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An introduction to variational quantum algorithms for combinatorial optimization problems

Author

Listed:
  • Camille Grange

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

  • Michael Poss

    (University of Montpellier, CNRS)

  • Eric Bourreau

    (University of Montpellier, CNRS)

Abstract

Noisy intermediate-scale quantum computers are now readily available, motivating many researchers to experiment with Variational Quantum Algorithms. Among them, the Quantum Approximate Optimization Algorithm 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, 2024. "An introduction to variational quantum algorithms for combinatorial optimization problems," Annals of Operations Research, Springer, vol. 343(2), pages 847-884, December.
    • Handle: RePEc:spr:annopr:v:343:y:2024:i:2:d:10.1007_s10479-024-06253-5
    DOI: 10.1007/s10479-024-06253-5
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. Lars Hellemo & Paul I. Barton & Asgeir Tomasgard, 2018. "Decision-dependent probabilities in stochastic programs with recourse," Computational Management Science, Springer, vol. 15(3), pages 369-395, October.
    4. 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.
    5. Samson Wang & Enrico Fontana & M. Cerezo & Kunal Sharma & Akira Sone & Lukasz Cincio & Patrick J. Coles, 2021. "Noise-induced barren plateaus in variational quantum algorithms," Nature Communications, Nature, vol. 12(1), pages 1-11, December.
    6. 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.
    7. 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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