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.
Similar content being viewed by others
Notes
In this paper a projective algebraic variety is understood as a subset \(X\subset \mathbb {P}(V)\) defined by the zero locus of a collection of homogeneous polynomials.
References
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)
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)
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)
Brylinski, J.L.: Algebraic measures of entanglement. In: Mathematics of Quantum Computation, p. 3–23. Chapman/Hall (CRC) (2002)
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)
Chakraborty, S., Banerjee, S., Adhikari, S., Kumar, A.: Entanglement in the Grover’s Search Algorithm. arXiv:1305.4454 (2013)
Cui, J., Fan, H.: Correlations in the Grover search. J. Phys. A Math. Theor. 43(4), 045305 (2010)
Dür, W., Vidal, G., Cirac, J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62(6), 062314 (2000)
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)
Fulton, W., Harris, J.: Representation Theory, vol. 129. Springer, Berlin (1991)
Galindo, A., Martin-Delgado, M.A.: Family of Grover’s quantum-searching algorithms. Phys. Rev. A 62(6), 062303 (2000)
Gelfand, I.M., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Springer, Berlin (2008)
Grover, L.K.: Quantum computers can search arbitrarily large databases by a single query. Phys. Rev. Lett. 79(23), 4709 (1997)
Harris, J.: Algebraic Geometry: A First Course, vol. 133. Springer, Berlin (2013)
Heydari, H.: Geometrical structure of entangled states and the secant variety. Quantum Inf. Process. 7(1), 43–50 (2008)
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)
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)
Holweck, F., Luque, J.G., Thibon, J.Y.: Geometric descriptions of entangled states by auxiliary varieties. J. Math. Phys. 53(10), 102203 (2012)
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)
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)
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)
Landsberg, J.M.: Tensors: Geometry and Applications. American Mathematical Society, Providence (2012)
Lavor, C., Manssur, L.R.U., Portugal, R.: Grover’s Algorithm: Quantum Database Search. arXiv:quant-ph/0301079 (2003)
Meyer, D.A., Wallach, N.R.: Global entanglement in multiparticle systems. arXiv:quant-ph/0108104 (2001)
Meyer, D.A.: Sophisticated quantum search without entanglement. Phys. Rev. Lett. 85(9), 2014 (2000)
Miyake, A.: Multipartite entanglement under stochastic local operations and classical communication. Int. J. Quantum Inf. 2(01), 65–77 (2004)
Parfenov, P.G.: Tensor products with finitely many orbits. Russ. Math. Surv. 53(3), 635–636 (1998)
Rieffel, E.G., Polak, W.H.: Quantum Computing: A Gentle Introduction. MIT Press, Cambridge (2011)
Rossi, M., Bruß, D., Macchiavello, C.: Scale invariance of entanglement dynamics in Grover’s quantum search algorithm. Phys. Rev. A 87(2), 022331 (2013)
Rossi, M., Bruß, D., Macchiavello, C.: Hypergraph states in Grover’s quantum search algorithm. Phys. Scr. 2014(T160), 014036 (2014)
Sanz, M., Braak, D., Solano, E., Egusquiza, I.L: Entanglement Classification with Algebraic Geometry. arXiv:1606.06621 (2016)
Sawicki, A., Tsanov, V.V.: A link between quantum entanglement, secant varieties and sphericity. J. Phys. A Math. Theor. 46(26), 265301 (2013)
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)
Zak, F.L.: Tangents and Secants of Algebraic Varieties, Translations of Mathematical Monographs, vol. 127. American Mathematical Society, Providence (1993)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Holweck, F., Jaffali, H. & Nounouh, I. Grover’s algorithm and the secant varieties. Quantum Inf Process 15, 4391–4413 (2016). https://doi.org/10.1007/s11128-016-1445-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-016-1445-2