithm, Phys. Rev. A 111, 022418 (2025).
  61. J. Villanueva, G. J. Mooney, B. R. Bardhan, J. Ghosh, C. D. Hill, and L. C. L. Hollenberg, Hybrid quantum optimization in the context of minimizing traffic congestion, arXiv:2504.08275.
  62. A. Dang, C. D. Hill, and L. C. L. Hollenberg, Optimising matrix product state simulations of Shor's algorithm, Quantum 3, 116 (2019).
  63. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  64. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  65. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  66. E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum computation by adiabatic evolution, arXiv:quant-ph/0001106.
  67. M. Born and V. Fock, Beweis des adiabatensatzes, Z. Phys. 51, 165 (1928).
  68. T. Kato, On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Jpn. 5, 435 (1950).

Outline

Information

Phys. Rev. Research 8, 013310– Published 24 March, 2026
  • Received 2 September 2025
  • Accepted 13 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation