Quantum Information Processing

, Volume 13, Issue 11, pp 2483–2498 | Cite as

Inexact and exact quantum searches with a preparation state in a three-dimensional subspace

  • C. Bautista-Ramos
  • C. Guillén-Galván
  • A. Rangel-Huerta
  • D. A. Valdes-Amaro
  • J. M. Amezcua-Ortega


It is well known that exact quantum searches can be performed by the quantum amplitude amplification algorithm with some phase matching condition. However, recently it was shown that for some preparation states in a three-dimensional subspace, an exact search is impossible to accomplish. We show this impossibility derives from two sources: a problem of state restriction to a cyclic subspace and the solution of a linear system of equations with a \(k\)-potent coefficient matrix. Furthermore, using said system of equations, we introduce a class of preparation states in a three-dimensional space that, even though the quantum amplitude amplification algorithm is unable to find the target state exactly, the same system of equations implies modifications to the quantum amplitude amplification algorithm under which exact solutions in three-dimensional subspaces can be found. We also prove that an inexact quantum search in the 3-potent case can find the target state with high probability if the Grover operator is iterated a number of times inversely proportional to the uncertainty of said 3-potent coefficient matrix as an observable operator.


Quantum amplitude amplification algorithm Grover algorithm  Exact quantum search \(k\)-potent matrix Cyclic subspace 



The authors would like to thank the anonymous reviewers for their helpful comments and suggestions.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • C. Bautista-Ramos
    • 1
  • C. Guillén-Galván
    • 1
  • A. Rangel-Huerta
    • 1
  • D. A. Valdes-Amaro
    • 1
  • J. M. Amezcua-Ortega
    • 1
  1. 1.Facultad de Ciencias de la ComputaciónBenemérita Universidad Autónoma de PueblaPueblaMexico

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