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Systematic analysis of majorization in quantum algorithms

  • R. Orús
  • J. I. Latorre
  • M. A. Martín-Delgado
Article

Abstract.

Motivated by the need to uncover some underlying mathematical structure of optimal quantum computation, we carry out a systematic analysis of a wide variety of quantum algorithms from the majorization theory point of view. We conclude that step-by-step majorization is found in the known instances of fast and efficient algorithms, namely in the quantum Fourier transform, in Grover’s algorithm, in the hidden affine function problem, in searching by quantum adiabatic evolution and in deterministic quantum walks in continuous time solving a classically hard problem. On the other hand, the optimal quantum algorithm for parity determination, which does not provide any computational speed-up, does not show step-by-step majorization. Lack of both speed-up and step-by-step majorization is also a feature of the adiabatic quantum algorithm solving the 2-SAT “ring of agrees” problem. Furthermore, the quantum algorithm for the hidden affine function problem does not make use of any entanglement while it does obey majorization. All the above results give support to a step-by-step Majorization Principle necessary for optimal quantum computation.

Keywords

Fourier Fourier Transform Continuous Time Efficient Algorithm Theory Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  • R. Orús
    • 1
  • J. I. Latorre
    • 1
  • M. A. Martín-Delgado
    • 2
  1. 1.Dept. d’Estructura i Constituents de la MatériaUniv. BarcelonaBarcelonaSpain
  2. 2.Departamento de Física Teórica IUniversidad ComplutenseMadridSpain

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