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  • Review Article
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Challenges and opportunities in quantum optimization

Nature Reviews Physics volume 6, pages 718–735 (2024)Cite this article

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Abstract

Quantum computers have demonstrable ability to solve problems at a scale beyond brute-force classical simulation. Interest in quantum algorithms has developed in many areas, particularly in relation to mathematical optimization — a broad field with links to computer science and physics. In this Review, we aim to give an overview of quantum optimization. Provably exact, provably approximate and heuristic settings are first explained using computational complexity theory, and we highlight where quantum advantage is possible in each context. Then, we outline the core building blocks for quantum optimization algorithms, define prominent problem classes and identify key open questions that should be addressed to advance the field. We underscore the importance of benchmarking by proposing clear metrics alongside suitable optimization problems, for appropriate comparisons with classical optimization techniques, and discuss next steps to accelerate progress towards quantum advantage in optimization.

Key points

  • Quantum computing is advancing rapidly, and quantum optimization is a promising area of application.

  • Quantum optimization algorithms — whether provably exact, provably approximate or heuristic — offer opportunities to demonstrate quantum advantage.

  • Systematic benchmarking is crucial to guide research, track progress and further advance understanding of quantum optimization.

  • Theoretical research and empirical research using real hardware can complement each other, in the move towards demonstrating quantum advantage.

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Fig. 1: Complexity classes for optimization problems.
Fig. 2: The optimization-platform ecosystem.

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Acknowledgements

The authors thank J. Eisert and M. Wilde for their valuable feedback and suggestions to further improve this paper. A. Ambainis acknowledges the support of the Latvian Quantum Initiative under European Union Recovery and Resilience Facility Project No. 2.3.1.1.i.0/1/22/I/CFLA/001 and the QuantERA II ERA-NET Cofund projects QOPT and HQCC. B.A. and S. Gupta acknowledge support from the US Defense Advanced Research Projects Agency (DARPA) Contract No. HR001120C0046. V.D. acknowledges the support of the Dutch Research Council (NWO/OCW), as part of the Quantum Software Consortium programme (Project No. 024.003.037) and of the Dutch National Growth Fund (NGF), as part of the Quantum Delta NL programme. S.H. was supported by NASA Academic Mission Services Contract No. NNA16BD14C and by DARPA under interagency agreement IAA 8839, Annex 114. T. Koch acknowledges support by the BMBF Research Campus MODAL (05M14ZAM and 05M20ZBM). S.L. was funded by Free State of Thuringia (Thüringer Aufbaubank) through project Quantum Hub Thüringen (2021 FGI 0047), by Bundesministerium für Wirtschaft und Energie, Germany through the project ‘EnerQuant’ (Project No. 03EI1025C) and by the European Union under Horizon Europe Programme. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or European Climate, Infrastructure and Environment Executive Agency (CINEA). Neither the European Union nor the granting authority can be held responsible for them, Grant Agreement 101080086 — NeQST. J.M. acknowledges the support of the Czech Science Foundation (23-07947S). S.M., E.S. and B.C.B.S. were supported by the Hartree National Centre for Digital Innovation, a UK Government-funded collaboration between STFC and IBM. (IBM, the IBM logo and ibm.com are trademarks of International Business Machines Corp., registered in many jurisdictions worldwide. Other product and service names might be trademarks of IBM or other companies. The current list of IBM trademarks is available at https://www.ibm.com/legal/copytrade.) G.N. acknowledges support by ONR (N000142312585). P.R. acknowledges support by the National Research Foundation, Singapore. J.Y. was supported in part by the NSERC Discovery under Grant No. RGPIN-2018-04742, the NSERC project FoQaCiA under Grant No. ALLRP-569582-21 and the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science (Los Alamos unlimited release LA-UR-24-22124).

Author information

Authors and Affiliations

  1. Institute of Physics, University of Amsterdam, Amsterdam, The Netherlands

    Amira Abbas & Harry Buhrman

  2. QuSoft, Amsterdam, The Netherlands

    Amira Abbas & Harry Buhrman

  3. Faculty of Computing, University of Latvia, Riga, Latvia

    Andris Ambainis

  4. Sloan School of Management, Massachussets Institute of Technology, Cambridge, MA, USA

    Brandon Augustino & Swati Gupta

  5. CCS-3 Information Sciences, Los Alamos National Laboratory, Los Alamos, NM, USA

    Andreas Bärtschi

  6. Centrum Wiskunde & Informatica (CWI), University of Amsterdam, Amsterdam, The Netherlands

    Harry Buhrman

  7. Advanced Network Science Initiative, Los Alamos National Laboratory, Los Alamos, NM, USA

    Carleton Coffrin

  8. E.ON Digital Technology GmbH, Hannover, Germany

    Giorgio Cortiana, Naeimeh Mohseni & Corey O’Meara

  9. Applied Quantum Algorithms Initiative, Leiden University, Leiden, The Netherlands

    Vedran Dunjko

  10. IBM Quantum, IBM Research Europe — Zurich, Rueschlikon, Switzerland

    Daniel J. Egger, Julien Gacon, Elena Peña Tapia, Manuel Proissl, Stefan Woerner, Mira L. Wolf-Bauwens & Christa Zoufal

  11. IBM Research, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    Bruce G. Elmegreen

  12. Quantum-Enhanced AI Department, Fraunhofer Institute for Cognitive Systems IKS, Munich, Germany

    Nicola Franco

  13. Erste Digital GmbH, Erste Group Bank AG, Vienna, Austria

    Filippo Fratini & Gerhard Kircher

  14. IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    Bryce Fuller

  15. Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

    Julien Gacon

  16. Advanced Technology, Wells Fargo, New York City, NY, USA

    Constantin Gonciulea & Vanio Markov

  17. Department of Econometrics and Operations Research, Tilburg University, Tilburg, The Netherlands

    Sander Gribling

  18. Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center, Moffett Field, CA, USA

