Quantum Branch-and-Bound Algorithm and its Application to the Travelling Salesman Problem
We propose a quantum branch-and-bound algorithm based on the general scheme of the branch-and-bound method and the quantum nested searching algorithm and examine its computational efficiency. We also compare this algorithm with a similar classical algorithm on the example of the travelling salesman...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 241; no. 2; pp. 168 - 184 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
09.08.2019
Springer Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 1072-3374 1573-8795 |
| DOI | 10.1007/s10958-019-04415-6 |
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| Abstract | We propose a quantum branch-and-bound algorithm based on the general scheme of the branch-and-bound method and the quantum nested searching algorithm and examine its computational efficiency. We also compare this algorithm with a similar classical algorithm on the example of the travelling salesman problem. We show that in the vast majority of problems, the classical algorithm is quicker than the quantum algorithm due to greater adaptability. However, the operation time of the quantum algorithm is constant for all problem, whereas the classical algorithm runs very slowly for certain problems. In the worst case, the quantum branch-and-bound algorithm is proved to be several times more efficient than the classical algorithm. |
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| AbstractList | We propose a quantum branch-and-bound algorithm based on the general scheme of the branch-and-bound method and the quantum nested searching algorithm and examine its computational efficiency. We also compare this algorithm with a similar classical algorithm on the example of the travelling salesman problem. We show that in the vast majority of problems, the classical algorithm is quicker than the quantum algorithm due to greater adaptability. However, the operation time of the quantum algorithm is constant for all problem, whereas the classical algorithm runs very slowly for certain problems. In the worst case, the quantum branch-and-bound algorithm is proved to be several times more efficient than the classical algorithm. We propose a quantum branch-and-bound algorithm based on the general scheme of the branch-and-bound method and the quantum nested searching algorithm and examine its computational efficiency. We also compare this algorithm with a similar classical algorithm on the example of the travelling salesman problem. We show that in the vast majority of problems, the classical algorithm is quicker than the quantum algorithm due to greater adaptability. However, the operation time of the quantum algorithm is constant for all problem, whereas the classical algorithm runs very slowly for certain problems. In the worst case, the quantum branch-and-bound algorithm is proved to be several times more efficient than the classical algorithm. Keywords and phrases: quantum computing, quantum computer, quantum search, Grover's algorithm, branch-and-bound method, travelling salesman problem. AMS Subject Classification: 81P68, 68Q12 |
| Audience | Academic |
| Author | Trushechkin, A. S. Markevich, E. A. |
| Author_xml | – sequence: 1 givenname: E. A. surname: Markevich fullname: Markevich, E. A. email: eva-markevich@mail.ru organization: National Research Nuclear University MEPhI, Steklov Mathematical Insitute of the Russian Academy of Sciences, National University of Science and Technology MISiS – sequence: 2 givenname: A. S. surname: Trushechkin fullname: Trushechkin, A. S. organization: National Research Nuclear University MEPhI |
| BackLink | https://www.osti.gov/biblio/22921158$$D View this record in Osti.gov |
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| Keywords | quantum computer quantum search 68Q12 81P68 quantum computing travelling salesman problem Grover’s algorithm branch-and-bound method |
