International Journal of Theoretical Physics

, Volume 58, Issue 1, pp 136–142 | Cite as

Necessary and Sufficient Condition for Quantum Computing

  • Koji NagataEmail author
  • Tadao Nakamura
  • Ahmed Farouk
  • Do Ngoc Diep


A necessary and sufficient condition for quantum computing performed with, for example, the Deutsch-Jozsa algorithm or the Bernstein-Vazirani algorithm, has theoretically been investigated. Assume a 2N qubit-quantum computing which starts with the state \(|\overbrace {0,0,...,0,1}^{N}\rangle |\overbrace {1,1,...,1}^{N}\rangle \) as follows: Uf|0,0,...,0,1〉|1,1,...,1〉 = |0,0,...,0,1〉 \( |\overline {f(0,0,...,0,1)}\rangle . \) Surprisingly the relation f(x) = f(−x) is the necessary and sufficient condition of holding this fundamental relation if local unitary operations can be used.


Quantum algorithms Quantum computation 



We thank Professor Han Geurdes, Professor Shahrokh Heidari, Professor Hamed Daei Kasmaei, and Professor Mark Behzad Doost for valuable comments.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsKorea Advanced Institute of Science and TechnologyDaejeonKorea
  2. 2.Department of Information and Computer ScienceKeio UniversityKohoku-ku, YokohamaJapan
  3. 3.Department of Physics and Computer Science, Faculty of ScienceWilfrid Laurier UniversityWaterlooCanada
  4. 4.TIMASThang Long UniversityHanoiVietnam
  5. 5.Institute of Mathematics, VASTHanoiVietnam

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