- 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

Lagrangian duality in quantum optimization: Overcoming QUBO limitations for constrained problems

Einar Gabbassov1,2,*, Gili Rosenberg1,†, and Artur Scherer1

  • *Contact author: egabbass@uwaterloo.ca
  • †Present address: Amazon Quantum Solutions Lab, Seattle, Washington 98170, USA.
PDF

Phys. Rev. Research 7, 023305 – Published 25 June, 2025

DOI: https://doi.org/10.1103/53gd-2374

Abstract

We propose an approach to solving constrained combinatorial optimization problems based on embedding the concept of Lagrangian duality into the framework of adiabatic quantum computation. Within the setting of circuit-model fault-tolerant quantum computation, we demonstrate that this approach achieves a quadratic improvement in circuit depth and maintains a constraint-independent circuit width in contrast with the prevalent approach of solving constrained problems via reformulations based on the quadratic unconstrained binary optimization (QUBO) framework. In this study, we include a detailed review of the limitations encountered when using QUBO for constrained optimization. We show that the proposed method overcomes these limitations by encoding the optimal solution at an energetically elevated level of a simpler problem Hamiltonian, which results in substantially more resource-efficient quantum circuits. We consolidate our strategy with a detailed analysis on how the concepts of Lagrangian duality such as duality gap and complementary slackness relate to the success probability of sampling the optimal solution. Our findings are illustrated by benchmarking the Lagrangian dual approach against the QUBO approach using the NP-complete binary knapsack problem.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (47)

