Abstract
Quantum computing (QC) is a rapidly growing technology in the field of computation that has garnered significant attention in recent years. This emerging technology has become particularly relevant due to the increasing complexity of optimization problems and their expanding search spaces. As a result, innovative solutions that can surpass the limitations of the current optimization paradigms executed on classic computers are becoming necessary. D-wave, a specialized quantum computer, presents a novel solution for addressing intricate optimization problems with remarkable speed advantages over traditional methods. However, a major hurdle in terms of utilizing the D-wave platform for topology optimization design is the conversion of an optimization problem into formulas that can be comprehended by a quantum annealing machine. This is because the D-wave platform is limited to solving quadratic unconstrained binary optimization problems or Ising model problems, making it necessary to find a way to adapt the task of interest to these specific types of optimization problems. This paper examines the current reality concerning the extremely limited availability of quantum computing resources. We focus on small-scale discrete structural topology optimization problems as a starting point and establish a mapping relationship between quantum bits and the cross-sectional area variables of truss elements. Utilizing this mapping, a quadratic unconstrained binary optimization model is developed with these variables. We propose a nested optimization process with dynamically adjusted cross-sectional areas, which enables the development of a quantum annealing approach for optimizing the topology of discrete variables. Our method is validated through numerical experiments, demonstrating its efficiency.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Airbus (2019) Airbus quantum computing challenge: bringing flight physics into the quantum era. On-line, Jan 2019. Airbus Quantum Computing Challenge | Airbus. Accessed 06 Nov 2019
Alberto P, McClean J, Shadbolt P, MH Yung, Zhou XQ, Love PJ, Aspuru-Guzik A, O'brien JL (2014) A variational eigenvalue solver on a photonic quantum processor. Nat Commun 5:4213. https://doi.org/10.1038/ncomms5213
Apolloni B, Cesa-Bianchi N, De Falco D (1998) Quantum tunneling in stochastic and combinatorial optimization. Parallel Architect Neural Netw 27–29:1–13. https://doi.org/10.1016/0304-4149(89)90040-9
Arute F, Arya K, Babbush R, Bacon D, Bardin JC, Barends R, Biswas R, Boixo S, Brandao FGSL, Buell DA, Burkett B, Chen Y, Chen Z, Chiaro B, Collins R, Courtney W, Dunsworth A, Farhi E, Foxen B, Fowler A, Gidney C, Giustina M, Graff R, Guerin K, Habegger S, Harrigan MP, Hartmann MJ, Ho A, Hoffmann M, Huang T, Humble TS, Isakov SV, Jeffrey E, Jiang Z, Kafri D, Kechedzhi K, Kelly J, Klimov PV, Knysh S, Korotkov A, Kostritsa F, Landhuis D, Lindmark M, Lucero E, Lyakh D, Mandrà S, McClean JR, McEwen M, Megrant A, Mi X, Michielsen K, Mohseni M, Mutus J, Naaman O, Neeley M, Neill C, Niu MY, Ostby E, Petukhov A, Platt JC, Quintana C, Rieffel EG, Roushan P, Rubin NC, Sank D, Satzinger KJ, Smelyanskiy V, Sung KJ, Trevithick MD, Vainsencher A, Villalonga B, White T, Yao ZJ, Yeh P, Zalcman A, Neven H, Martinis JM (2019) Quantum supremacy using a programmable superconducting processor. Nature 574:505–510. https://doi.org/10.1038/s41586-019-1666-5
Bauer J, Gutkowski W, Iwanow Z (1981) Discrete method for lattice structures optimization. Eng Optim 5(2):121–127
Bendsøe M, Kikuchi N (1998) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bollapragada S, Ghattas O, Hooker J (2001) Optimal design of truss structures by logical-based branch and cut. Oper Res 49(1):42–51
Carugno C, Ferrari Dacrema M, Cremonesi P (2022) Evaluating the job shop scheduling problem on a D-wave quantum annealer. Sci Rep 12:6539. https://doi.org/10.1038/s41598-022-10169-0
Chai S, Shi L, Sun H (1999) An application of relative difference quotient algorithm to topology optimization of truss structures with discrete variables. Struct Optim 18(1):48–55
Chapman CD, Jakiela MJ (1996) Genetic algorithm-based structural topology design with compliance and topology simplification considerations. J Mech Des 118(1):89–98. https://doi.org/10.1115/1.2826862
