Quantum Information Processing

, Volume 12, Issue 4, pp 1719–1735 | Cite as

An improved formalism for quantum computation based on geometric algebra—case study: Grover’s search algorithm

  • James M. ChappellEmail author
  • Azhar Iqbal
  • M. A. Lohe
  • Lorenz von Smekal
  • Derek Abbott


The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford’s geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search analogous to the precession of a spin-\({\frac{1}{2}}\) particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations. We do not claim the development of an improved algorithm, but show in a tutorial paper that geometric algebra provides extremely compact and elegant expressions with improved clarity for the Grover search algorithm. Being a key algorithm in quantum computing and one of the most studied, it forms an ideal basis for a tutorial on how to elucidate quantum operations in terms of geometric algebra—this is then of interest in extending the applicability of geometric algebra to more complicated problems in fields of quantum computing, quantum decision theory, and quantum information.


Grover search algorithm Quantum algorithms Geometric algebra Quantum computing 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • James M. Chappell
    • 1
    Email author
  • Azhar Iqbal
    • 1
  • M. A. Lohe
    • 2
  • Lorenz von Smekal
    • 3
  • Derek Abbott
    • 1
  1. 1.School of Electrical and Electronic EngineeringUniversity of AdelaideAdelaideAustralia
  2. 2.School of Chemistry and PhysicsUniversity of AdelaideAdelaideAustralia
  3. 3.Institut für KernphysikTechnische Universität DarmstadtDarmstadtGermany

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