Skip to main content

Advertisement

SpringerLink
Log in

A quantum genetic algorithm for optimization problems on the Bloch sphere

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Optimization problems on the surface of a unit sphere are addressed using a quantum genetic algorithm. That a point on the surface of the Bloch sphere is representative of a pure state qubit is effectively used. Qubits are thought of as genes, and a sequence of qubits as a chromosome, and an ensemble of chromosomes as the population. The crossover and mutation of the genes are implemented using the superposition principle, and mutation is achieved through random phases in the superposition. As illustrations, examples pertaining to the Thomson optimization problem, the logarithmic Thomson optimization problem, and the evaluation of the geometric measure of entanglement are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Nocedal, J., Wright, S.: Numerical Optimization. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Bonnans, J.-F., Gilbert, J.C., Lemaréchal, C., Sagastizábal, C.A.: Numerical Optimization: Theoretical and Practical Aspects. Springer, Berlin (2006)

    MATH  Google Scholar 

  3. Fraser, A.S.: Simulation of genetic systems by automatic digital computers II. Effects of linkage on rates of advance under selection. Aust. J. Biol. Sci. 10(4), 492–500 (1957)

    Article  Google Scholar 

  4. Bremermann, H.J.: The Evolution of Intelligence: The Nervous System as a Model of Its Environment. Department of Mathematics, University of Washington, Washington (1958)

    Google Scholar 

  5. Holland, J.H.: Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. MIT Press, Cambridge (1992)

    Book  Google Scholar 

  6. Gutiérrez, J.A.G., Cotta, C., Fernández-Leiva, A.J.: Evolutionary computation in astronomy and astrophysics: a review (2012). arXiv preprint arXiv:1202.2523

  7. Honda, M.: Application of genetic algorithms to modelings of fusion plasma physics. Comput. Phys. Commun. 231, 94–106 (2018)

    Article  ADS  Google Scholar 

  8. Höschel, K., Lakshminarayanan, V.: Genetic algorithms for lens design: a review. J. Opt. 48(1), 134–144 (2019)

    Article  Google Scholar 

  9. Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison Wesley Publishing Co. Inc., Boston (1989)

    MATH  Google Scholar 

  10. Goldberg, D.E.: Genetic Algorithms. Pearson Education India, New Delhi (2006)

    Google Scholar 

  11. Narayanan, A., Moore, M.: Quantum-inspired genetic algorithms. In: Proceedings of IEEE International Conference on Evolutionary Computation, pp. 61–66. IEEE (1996)

  12. Malossini, A., Blanzieri, E., Calarco, T.: Quantum genetic optimization. IEEE Trans. Evol. Comput. 12(2), 231–241 (2008)

    Article  Google Scholar 

  13. SaiToh, A., Rahimi, R., Nakahara, M.: A quantum genetic algorithm with quantum crossover and mutation operations. Quantum Inf. Process. 13(3), 737–755 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. 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, 2004. IEEE ICIT’04., vol. 3, pp. 1192–1197. IEEE (2004)

  15. Wang, H., Liu, J., Zhi, J., Fu, C.: The improvement of quantum genetic algorithm and its application on function optimization. Probl. Eng. 2013, 10 (2013)

    Google Scholar 

  16. Wang, L., Wu, H., Zheng, D.: A quantum-inspired genetic algorithm for scheduling problems. In: International Conference on Natural Computation, pp. 417–423. Springer (2005)

  17. Tkachuk, V.: Quantum genetic algorithm based on qutrits and its application. Probl. Eng. 2018, 8 (2018)

    MathSciNet  MATH  Google Scholar 

  18. Montiel, O., Rubio, Y., Olvera, C., Rivera, A.: Quantum-inspired acromyrmex evolutionary algorithm. Sci. Rep. 9(1), 1–10 (2019)

    Article  Google Scholar 

  19. Zhou, S., Sun, Z.: A new approach belonging to EDAs: quantum-inspired genetic algorithm with only one chromosome. In: International Conference on Natural Computation, pp. 141–150. Springer (2005)

  20. Reichardt, B.W.: The quantum adiabatic optimization algorithm and local minima. In: Proceedings of the Thirty-sixth Annual ACM Symposium on Theory of Computing, pp. 502–510 (2004)

  21. Farhi, E., Goldstone, J., Gutmann, S.: A quantum approximate optimization algorithm (2014). arXiv preprint arXiv:1411.4028

  22. Peruzzo, A., McClean, J., Shadbolt, P., Yung, M.-H., Zhou, X.-Q., Love, P.J., Aspuru-Guzik, A., O’brien, J.L.: A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5(1), 1–7 (2014)

  23. Farhi, E., Goldstone, J., Gutmann, S.: A Quantum approximate optimization algorithm applied to a bounded occurrence constraint problem (2014). arXiv preprint arXiv:1412.6062

  24. Wecker, D., Hastings, M.B., Troyer, M.: Progress towards practical quantum variational algorithms. Phys. Rev. A 92, 042303 (2015)