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Yonge-Mallo, “Quantum algorithms for evaluating min-max trees,” in: Theory of Quantum Computation, Communication, and Cryptography, Lect. Notes Comput. Sci., 5106, (2008), pp. 11–15. – reference: BennettCBernsteinEBrassardGVaziraniUStrengths and weaknesses of quantum computingSIAM J. Comput.199726515101523147199110.1137/S00975397963009330895.68044 – reference: BillingsleyPProbability and Measure1995New YorkWiley0822.60002 – reference: KowadaLABLavorCPortugalRde FigueiredoCMHA new quantum algorithm for solving the minimum searching problemInt. J. Quantum Inform.20086342743610.1142/S021974990800361X1192.81085 – reference: DürrCHeiligmanMHøyerPMhallaMQuantum query complexity of some graph problemsSIAM J. Comput.200635613101328221714810.1137/0506447191101.81024 – reference: KestenHLeeSThe central limit theorem for weighted minimal spanning trees on random pointsAnn. Probab.199662495527139805510.1214/aoap/10349681410862.60008 – reference: W. Zhang, State-Space Search: Algorithms, Complexity, Extensions, and Applications, Springer-Verlag (1999). – reference: WagnerHPrinciples of Operations Research1969Englewood Cliffs, New JerseyPrentice Hall0193.18402 – reference: FarhiEGoldstoneJGutmannSA quantum algorithm for the Hamiltonian NAND treeTheory Comput.20084169190248023710.4086/toc.2008.v004a0081213.68284 – reference: B. W. Reichardt, “Reflections for quantum query algorithms,” in: Proc. 22 ACM-SIAM Symp. on Discrete Algorithms (2011), pp. 560–569; arxiv.org/abs/1005.1601 – reference: CerfNJGroverLWilliamsCPNested quantum search and structured problemsPhys. Rev. A200061303230310.1103/PhysRevA.61.032303 – reference: HøyerPNeerbekJShiYQuantum complexities of ordered searching, sorting, and element distinctnessAlgorithmica2002344429448194351610.1007/s00453-002-0976-31012.68071 – reference: AmbainisAQuantum search algorithmSIGACT News2004352223510.1145/992287.992296 – reference: SteeleMGrowth rates of Euclidean minimal spanning trees with power weighted edgesAnn. Probab.19881641767178795821510.1214/aop/11769915960655.60023 – reference: BrassardGHøyerPMoscaMTappAQuantum amplitude amplification and estimationQuantum Comput. Quantum Inform. Sci., Contemp. Math. Ser.2002305537419473321063.81024 – reference: MandràSGuerreschiGGAspuru-GuzikAFaster than classical quantum algorithm for dense formulas of exact satisfiability and occupation problemsNew J. Phys.201618707300310.1088/1367-2630/18/7/073003 – reference: E. Farhi, J. Goldstone, S. Gutmann, M. Sipser, “Invariant quantum algorithms for insertion into an ordered list,” arxiv.org/abs/quant-ph/9901059 – reference: L. K. Grover, “A fast quantum mechanical algorithm for database search,” in: Proc. 28 Ann. Symp. on the Theory of Computing, ACM Press, New York (1996), pp. 212–219. – reference: T. G. Crainic, B. Le Cun, and C. Roucairol, “Parallel branch-and-bound algorithms,” in: Parallel Combinatorica and Optimization, Wiley, New York (2006), pp. 1–28. – reference: GentIPWalshTThe TSP phase transitionArtificial Intelligence1996881349358142642510.1016/S0004-3702(96)00030-60907.68177 – reference: A. Ahuja, S. Kapoor, “A quantum algorithm for finding the maximum,” e-print arxiv.org/abs/quant-ph/9911082 – volume: 16 start-page: 1767 issue: 4 year: 1988 ident: 4415_CR23 publication-title: Ann. Probab. doi: 10.1214/aop/1176991596 – volume: 18 start-page: 073003 issue: 7 year: 2016 ident: 4415_CR21 publication-title: New J. 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A doi: 10.1103/PhysRevA.75.032335 – ident: 4415_CR26 – volume: 34 start-page: 429 issue: 4 year: 2002 ident: 4415_CR18 publication-title: Algorithmica doi: 10.1007/s00453-002-0976-3 – volume: 4 start-page: 169 year: 2008 ident: 4415_CR14 publication-title: Theory Comput. doi: 10.4086/toc.2008.v004a008 – volume: 6 start-page: 427 issue: 3 year: 2008 ident: 4415_CR20 publication-title: Int. J. Quantum Inform. doi: 10.1142/S021974990800361X – volume: 113 start-page: 210501 year: 2014 ident: 4415_CR25 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.113.210501 – ident: 4415_CR15 – ident: 4415_CR17 – volume: 35 start-page: 22 issue: 2 year: 2004 ident: 4415_CR2 publication-title: SIGACT News doi: 10.1145/992287.992296 – ident: 4415_CR13 – volume: 305 start-page: 53 year: 2002 ident: 4415_CR5 publication-title: Quantum Comput. Quantum Inform. Sci., Contemp. Math. Ser. – ident: 4415_CR11 – volume: 6 start-page: 495 issue: 2 year: 1996 ident: 4415_CR19 publication-title: Ann. Probab. doi: 10.1214/aoap/1034968141 – volume: 61 start-page: 032303 issue: 3 year: 2000 ident: 4415_CR6 publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.61.032303 – ident: 4415_CR9 |
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| SubjectTerms | ALGORITHMS COMPARATIVE EVALUATIONS EFFICIENCY Management science MATHEMATICAL METHODS AND COMPUTING Mathematics Mathematics and Statistics Production scheduling QUANTUM COMPUTERS Search algorithms Traveling salesman problem |
| Title | Quantum Branch-and-Bound Algorithm and its Application to the Travelling Salesman Problem |
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