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An introduction to variational quantum algorithms for combinatorial optimization problems

Author

Listed:
  • Camille Grange

    (University of Montpellier
    SNCF, Technology, Innovation and Group Projects Department)

  • Michael Poss

    (University of Montpellier)

  • Eric Bourreau

    (University of Montpellier)

Abstract

Noisy intermediate-scale quantum computers (NISQ computers) are now readily available, motivating many researchers to experiment with Variational Quantum Algorithms (VQAs). Among them, the Quantum Approximate Optimization Algorithm (QAOA) is one of the most popular one studied by the combinatorial optimization community. In this tutorial, we provide a mathematical description of the class of Variational Quantum Algorithms, assuming no previous knowledge of quantum physics from the readers. We introduce precisely the key aspects of these hybrid algorithms on the quantum side (parametrized quantum circuit) and the classical side (guiding function, optimizer). We devote a particular attention to QAOA, detailing the quantum circuits involved in that algorithm, as well as the properties satisfied by its possible guiding functions. Finally, we discuss the recent literature on QAOA, highlighting several research trends.

Suggested Citation

  • Camille Grange & Michael Poss & Eric Bourreau, 2023. "An introduction to variational quantum algorithms for combinatorial optimization problems," 4OR, Springer, vol. 21(3), pages 363-403, September.
    • Handle: RePEc:spr:aqjoor:v:21:y:2023:i:3:d:10.1007_s10288-023-00549-1
    DOI: 10.1007/s10288-023-00549-1
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    References listed on IDEAS

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    1. Alberto Peruzzo & Jarrod McClean & Peter Shadbolt & Man-Hong Yung & Xiao-Qi Zhou & Peter J. Love & Alán Aspuru-Guzik & Jeremy L. O’Brien, 2014. "A variational eigenvalue solver on a photonic quantum processor," Nature Communications, Nature, vol. 5(1), pages 1-7, September.
    2. Fred Glover & Gary Kochenberger & Yu Du, 2019. "Quantum Bridge Analytics I: a tutorial on formulating and using QUBO models," 4OR, Springer, vol. 17(4), pages 335-371, December.
    3. Kurowski, Krzysztof & Pecyna, Tomasz & Slysz, Mateusz & Różycki, Rafał & Waligóra, Grzegorz & Wȩglarz, Jan, 2023. "Application of quantum approximate optimization algorithm to job shop scheduling problem," European Journal of Operational Research, Elsevier, vol. 310(2), pages 518-528.
    4. Jarrod R. McClean & Sergio Boixo & Vadim N. Smelyanskiy & Ryan Babbush & Hartmut Neven, 2018. "Barren plateaus in quantum neural network training landscapes," Nature Communications, Nature, vol. 9(1), pages 1-6, December.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Yves Crama & Michel Grabisch & Silvano Martello, 2024. "21 volumes for the 21st century," 4OR, Springer, vol. 22(1), pages 1-16, March.
    2. Grange, Camille & Poss, Michael & Bourreau, Eric & T’kindt, Vincent & Ploton, Olivier, 2025. "Moderate exponential-time quantum dynamic programming across the subsets for scheduling problems," European Journal of Operational Research, Elsevier, vol. 320(3), pages 516-526.
    3. Camille Grange & Michael Poss & Eric Bourreau, 2024. "An introduction to variational quantum algorithms for combinatorial optimization problems," Annals of Operations Research, Springer, vol. 343(2), pages 847-884, December.
    4. Yves Crama & Michel Grabisch & Silvano Martello, 2024. "Still more surveys in operations research…," Annals of Operations Research, Springer, vol. 343(2), pages 559-571, December.

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