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Grover’s algorithm and the secant varieties

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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.

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Notes

  1. 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.

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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.

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Authors and Affiliations

  1. IRTES/UTBM, Université de Bourgogne-Franche-Comté, 90010, Belfort Cedex, France

    Frédéric Holweck, Hamza Jaffali & Ismaël Nounouh

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  1. Frédéric Holweck
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  2. Hamza Jaffali
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  3. Ismaël Nounouh
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Correspondence to Frédéric Holweck.

Appendix: Examples of marked elements

Appendix: Examples of marked elements

The following tables provide examples of sets of marked elements which allows to reach the corresponding orbits by running Grover’s algorithm in the standard regime \(|S|<\dfrac{N}{4}\) (See Tables 4, 5, 6).

Table 4 Examples of family of marked elements S and the corresponding orbits reached by the algorithm in the \(2\times 2\times 2\) case
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Table 5 Examples of family of marked elements S and the corresponding orbits reached by the algorithm in the \(2\times 2\times 3\) case
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Table 6 Examples of family of marked elements S and the corresponding orbits reached by the algorithm in the \(2\times 3\times 3\) case
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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

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