    Stuart Hadfield

  19. USRA Research Institute for Advanced Computer Science (RIACS), Universities Space Research Association, Mountain View, CA, USA

    Stuart Hadfield

  20. Optimization Department, Fraunhofer ITWM, Kaiserslautern, Germany

    Raoul Heese

  21. Quantagonia GmbH, Bad Homburg, Germany

    Thomas Kleinert & Dirk Zechiel

  22. Applied Algorithmic Intelligence Methods, Zuse Institute Berlin, Berlin, Germany

    Thorsten Koch

  23. Software and Algorithms for Discrete Optimization, Technische Universität Berlin, Berlin, Germany

    Thorsten Koch

  24. Quantum Technologies Group, Emerging Technologies, Innovation and Ventures, HSBC, Singapore, Singapore

    Georgios Korpas

  25. Department of Computer Science, Czech Technical University in Prague, Prague, Czech Republic

    Georgios Korpas & Jakub Marecek

  26. Archimedes Research Unit on AI, Data Science and Algorithms, Athena Research Center, Marousi, Greece

    Georgios Korpas

  27. Cognitive Energy Systems Department, Applied Systems Technology Division (AST), Fraunhofer Institute of Optronics, System Technologies and Image Exploitation IOSB, Ilmenau, Germany

    Steve Lenk

  28. Institute of Quantum Technologies, German Aerospace Centre (DLR), Ulm, Germany

    Steve Lenk

  29. Institute for Computational Science, University of Zurich, Zurich, Switzerland

    Guglielmo Mazzola

  30. The Hartree Centre, STFC, Sci-Tech Daresbury, Daresbury, UK

    Stefano Mensa, Emre Sahin & Benjamin C. B. Symons

  31. Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, CA, USA

    Giacomo Nannicini

  32. Zuse Institute Berlin, Berlin, Germany

    Sebastian Pokutta

  33. Institute for Mathematics, Technische Universität Berlin, Berlin, Germany

    Sebastian Pokutta

  34. Centre for Quantum Technologies, National University of Singapore, Singapore, Singapore

    Patrick Rebentrost

  35. Research Institute CODE, University of the Bundeswehr Munich, Munich, Germany

    Sabine Tornow

  36. IBM Quantum, IBM Research Europe — Dublin, Dublin, Ireland

    Víctor Valls & Sergiy Zhuk

  37. Institute for Quantum Computing, University of Waterloo, Waterloo, Canada

    Jon Yard

  38. Volkswagen Data:Lab, Volkswagen AG, Munich, Germany

    Sheir Yarkoni

Authors
  1. Amira Abbas
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  2. Andris Ambainis
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  3. Brandon Augustino
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  4. Andreas Bärtschi
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  5. Harry Buhrman
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  6. Carleton Coffrin
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  7. Giorgio Cortiana
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  8. Vedran Dunjko
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  9. Daniel J. Egger
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  10. Bruce G. Elmegreen
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  11. Nicola Franco
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  12. Filippo Fratini
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  13. Bryce Fuller
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  14. Julien Gacon
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  15. Constantin Gonciulea
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  16. Sander Gribling
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  17. Swati Gupta
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  18. Stuart Hadfield
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  19. Raoul Heese
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  20. Gerhard Kircher
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  21. Thomas Kleinert
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  22. Thorsten Koch
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  23. Georgios Korpas
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  24. Steve Lenk
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  25. Jakub Marecek
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  26. Vanio Markov
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  27. Guglielmo Mazzola
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  28. Stefano Mensa
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  29. Naeimeh Mohseni
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  30. Giacomo Nannicini
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  31. Corey O’Meara
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  32. Elena Peña Tapia
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  33. Sebastian Pokutta
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  34. Manuel Proissl
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  35. Patrick Rebentrost
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  36. Emre Sahin
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  37. Benjamin C. B. Symons
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  38. Sabine Tornow
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  39. Víctor Valls
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  40. Stefan Woerner
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  41. Mira L. Wolf-Bauwens
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  42. Jon Yard
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  43. Sheir Yarkoni
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  44. Dirk Zechiel
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  45. Sergiy Zhuk
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  46. Christa Zoufal
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Contributions

This review was written as part of the Quantum Optimization Working Group initiated in July 2023 by IBM Quantum and its partners. The authors are listed alphabetically. All authors contributed to the discussions as part of the working group meetings and to writing the manuscript and Supplementary information. The discussions on and writing of the different sections were led by the following co-authors: A.A. and J.M. coordinated the writing of the section ‘Quantum advantage and complexity theory’ (in main text and Supplementary information). J.M. and S.W. oversaw Box 1 (and the corresponding section in Supplementary information) and the section ‘Problem classes and algorithms’ (in main text and Supplementary information). T. Koch and C.Z. coordinated sections ‘Benchmarks’ (in main text and Supplementary information), ‘Benchmarking problems’ (in Supplementary information) and ‘Demonstrations’ (in Supplementary information). S.W. led the writing of the section ‘The path towards quantum advantage’. D.J.E. and S.M. were in charge of the section ‘Execution on noisy hardware at scale’ (in Supplementary information). C.O’M., M.P., V.V. and C.Z. oversaw the section ‘Illustrative industry problems’. S.W. coordinated the overall project.

Corresponding author

Correspondence to Stefan Woerner.

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Nature Review Physics thanks James Ostrowski, Leo Zhou and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Abbas, A., Ambainis, A., Augustino, B. et al. Challenges and opportunities in quantum optimization. Nat Rev Phys 6, 718–735 (2024). https://doi.org/10.1038/s42254-024-00770-9

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