  1. M. W. Johnson, M. H. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk et al., Quantum annealing with manufactured spins, Nature (London) 473, 194 (2011).
  2. S. Morita and H. Nishimori, Mathematical foundation of quantum annealing, J. Math. Phys. 49, 125210 (2008).
  3. 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).
  4. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  5. J. Montanez-Barrera and K. Michielsen, Towards a universal QAOA protocol: Evidence of a scaling advantage in solving some combinatorial optimization problems, arXiv:2405.09169.
  6. R. Kashapogu, S. Hasib, and A. Rasool, Exploring the versatility of QAOA: A comprehensive review, in Proceedings of the 2024 15th International Conference on Computing Communication and NetworkingTechnologies (ICCCNT) (IEEE, Kamand, India, 2024), pp. 1–8.
  7. M. Lewis and F. Glover, Quadratic unconstrained binary optimization problem preprocessing: Theory and empirical analysis, Networks 70, 79 (2017).
  8. A. Geoffrion, Lagrangean relaxation for integer programming, in Approaches to Integer Programming, edited by M. L. Balinski (Springer, Berlin, 1974), pp. 82–114.
  9. J. Roland and N. Cerf, Quantum search by local adiabatic evolution, Phys. Rev. A 65, 042308 (2002).
  10. T. Albash and D. Lidar, Adiabatic quantum computation, Rev. Mod. Phys. 90, 015002 (2018).
  11. D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, Adiabatic quantum computation is equivalent to standard quantum computation, SIAM Rev. 50, 755 (2008).
  12. E. Gabbassov, Transit facility allocation: Hybrid quantum-classical optimization, PLoS ONE 17, e0274632 (2022).
  13. K. Sankar, A. Scherer, S. Kako, S. Reifenstein, N. Ghadermarzy, W. Krayenhoff, Y. Inui, E. Ng, T. Onodera, P. Ronagh et al., A benchmark study of quantum algorithms for combinatorial optimization, npj Quantum Inf. 10, 64 (2024).
  14. A. Gilyén, S. Arunachalam, and N. Wiebe, Optimizing quantum optimization algorithms via faster quantum gradient computation, in Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms (SIAM, San Diego California, 2019), pp. 1425–1444.
  15. B. Apolloni, C. Carvalho, and D. De Falco, Quantum stochastic optimization, Stoch. Process. Appl. 33, 233 (1989).
  16. Y. Seki and H. Nishimori, Quantum annealing with antiferromagnetic fluctuations, Phys. Rev. E 85, 051112 (2012).
  17. O. Galindo and V. Kreinovich, What is the optimal annealing schedule in quantum annealing, in Proceedings of the 2020 IEEE Symposium Series On Computational Intelligence (SSCI) (IEEE, Canberra, ACT, Australia, 2020), pp. 963–967.
  18. S. Matsuura, T. Yamazaki, V. Senicourt, L. Huntington, and A. Zaribafiyan, Vanqver: The variational and adiabatically navigated quantum eigensolver, New J. Phys. 22, 053023 (2020).
  19. S. Matsuura, S. Buck, V. Senicourt, and A. Zaribafiyan, Variationally scheduled quantum simulation, Phys. Rev. A 103, 052435 (2021).
  20. J. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New J. Phys. 18, 023023 (2016).
  21. P. Ronagh, B. Woods, and E. Iranmanesh, Solving constrained quadratic binary problems via quantum adiabatic evolution, Quantum Inf. Comput. 16, 1029 (2016).
  22. K. Yonaga, M. J. Miyama, and M. Ohzeki, Solving inequality-constrained binary optimization problems on quantum annealer, arXiv:2012.06119.
  23. S. Karimi and P. Ronagh, A subgradient approach for constrained binary optimization via quantum adiabatic evolution, Quantum Inf. Process. 16, 1 (2017).
  24. I. Hen and F. Spedalieri, Quantum annealing for constrained optimization, Phys. Rev. Appl. 5, 034007 (2016).
  25. S. Boixo, T. Rønnow, S. Isakov, Z. Wang, D. Wecker, D. Lidar, J. Martinis, and M. Troyer, Evidence for quantum annealing with more than one hundred qubits, Nat. Phys. 10, 218 (2014).
  26. T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. McMahon, T. Umeki, K. Enbutsu et al., A coherent Ising machine for 2000-node optimization problems, Science 354, 603 (2016).
  27. M. Aramon, G. Rosenberg, E. Valiante, T. Miyazawa, H. Tamura, and H. Katzgraber, Physics-inspired optimization for quadratic unconstrained problems using a digital annealer, Front. Phys. 7, 48 (2019).
  28. H. Goto, K. Tatsumura, and A. Dixon, Combinatorial optimization by simulating adiabatic bifurcations in nonlinear Hamiltonian systems, Sci. Adv. 5, eaav2372 (2019).
  29. S. Brush, History of the Lenz-Ising model, Rev. Mod. Phys. 39, 883 (1967).
  30. F. Glover, J. Hao, and G. Kochenberger, Polynomial unconstrained binary optimisation—Part 2., Int. J. Metaheuristics 1, 317 (2011).
  31. E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum computation by adiabatic evolution, arXiv:quant-ph/0001106.
  32. A. Ambainis and O. Regev, An elementary proof of the quantum adiabatic theorem, arXiv:quant-ph/0411152.
  33. R. Jozsa and N. Linden, On the role of entanglement in quantum-computational speed-up, Proc. R. Soc. Lond. A 459, 2011 (2003).
  34. S. Ding and Z. Jin, Review on the study of entanglement in quantum computation speedup, Chin. Sci. Bull. 52, 2161 (2007).
  35. J. Biamonte and P. Love, Realizable Hamiltonians for universal adiabatic quantum computers, Phys. Rev. A 78, 012352 (2008).
  36. N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Vol. 3 (Springer Science & Business Media, Berlin, 1985).
  37. J. Bergstra and Y. Bengio, Random search for hyper-parameter optimization, J. Mach. Learn. Res. 13, 281 (2012).
  38. H. Kellerer, U. Pferschy, and D. Pisinger, Introduction to NP—Completeness of knapsack problems, in Knapsack Problems (Springer, Berlin, 2004), pp. 483–493.
  39. L. Perron and F. Didier, CP-SAT, https://developers.google.com/optimization/cp/cp_solver.
  40. M. Suchara, A. Faruque, C.-Y. Lai, G. Paz, F. T. Chong, and J. Kubiatowicz, Comparing the overhead of topological and concatenated quantum error correction, arXiv:1312.2316.
  41. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012).
  42. Google Quantum AI, Suppressing quantum errors by scaling a surface code logical qubit, Nature (London) 614, 676 (2023).
  43. Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2025).
  44. C. A. Floudas, Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications (Oxford University Press, Oxford, 1995).
  45. D. Li and X. Sun, Nonlinear Integer Programming (Springer Science & Business Media, Berlin, 2006), Vol. 84.
  46. S. Burer and A. N. Letchford, Non-convex mixed-integer nonlinear programming: A survey, Surv. Oper. Res. Manage. Sci. 17, 97 (2012).
  47. F. Khosravi, A. Scherer, and P. Ronagh, Mixed-integer programming using a bosonic quantum computer, in Proceedings of the 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, Bellevue, WA, USA, 2023), pp. 184–195.

Outline

Information

Phys. Rev. Research 7, 023305– Published 25 June, 2025
  • Received 10 January 2025
  • Accepted 23 May 2025
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