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Quantum Information Processing

, Volume 1, Issue 4, pp 283–302 | Cite as

Natural Majorization of the Quantum Fourier Transformation in Phase-Estimation Algorithms

  • Román Orús
  • José I. Latorre
  • Miguel A. Martín-Delgado
Article

Abstract

We prove that majorization relations hold step by step in the Quantum Fourier Transformation (QFT) for phase-estimation algorithms. Our result relies on the fact that states which are mixed by Hadamard operators at any stage of the computation only differ by a phase. This property is a consequence of the structure of the initial state and of the QFT, based on controlled-phase operators and a single action of a Hadamard gate per qubit. The detail of our proof shows that Hadamard gates sort the probability distribution associated to the quantum state, whereas controlled-phase operators carry all the entanglement but are immaterial to majorization. We also prove that majorization in phase-estimation algorithms follows in a most natural way from unitary evolution, unlike its counterpart in Grover's algorithm.

PACS: 03.67.-a, 03.67.Lx

majorization quantum Fourier transformation quantum phase-estimation quantum algorithms 

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Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Román Orús
    • 1
  • José I. Latorre
    • 1
  • Miguel A. Martín-Delgado
    • 2
  1. 1.Department d'Estructura i Constituents de la MatèriaUniv. BarcelonaBarcelonaSpain
  2. 2.Departamento de Física Teórica IUniversidad ComplutenseMadridSpain

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