Skip to main content
SpringerLink
Log in

Global multipartite entanglement dynamics in Grover’s search algorithm

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Entanglement is considered to be one of the primary reasons for why quantum algorithms are more efficient than their classical counterparts for certain computational tasks. The global multipartite entanglement of the multiqubit states in Grover’s search algorithm can be quantified using the geometric measure of entanglement (GME). Rossi et al. (Phys Rev A 87:022331, 2013) found that the entanglement dynamics is scale invariant for large n. Namely, the GME does not depend on the number n of qubits; rather, it only depends on the ratio of iteration k to the total iteration. In this paper, we discuss the optimization of the GME for large n. We prove that “the GME is scale invariant” does not always hold. We show that there is generally a turning point that can be computed in terms of the number of marked states and their Hamming weights during the curve of the GME. The GME is scale invariant prior to the turning point. However, the GME is not scale invariant after the turning point since it also depends on n and the marked states.

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. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  2. Bruß, D.: Characterizing entanglement. J. Math. Phys. 43(9), 4237–4251 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Shor, P.W.: Algorithms for quantum computer: discrete logarithm and factoring. In: Proceedings of the 35th IEEE Symposium on the Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, pp. 56–65 (1994)

  4. Grover, L.K.: A fast quantum mechanical algorithm for database search. Phys. Rev. Lett. 79, 325 (1997)

    Article  ADS  Google Scholar 

  5. Jozsa, R., Linden, N.: On the role of entanglement in quantum-computational speed-up. Proc. R. Soc. Lond. A 459, 2011–2023 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003)

    Article  ADS  Google Scholar 

  7. Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. A 439, 553 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Simon, D.: On the power of quantum computation. SIAM J. Comput. 26, 1474–1483 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bruß, D., Macchiavello, C.: Multipartite entanglement in quantum algorithms. Phys. Rev. A 83, 052313 (2011)

    Article  ADS  Google Scholar 

  10. Meyer, D.A.: Sophisticated quantum search without entanglement. Phys. Rev. Lett. 85, 2014 (2000)

    Article  ADS  Google Scholar 

  11. Biham, O., Nielsen, M.A., Osborne, T.J.: Entanglement monotone derived from Grover’s algorithm. Phys. Rev. A 65, 062312 (2002)

    Article  ADS  Google Scholar 

  12. Shimoni, Y., Biham, O.: Groverian entanglement measure of pure quantum states with arbitrary partitions. Phys. Rev. A 75, 022308 (2007)

    Article  ADS  Google Scholar 

  13. Wen, J.Y., Qiu, D.W.: Entanglement in adiabatic quantum searching algorithms. Int. J. Quantum Inf. 6(5), 997–1009 (2008)

    Article  MATH  Google Scholar 

  14. Shimoni, Y., Shapira, D., Biham, O.: Characterization of pure quantum states of multiple qubits using the Groverian entanglement measure. Phys. Rev. A 69, 062303 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Fang, Y., Kaszlikowski, D., Chin, C., Tay, K., Kwek, L.C., Oh, C.H.: Entanglement in the Grover search algorithm. Phy. Lett. A 345, 265–272 (2005)

    Article  ADS  MATH  Google Scholar 

  16. Rungta, P.: The quadratic speedup in Grover’s search algorithm from the entanglement perspective. Phys. Let. A 373, 2652–2659 (2009)

    Article  ADS  MATH  Google Scholar 

  17. Cui, J., Fan, H.: Correlations in Grover search. J. Phys. A Math. Theor. 43(4), 045305 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Rossi, M., Bruß, D., Macchiavello, C.: Scale invariance of entanglement dynamics in Grover’s quantum search algorithm. Phys. Rev. A 87, 022331 (2013)

    Article  ADS  Google Scholar 

  19. Chakraborty, S., Banerjee, S., Adhikari, S., Kumar, A.: Entanglement in the Grover’s search algorithm. arXiv: 1305.4454 (2013)