Cho A (2018) DOE pushes for useful quantum computing. Science 359(6372):141–142. https://doi.org/10.1126/science.359.6372.141
Desai SB, Madhvapathy SR, Sachid AB, Llinas JP, Wang Q, Ahn GH, Pitner G, Kim MJ, Bokor J, Hu C, Wong HS (2016) MoS2 transistors with 1-nanometer gate lengths. Science 354:96–99. https://doi.org/10.1126/science.aah4698
D-Wave Systems Inc (2019). D-wave systems Inc. repositories. https://github.com/dwavesystems. Accessed 08 Nov 2019
D-Wave Systems Inc. (2020) D-wave solver properties and parameters reference. https://docs.dwavesys.com/docs/latest/doc_solver_ref.html
Dyakonov M (2018) The case against quantum computing. the case against quantum computing-IEEE Spectrum
Edward F, Jeffrey G, Sam G (2014) A quantum approximate optimization algorithm. https://doi.org/10.48550/arXiv.1411.4028
Elperin T (1988) Monte Carlo structural optimization in discrete variables with annealing algorithm. Int J Numer Methods Eng 26:815821
Gibney Elizabeth (2014) Physics: quantum computer quest. Nature 516(7529):24–6. https://doi.org/10.1038/516024a
Gutkowski W, Bauer J, Zawidzka J (2000) An effective method fordiscrete structural optimization. Eng Comput 17(4):417–426
Hajela P, Lee E (1995) Genetic algorithms in truss topological optimization. Int J Solids Struct 32(22):3341–335. https://doi.org/10.1016/0020-7683(94)00306-H
Harrow AW, Hassidim A, Lloyd S (2009) Quantum algorithm for solving linear systems of equations. Phys Rev Lett. https://doi.org/10.1103/PhysRevLett.103.150502
Huang X, Xie YM (2007) Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method. Finite Elem Anal Des 43(14):1039–1049. https://doi.org/10.1016/j.finel.2007.06.006
Ikeda K, Nakamura Y, Humble TS (2019) Application of quantum annealing to nurse scheduling problem. Sci Rep 9:12837. https://doi.org/10.1038/s41598-019-49172-3
Ising E (1925) Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik 31(1):253–258. https://doi.org/10.1007/BF02980577
Jenkins W (2002) A decimal-coded evolutionary algorithm for constrained optimization. Comput Struct 80(5–6):471–480
Kaveh A, Talatahari S (2009) A particle swarm ant colony optimization for truss structures with discrete variables. J Constr Steel Res 65(8–9):1558–1568
Kirkpatrick S, Gelatt CD Jr, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680. https://doi.org/10.1126/science.220.4598.671
Kitai K, Guo J, Ju S, Tanaka S, Tsuda K, Shiomi J, Tamura R (2020) Designing metamaterials with quantum annealing and factorization machines. Phys Rev Res 2(1):013319. https://doi.org/10.1103/PhysRevResearch.2.013319
Liang Y, Sun K, Cheng G (2020) Discrete variable topology optimization for compliant mechanism design via sequential approximate integer programming with trust region (SAIP-TR). Struct Multidisc Optim 62:2851–2879. https://doi.org/10.1007/s00158-020-02693-2
Metropolis N, Rosenbluth AW, Rosenbluth MN (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092. https://doi.org/10.1063/1.1699114
Morita S, Nishimori H (2008) Mathematical foundation of quantum annealing. J Math Phys 49(12):125210. https://doi.org/10.1063/1.2995837
Mu L, Wang P, Xin G (2020) Quantum-inspired algorithm with fitness landscape approximation in reduced dimensional spaces for numerical function optimization. Inform Sci 527:253–278. https://doi.org/10.1016/j.ins.2020.03.035
Munoz E, Stolpe M (2011) Generalized Benders’ decomposition for topology optimization problems. J Glob Optim 51(1):149–183
Neukart F, Compostella G, Seidel C, Von Dollen D, Yarkoni S, Parney B (2017) Traffic flow optimization using a quantum annealer. Front ICT 4:29. https://doi.org/10.3389/fict.2017.00029
Ohzeki M, Takahashi C, Okada S, Terabe M, Taguchi S, Tanaka K (2018) Quantum annealing: next-generation computation and how to implement it when information is missing. Nonlinear Theory Appl 9(4):392–405. https://doi.org/10.1587/nolta.9.392
Querin OM, Steven GP, Xie YM (2000) Evolutionary structural optimization using an additive algorithm. Finite Elem Anal Des 34(3–4):291–308. https://doi.org/10.1016/S0168-874X(99)00044-X
Ringertz U (1988) On methods for discrete structural optimization. Eng Optim 13:44–64
Ruiter MJD, Keulen FV (2004) Topology optimization using a topology description function. Struct Multidisc Optim 26(6):406–416. https://doi.org/10.1007/s00158-003-0375-7
