From orthogonal projections to a generalized quantum search

  • Authors:

  • César Bautista-Ramos;Carlos Guillén-Galván;Alejandro Rangel-Huerta

  • Affiliations:

  • Facultad de Ciencias de la Computación, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico 72570;Facultad de Ciencias de la Computación, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico 72570;Facultad de Ciencias de la Computación, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico 72570

  • Venue:
  • Quantum Information Processing
  • Year:
  • 2013

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Abstract

A quantum algorithm with certainty is introduced in order to find a marked pre-image of an element which is known to be in the image domain of an orthogonal projection operator. The analysis of our algorithm is made by using properties of the Moebius transformations acting on the complex projective line. This new algorithm closely resembles the quantum amplitude amplification algorithm, however it is proven that our algorithm is a proper generalization of the latter (with generalized phases), in such a way that the quantum search engine of the main operator of quantum amplification is included as a particular case. In order to show that there exist search problems that can be solved by our proposal but cannot be by applying the quantum amplitude amplification algorithm, we modify our algorithm as a cryptographic authentification protocol. This protocol results to be robust enough against attacks based on the quantum amplitude amplification algorithm. As a byproduct, we show a condition where it is impossible to find exactly a pre-image of an orthoghonal projection. This result generalizes the fact that, it is impossible to find a target state exactly by using quantum amplification on a three dimensional invariant subspace.