  20. Qu, R., Shang, B.J., Bao, Y.R., Song, D.W., Teng, C.M., Zhou, Z.W.: Multipartite entanglement in Grover’s search algorithm. Nat. Comput. 14, 683–689 (2015)

    Article  MathSciNet  Google Scholar 

  21. Batle, J., Raymond Ooi, C.H., Farouk, A., Alkhambash, M.S., Abdalla, S.: Global versus local quantum correlations in the Grover. Quantum Inf. Process. 15(2), 833–850 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Holweck, F., Jaffali, H., Nounouh, I.: Grover’s algorithm and the secant varieties. Quantum Inf. Process. 15(11), 4391–4413 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Chamoli, A., Bhandari, C.M.: Success rate and entanglement measure in Grover’s search algorithm for certain kinds of four qubit states. Phys. Lett. A 346, 17–26 (2005)

    Article  ADS  MATH  Google Scholar 

  24. Meyer, D.A., Wallach, N.R.: Global entanglement in multiparticle systems. J. Math. Phys. 43(9), 4273–4278 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Greenberger, D.M., Horne, M.A., Zeilinger, A.: Going beyond Bell’s theorem, arXiv: 0712.0921 (2007)

  26. Shimony, A.: Degree of entanglement. Ann. N. Y. Acad. Sci. 755, 675–679 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  27. 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 

  28. Blasone, M., Dell’Anno, F., Siena, S.D., Illuminati, F.: Hierarchies of geometric entanglement. Phys. Rev. A 77, 062304 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  29. Cavalcanti, D.: Connecting the generalized robustness and the geometric measure of entanglement. Phys. Rev. A 73, 044302 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  30. Ghne, O., Reimpell, M., Werner, R.F.: Estimating entanglement measures in experiments. Phys. Rev. Lett. 98, 110502 (2007)

    Article  ADS  Google Scholar 

  31. H\(\ddot{u}\)bener, R., Kleinmann, M., Wei, T.C., Gonz\(\acute{a}\)lez-Guill\(\acute{e}\)n, C., G\(\ddot{u}\)hne, O.: Geometric measure of entanglement for symmetric states. Phys. Rev. A 80, 032324 (2009)

  32. Martin, J., Giraud, O., Braun, P.A., Braun, D., Bastin, T.: Multiqubit symmetric states with high geometric entanglement. Phys. Rev. A 81, 062347 (2010)

    Article  ADS  Google Scholar 

  33. Stockton, J.K., Geremia, J.M., Doherty, A.C., Mabuchi, H.: Characterizing the entanglement of symmetric many-particle spin-\(\frac{1}{2}\) systems. Phys. Rev. A 67, 022112 (2003)

    Article  ADS  Google Scholar 

  34. Dicke, R.H.: Coherence in spontaneous radiation processes. Phys. Rev. 93, 99 (1954)

    Article  ADS  MATH  Google Scholar 

  35. Dur, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are thankful to the anonymous referees and editor for their comments and suggestions that have greatly helped to improve the quality of the manuscript. This work is supported in part by the National Natural Science Foundation of China (Nos. 61572532, 61272058, 61602532) and the Fundamental Research Funds for the Central Universities of China (Nos. 17lgjc24, 161gpy43).

Author information

Authors and Affiliations

  1. Institute of Computer Science Theory, School of Data and Computer Science, Sun Yat-sen University, Guangzhou, 510006, China

    Minghua Pan, Daowen Qiu & Shenggen Zheng

  2. School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou, 510006, China

    Minghua Pan

  3. School of Information and Electronic Engineering, Wuzhou University, Wuzhou, 543002, China

    Minghua Pan

Authors
  1. Minghua Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daowen Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shenggen Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daowen Qiu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, M., Qiu, D. & Zheng, S. Global multipartite entanglement dynamics in Grover’s search algorithm. Quantum Inf Process 16, 211 (2017). https://doi.org/10.1007/s11128-017-1661-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-017-1661-4

Keywords

Search

Navigation