Advertisement

The travelling salesman problem and adiabatic quantum computation: an algorithm

  • Tien D. KieuEmail author
Article
  • 29 Downloads

Abstract

An explicit algorithm for the travelling salesman problem is constructed in the framework of adiabatic quantum computation, AQC. The initial Hamiltonian for the AQC process admits canonical coherent states as the ground state, and the target Hamiltonian has the shortest tour as the desirable ground state. Some estimates/bounds are also given for the computational complexity of the algorithm with particular emphasis on the required energy resources, besides the space and time complexity, for the physical process of (quantum) computation in general.

Keywords

Travelling salesman problem Quantum algorithms Adiabatic quantum computation 

Notes

References

  1. 1.
    Lawler, E., Lenstra, J., RinooyKan, A., Shmoys, D.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, vol. 3. Wiley, New York (1985)Google Scholar
  2. 2.
    Applegate, D., Bixby, R., Chvatal, V., Cook, W.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2011)zbMATHGoogle Scholar
  3. 3.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information, 2nd edn. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  4. 4.
    Albash, T., Lidar, D.: Adiabatic quantum computing. arXiv:1611.04471 [quant-ph] (2016)
  5. 5.
    Moser, R.: The quantum mechanical solution of the traveling salesman problem. Physica E 16, 280–285 (2003)ADSCrossRefGoogle Scholar
  6. 6.
    Talbi, H., Draa, A., Batouche, M.: A new quantum-inspired genetic algorithm for solving the travelling salesman problem. In: 2004 IEEE International Conference on Industrial Technology. IEEE ICIT’04 (2004)Google Scholar
  7. 7.
    Martonak, R., Santoro, G.E., Tosatti, E.: Quantum annealing of the traveling salesman problem. arXiv:cond-mat/0402330 [cond-mat.dis-nn] (2004)
  8. 8.
    Moylett, D.J., Linden, N., Montanaro, A.: Quantum speedup of the travelling salesman problem for bounded-degree graphs. arXiv:1612.06203 [quant-ph] (2017)
  9. 9.
    Heim, B., Brown, E.W., Wecker, D., Troyer, M.: Designing adiabatic quantum optimization: a case study for the traveling salesman problem. arXiv:1702.06248 [quant-ph] (2017)
  10. 10.
    Goswami, D., Karnick, H., Jain, P., Maji, H.K.: Towards efficiently solving quantum traveling salesman problem. arXiv:quant-ph/0411013 (2004)
  11. 11.
    Kieu, T.D.: Quantum adiabatic computation and the travelling salesman problem. arXiv:quant-ph/0601151 (2006)
  12. 12.
    Kieu, T.D.: A factorisation algorithm in adiabatic quantum computation. arXiv:1808.02781 [quant-ph] (2018)
  13. 13.
    Lokshtanov, D., Narayanaswamy, N., Raman, V., Ramnujan, M., Saurabh, S.: Faster parameterized algorithms using linear programming. arXiv:1203.0833 [cs.DS] (2012)
  14. 14.
    Kieu, T.D.: A new class of time-energy uncertainty relations for time-dependent Hamiltonians. arXiv:1702.00603 [quant-ph] (2017)
  15. 15.
    Messiah, A.: Quantum Mechanics. Wiley, New York (1966)zbMATHGoogle Scholar
  16. 16.
    Das, S., Kobes, R., Kunstatter, G., Zaraket, H.: Energy and efficiency of adiabatic quantum search algorithms.arXiv:quant-ph/0204044 (2002)
  17. 17.
    Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002)ADSCrossRefGoogle Scholar
  18. 18.
    Wei, Z., Ying, M.: Quantum search algorithm by adiabatic evolution under a priori probability. arXiv:quant-ph/0412117 (2004)
  19. 19.
    Grover, L.: In: Proceedings, 28th Annual ACM Symposium on the Theory of Computing, p. 212. (May 1996)Google Scholar
  20. 20.
    Bae, J., Kwon, Y.: Generalized quantum search hamiltonian. arXiv:quant-ph/0110020 (2002)
  21. 21.
    Bae, J., Kwon, Y.: Maximum speedup in quantum search: \({\cal{O}}(1)\) running time. arXiv:quant-ph/0204087 (2003)
  22. 22.
    Bae, J., Kwon, Y.: Speedup in quantum adiabatic evolution algorithm. arXiv:quant-ph/0205048 (2002)
  23. 23.
    Farhi, E., Goldstone, J., Gutmann, S.: Quantum adiabatic evolution algorithms with different paths. arXiv:quant-ph/0208135 (2002)
  24. 24.
    Eryigit, R., Gunduc, Y., Eryigit, R.: Local adiabatic quantum search with different paths. arXiv:quant-ph/0309201 (2004)

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Centre for Quantum and Optical ScienceSwinburne University of TechnologyHawthornAustralia

Personalised recommendations