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Quinta Escuela de Física Matemática\\
The Mathematics of Entanglement\\
Universidad de los Andes\\
Mayo 27 al 31 de 2013\\
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\textbf{{\Large Homothety of entanglement in Grover's quantum algorithm}}\\
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{\scriptsize VLADIMIR BALLESTEROS BALLESTEROS}$^{1,}$ \footnote[1]{E-mail:\texttt{ ballesterosv@javeriana.edu.co}}, {\scriptsize JOSÉ ALFONSO LEYVA ROJAS}$^{1,}$ \footnote[2]{E-mail:\texttt{ leyvaa@javeriana.edu.co}}

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{\tiny $^{1}$GRUPO DE BIOFÍSICA Y BIOQUÍMICA ESTRUCTURAL, DEPARTAMENTO DE FÍSICA, PONTIFICIA UNIVERSIDAD JAVERIANA, BOGOTÁ, D.C.}
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\textbf{Abstract}
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Entanglement represents an important factor in the speed-up of quantum computational processes [1-4].Recently, it is of interest to analyze the function of multipartite entanglement in quantum algorithms.On the other hand, it was shown that multipartite entangled states are used in the first phase of the Grover's quantum search algorithm\cite{brub}.In this work, we show the behavior  of entanglement in this algorithm, computing it through GME (Geometric Measure of Entanglement).Then, we demonstrate that the behavior of any type of entanglement is independient of the quantity $q$ of qubits for large $q$, thus showing a homothetic transformation.Finally, we compare this outcome with the one obtained by a fixed-point quantum algorithm and we propose some guidelines for efficient simulation in the context of this quantum algorithm.

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\bibitem{wei} T.C. Wei, P.M. Goldbart, Phys. Rev. A 68 (2003) 042307, quant-ph/0307219.

\bibitem{lew} M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, U. Sen, Adv. Phys. 56 (2007) 243.
 
\bibitem{sach} S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, Cambridge, 2011.
  
\bibitem{petz} D. Petz, Quantum Information Theory and Quantum Statistics, Springer-Verlag, Berlin, 2008.
\bibitem{brub} D. Brub, C. Macchiavello, Phys. Rev. A 83, 052313 (2011).
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