Quantum Information Processing

, Volume 15, Issue 11, pp 4391–4413 | Cite as

Grover’s algorithm and the secant varieties

  • Frédéric Holweck
  • Hamza Jaffali
  • Ismaël Nounouh
Article

Abstract

In this paper we investigate the entanglement nature of quantum states generated by Grover’s search algorithm by means of algebraic geometry. More precisely we establish a link between entanglement of states generated by the algorithm and auxiliary algebraic varieties built from the set of separable states. This new perspective enables us to propose qualitative interpretations of earlier numerical results obtained by M. Rossi et al. We also illustrate our purpose with a couple of examples investigated in details.

Keywords

Quantum algorithm Entangled states Secant varieties 

Notes

Acknowledgments

The authors would like to thank Prof. Jean-Gabriel Luque for kindly providing them his Maple code to compute the invariants/covariants used in the calculation of Sect. 5.

References

  1. 1.
    Batle, J., Ooi, C.R., Farouk, A., Alkhambashi, M.S., Abdalla, S.: Global versus local quantum correlations in the Grover search algorithm. Quantum Inf. Process. 15(2), 833–849 (2016)ADSCrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bennett, C.H., Popescu, S., Rohrlich, D., Smolin, J.A., Thapliyal, A.V.: Exact and asymptotic measures of multipartite pure-state entanglement. Phys. Rev. A 63(1), 012307 (2000)ADSCrossRefGoogle Scholar
  3. 3.
    Brody, D.C., Gustavsson, A.C., Hughston, L.P.: Entanglement of three-qubit geometry. In: Journal of Physics: Conference Series, vol. 67, No. 1, p. 012044. IOP Publishing (2007)Google Scholar
  4. 4.
    Brylinski, J.L.: Algebraic measures of entanglement. In: Mathematics of Quantum Computation, p. 3–23. Chapman/Hall (CRC) (2002)Google Scholar
  5. 5.
    Catalisano, M.V., Geramita, A., Gimigliano, A.: Secant varieties of \(({\mathbb{P}}^1) \times ....\times ({\mathbb{P}}^1)\) (n-times) are NOT Defective for \(n\ge 5\). arXiv:0809.1701 (2008)
  6. 6.
    Chakraborty, S., Banerjee, S., Adhikari, S., Kumar, A.: Entanglement in the Grover’s Search Algorithm. arXiv:1305.4454 (2013)
  7. 7.
    Cui, J., Fan, H.: Correlations in the Grover search. J. Phys. A Math. Theor. 43(4), 045305 (2010)ADSCrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62(6), 062314 (2000)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fang, Y., Kaszlikowski, D., Chin, C., Tay, K., Kwek, L.C., Oh, C.H.: Entanglement in the Grover search algorithm. Phys. Lett. A 345(4), 265–272 (2005)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Fulton, W., Harris, J.: Representation Theory, vol. 129. Springer, Berlin (1991)MATHGoogle Scholar
  11. 11.
    Galindo, A., Martin-Delgado, M.A.: Family of Grover’s quantum-searching algorithms. Phys. Rev. A 62(6), 062303 (2000)ADSCrossRefGoogle Scholar
  12. 12.
    Gelfand, I.M., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Springer, Berlin (2008)MATHGoogle Scholar
  13. 13.
    Grover, L.K.: Quantum computers can search arbitrarily large databases by a single query. Phys. Rev. Lett. 79(23), 4709 (1997)ADSCrossRefGoogle Scholar
  14. 14.
    Harris, J.: Algebraic Geometry: A First Course, vol. 133. Springer, Berlin (2013)MATHGoogle Scholar
  15. 15.
    Heydari, H.: Geometrical structure of entangled states and the secant variety. Quantum Inf. Process. 7(1), 43–50 (2008)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Holweck, F., Lévay, P.: Classification of multipartite systems featuring only \(|W{\rangle }\) and \(|GHZ{\rangle }\) genuine entangled states. J. Phys. A Math. General 49(8), 085201 (2016)ADSCrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Holweck, F., Luque, J.G., Thibon, J.Y.: Entanglement of four qubit systems: a geometric atlas with polynomial compass II (the tame world). arXiv:1606.05569 (2016)
  18. 18.
    Holweck, F., Luque, J.G., Thibon, J.Y.: Geometric descriptions of entangled states by auxiliary varieties. J. Math. Phys. 53(10), 102203 (2012)ADSCrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Holweck, F., Luque, J.G., Thibon, J.Y.: Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world). J. Math. Phys. 55(1), 012202 (2014)ADSCrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Holweck, F., Luque, J.G., Planat, M.: Singularity of type D4 arising from four-qubit systems. J. Phys. A Math. Theor. 47(13), 135301 (2014)ADSCrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Hübener, R., Kleinmann, M., Wei, T.C., González-Guillén, C., Gühne, O.: Geometric measure of entanglement for symmetric states. Phys. Rev. A 80(3), 032324 (2009)ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    Landsberg, J.M.: Tensors: Geometry and Applications. American Mathematical Society, Providence (2012)MATHGoogle Scholar
  23. 23.
    Lavor, C., Manssur, L.R.U., Portugal, R.: Grover’s Algorithm: Quantum Database Search. arXiv:quant-ph/0301079 (2003)
  24. 24.
    Meyer, D.A., Wallach, N.R.: Global entanglement in multiparticle systems. arXiv:quant-ph/0108104 (2001)
  25. 25.
    Meyer, D.A.: Sophisticated quantum search without entanglement. Phys. Rev. Lett. 85(9), 2014 (2000)ADSCrossRefGoogle Scholar
  26. 26.
    Miyake, A.: Multipartite entanglement under stochastic local operations and classical communication. Int. J. Quantum Inf. 2(01), 65–77 (2004)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Parfenov, P.G.: Tensor products with finitely many orbits. Russ. Math. Surv. 53(3), 635–636 (1998)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Rieffel, E.G., Polak, W.H.: Quantum Computing: A Gentle Introduction. MIT Press, Cambridge (2011)MATHGoogle Scholar
  29. 29.
    Rossi, M., Bruß, D., Macchiavello, C.: Scale invariance of entanglement dynamics in Grover’s quantum search algorithm. Phys. Rev. A 87(2), 022331 (2013)ADSCrossRefGoogle Scholar
  30. 30.
    Rossi, M., Bruß, D., Macchiavello, C.: Hypergraph states in Grover’s quantum search algorithm. Phys. Scr. 2014(T160), 014036 (2014)CrossRefGoogle Scholar
  31. 31.
    Sanz, M., Braak, D., Solano, E., Egusquiza, I.L: Entanglement Classification with Algebraic Geometry. arXiv:1606.06621 (2016)
  32. 32.
    Sawicki, A., Tsanov, V.V.: A link between quantum entanglement, secant varieties and sphericity. J. Phys. A Math. Theor. 46(26), 265301 (2013)ADSCrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Verstraete, F., Dehaene, J., De Moor, B., Verschelde, H.: Four qubits can be entangled in nine different ways. Phys. Rev. A 65(5), 052112 (2002)ADSCrossRefMathSciNetGoogle Scholar
  34. 34.
    Zak, F.L.: Tangents and Secants of Algebraic Varieties, Translations of Mathematical Monographs, vol. 127. American Mathematical Society, Providence (1993)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.IRTES/UTBMUniversité de Bourgogne-Franche-ComtéBelfort CedexFrance

Personalised recommendations