    Article  ADS  Google Scholar 

  25. Farhi, E., Harrow, A.W.: Quantum supremacy through the quantum approximate optimization algorithm (2016). arXiv preprint arXiv:1602.07674

  26. McClean, J.R., Romero, J., Babbush, R., Aspuru-Guzik, A.: The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016)

    Article  ADS  MATH  Google Scholar 

  27. Li, Y., Benjamin, S.C.: Efficient variational quantum simulator incorporating active error minimization. Phys. Rev. X 7, 021050 (2017)

    Google Scholar 

  28. Moll, N., Barkoutsos, P., Bishop, L.S., Chow, J.M., Cross, A., Egger, D.J., Filipp, S., Fuhrer, A., Gambetta, J.M., Ganzhorn, M.: Quantum optimization using variational algorithms on near-term quantum devices. Quantum Sci. Technol. 3, 030350 (2018)

    Article  Google Scholar 

  29. Thomson, J.J.: XXIV. On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; ; with application of the results to the theory of atomic structure. Lond. Edinb. Dubl. Philos. Mag. 7(39), 237–265 (1904)

    Article  MATH  Google Scholar 

  30. Berezin, A.A.: An unexpected result in classical electrostatics. Nature 315(6015), 104–104 (1985)

    Article  ADS  Google Scholar 

  31. Erber, T., Hockney, G.: Equilibrium configurations of N equal charges on a sphere. J. Phys. A Math. Gen. 24(23), L1369 (1991)

    Article  ADS  Google Scholar 

  32. Erber, T., Hockney, G.: Complex systems: equilibrium configurations of n equal charges on a sphere (2\(<\) n\(<\) 112). Adv. Chem. Phys 98, 495–594 (1997)

  33. Altschuler, E.L., Williams, T.J., Ratner, E.R., Tipton, R., Stong, R., Dowla, F., Wooten, F.: Possible global minimum lattice configurations for Thomson’s problem of charges on a sphere. Phys. Rev. Lett. 78, 2681 (1997)

  34. Whyte, L.: Unique arrangements of points on a sphere. Am. Math. Mon. 59(9), 606–611 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  35. Bergersen, B., Boal, D., Palffy-Muhoray, P.: Equilibrium configurations of particles on a sphere: the case of logarithmic interactions. J. Phys. A Math. Gen. 27, 2579 (1994)

    Article  ADS  MATH  Google Scholar 

  36. Dragnev, P.D., Legg, D.A., Townsend, D.W.: Discrete logarithmic energy on the sphere. Pac. J. Math. 207(2), 345–358 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shimony, A.: Degree of Entanglement a. Ann. N. Y. Acad. Sci. 755(1), 675–679 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  38. Wei, T.-C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)

    Article  ADS  Google Scholar 

  39. Chen, L., Xu, A., Zhu, H.: Geometric measure of entanglement for pure multipartite states (2009). arXiv preprint arXiv:0911.1493

  40. Hübener, R., Kleinmann, M., Wei, T.-C., González-Guillén, C., Gühne, O.: Geometric measure of entanglement for symmetric states. Phys. Rev. A 80, 032324 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  41. Hayashi, M., Markham, D., Murao, M., Owari, M., Virmani, S.: The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes. J. Math. Phys. 50, 122104 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Wei, T.-C., Severini, S.: Matrix permanent and quantum entanglement of permutation invariant states. J. Math. Phys. 51, 092203 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. ‘Thomson problem’: Wikipedia (2021). Available at: https://en.wikipedia.org/wiki/Thomson_problem#Configurations_of_smallest_known_energy

  44. Băutu, A., Băutu, E.: Energy minimization of point charges on a sphere with particle swarms. In: Paper Presented at the 8th International Balkan Workshop on Applied Physics, vol. 5, pp. 7 (2007)

  45. Lakhbab, H., El Bernoussi, S., El Harif, A.: Energy minimization of point charges on a sphere with a hybrid approach. Appl. Math. Sci. 6(30), 1487–1495 (2012)

    MathSciNet  MATH  Google Scholar 

  46. Morris, J.R., Deaven, D.M., Ho, K.M.: Genetic-algorithm energy minimization for point charges on a sphere. Phys. Rev. B 53(4), R1740 (1996)

    Article  ADS  Google Scholar 

  47. Wales, D.J., Ulker, S.: Structure and dynamics of spherical crystals characterized for the Thomson problem. Phys. Rev. B 74, 212101 (2006)

    Article  ADS  Google Scholar 

  48. Wales, D.J., McKay, H., Altschuler, E.L.: Defect motifs for spherical topologies. Phys. Rev. B 79(22), 224115 (2009)

    Article  ADS  Google Scholar 

  49. Acín, A., Andrianov, A., Costa, L., Jané, E., Latorre, J., Tarrach, R.: Generalized Schmidt decomposition and classification of three-quantum-bit states. Phys. Rev. Lett. 85, 1560 (2000)