Sato Y, Kondo R, Koide S, Kajita S (2022) Quantum topology optimization of ground structures using noisy intermediate-scale quantum devices. https://doi.org/10.48550/arXiv.2207.09181
Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidisc Optim 21(2):120–127. https://doi.org/10.1007/s001580050176
Sivapuram R, Picelli R (2018) Topology optimization of binary structures using Integer Linear Programming. Finite Elem Anal Des 139:49–61. https://doi.org/10.1016/j.finel.2017.10.006
Stolpe M (2007) On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs. Optim Eng 8:163–192
Stolpe M (2015) Truss topology optimization with discrete design variables by outer approximation. J Glob Optim 61(1):139–163
Stolpe M (2016) Truss optimization with discrete design variables: a critical review. Struct Multidisc Optim 53:349–374. https://doi.org/10.1007/s00158-015-1333-x
Toakley A (1968) Optimum design using available sections. American society of civil engineers proceedings. J Struct Div 94:1219–1241
Van Vreumingen D, Neukart F, Von Dollen D, Othmer C, Hartmann M, Voigt A-C, Bäck T (2019) Quantum-assisted finite-element design optimization. https://doi.org/10.48550/arXiv.1908.03947
Volkswagen (2019) Volkswagen optimizes traffic flow with quantum computers. Online, Oct. 2019. Volkswagen optimizes traffic flow with quantum computers | Volkswagen Newsroom (volkswagen-newsroom.com). Accessed 06 Nov 2019
Wang MY, Wang XM (2004) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 35(7):415–441. https://doi.org/10.1016/S0045-7825(02)00559-5
Wang L, Niu Q, Fei M (2007) A novel quantum ant colony optimization algorithm, bio-inspired computational intelligence and applications: international conference on life system modeling and simulation. LSMS 2007. Lecture notes in computer science, vol 4688. Springer, Berlin. https://doi.org/10.1007/978-3-540-74769-7_31
Wils K, Chen B (2023) A symbolic approach to discrete structural optimization using quantum annealing. Mathematics 11:3451. https://doi.org/10.3390/math11163451
Yarkoni S, Raponi E, Bäck T, Schmitt S (2022) Quantum annealing for industry applications: introduction and review. Rep Prog Phys. https://doi.org/10.1088/1361-6633/ac8c54
Zhang ZQ et al (2011) Two-level optimization method of transmission tower structure based on ant colony algorithm. Adv Mat Res 243–249:5849–5853
Zhang Y, Hou Y, Liu S (2014) A new method of discrete optimization for cross-section selection of truss structures. Eng Optim 46(8):1052–1073
Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model. Struct Multidisc Optim 53:1243–1260. https://doi.org/10.1007/s00158-015-1372-3
Zisheng Ye, Xiaoping Qian, Wenxiao Pan (2023) Quantum topology optimization via quantum annealing. IEEE Trans Quantum Eng 4:1–15. https://doi.org/10.1109/TQE.2023.3266410
Acknowledgements
The authors extend their heartfelt thanks to the editors and reviewers for their invaluable insights and guidance, which have significantly improved both the rigor and coherence of this work. We are truly grateful for their dedicated efforts throughout this process.
Funding
Funding was provided by National Natural Science Foundation of China (Grant No. 12072006), National Natural Science Foundation of China (Grant No. 12132001) and National Nature Science Foundation of China (Grant No. 52192632).
Author information
Authors and Affiliations
Contributions
Wang Xiaojun was instrumental in conceptualizing the overall framework and methodology of the article, in addition to reviewing and revising the manuscript. Wang Zhenghuan and Ni Bowen jointly contributed by writing the main content of the article, designing the methods, and conducting the data analysis.
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all the authors, the corresponding author states that there are no conflicts of interest.
Replication of the results
The Python codes generated in the current study are available from the corresponding author upon request.
Additional information
Responsible Editor: Josephine Carstensen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Wang, X., Wang, Z. & Ni, B. Mapping structural topology optimization problems to quantum annealing. Struct Multidisc Optim 67, 74 (2024). https://doi.org/10.1007/s00158-024-03791-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00158-024-03791-1