    Article  ADS  Google Scholar 

  50. Szalay, S., Pfeffer, M., Murg, V., Barcza, G., Verstraete, F., Schneider, R., Legeza, Ö.: Tensor product methods and entanglement optimization for ab initio quantum chemistry. Int. J. Quantum Chem. 115(19), 1342–1391 (2015)

    Article  Google Scholar 

  51. Kim, J.H., Aghaeimeibodi, S., Carolan, J., Englund, D., Waks, E.: Hybrid integration methods for on-chip quantum photonics. Optica 7(4), 291–308 (2020)

    Article  ADS  Google Scholar 

  52. Saleh, B.E., Teich, M.C.: Fundamentals of Photonics. Wiley, Hoboken (2019)

    Google Scholar 

  53. Caspar, D.L., Klug, A.: Physical principles in the construction of regular viruses. In: Cold Spring Harbor Symposia on Quantitative Biology, vol. 27, pp. 1–24. Cold Spring Harbor Laboratory Press (1962)

  54. Marzec, C.J., Day, L.A.: Pattern formation in icosahedral virus capsids: the papova viruses and Nudaurelia capensis beta virus. Biophys. J. 65(6), 2559–2577 (1993)

    Article  ADS  Google Scholar 

  55. Leiderer, P.: Ions at helium interfaces. Z. Phys. B 98(3), 303–308 (1995)

    Article  ADS  Google Scholar 

  56. Guo, W., Jin, D., Maris, H.J.: Theory of the stability of multielectron bubbles in liquid helium. J. Phys. Conf. Ser. 150, 032027 (2009)

    Article  Google Scholar 

  57. Davis, E.J.: A history of single aerosol particle levitation. Aerosol. Sci. Technol. 26, 212–254 (1997)

    Article  ADS  Google Scholar 

  58. Bowick, M., Cacciuto, A., Nelson, D.R., Travesset, A.: Crystalline order on a sphere and the generalized Thomson problem. Phys. Rev. Lett. 89, 185502 (2002)

    Article  ADS  Google Scholar 

  59. Kroto, H.W., Heath, J.R., O’Brien, S.C., Curl, R.F., Smalley, R.E.: C 60: buckminsterfullerene. Nature 318(6042), 162–163 (1985)

    Article  ADS  Google Scholar 

  60. Dinsmore, A., Hsu, M.F., Nikolaides, M., Marquez, M., Bausch, A., Weitz, D.: Colloidosomes: selectively permeable capsules composed of colloidal particles. Science 298(5595), 1006–1009 (2002)

    Article  ADS  Google Scholar 

  61. LaFave, T., Jr.: Correspondences between the classical electrostatic Thomson problem and atomic electronic structure. J. Electrostat. 71(6), 1029–1035 (2013)

    Article  Google Scholar 

  62. Dodgson, M., Moore, M.: Vortices in a thin-film superconductor with a spherical geometry. Phys. Rev. B 55, 3816 (1997)

    Article  ADS  Google Scholar 

  63. Feynman, R.P.: Quantum mechanical computers. Found. Phys. 16(6), 507–531 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  64. Linke, N.M., Maslov, D., Roetteler, M., Debnath, S., Figgatt, C., Landsman, K.A., Wright, K., Monroe, C.: Experimental comparison of two quantum computing architectures. Proc. Natl. Acad. Sci. 114(13), 3305–3310 (2017)

    Article  Google Scholar 

  65. Preskill, J.: Quantum computing in the NISQ era and beyond. Quantum 2, 79 (2018)

    Article  Google Scholar 

  66. Zulehner, A., Paler, A., Wille, R.: An efficient methodology for mapping quantum circuits to the IBM QX architectures. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 38(7), 1226–1236 (2018)

    Article  Google Scholar 

  67. Behera, B.K., Seth, S., Das, A., Panigrahi, P.K.: Demonstration of entanglement purification and swapping protocol to design quantum repeater in IBM quantum computer. Quantum Inf. Process. 18(4), 1–13 (2019)

    MATH  Google Scholar 

  68. Behera, B.K., Reza, T., Gupta, A., Panigrahi, P.K.: Designing quantum router in IBM quantum computer. Quantum Inf. Process. 18(11), 1–13 (2019)

    Article  Google Scholar 

Download references

Acknowledgements

Amal R. S. acknowledges the insightful discussions on classical genetic algorithms with Dr. Sumitra S. Nair.

Author information

Authors and Affiliations

  1. Department of physics, Indian Institute of Space Science and Technology, Valiamala, Trivandrum, 695547, India

    R. S. Amal & J. Solomon Ivan

Authors
  1. R. S. Amal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Solomon Ivan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Solomon Ivan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amal, R.S., Ivan, J.S. A quantum genetic algorithm for optimization problems on the Bloch sphere. Quantum Inf Process 21, 43 (2022). https://doi.org/10.1007/s11128-021-03368-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-021-03368-7

Keywords

Search

Navigation