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Abstract Quantum Automata as Formal Models of Quantum Information Processing Systems

  • Mizal Alobaidi
  • Andriy Batyiv
  • Grygoriy Zholtkevych
Conference paper
  • 544 Downloads
Part of the Communications in Computer and Information Science book series (CCIS, volume 347)

Abstract

Nowadays, quantum computation is considered as a perspective way to overcome the computational complexity barrier. Development of a quantum programming technology requires to build theoretical background of quantum computing similarly to the classical computability theory and the classical computational complexity theory. The challenge to develop such theoretical background was posed by Yu.I. Manin. An attempt to build a mathematically rigorous model for quantum information processing systems in compliance with the concept of Yu.I. Manin is presented in the chapter. The attempt carries out by identifying elementary constituents of quantum computational processes. They are called quantum actions and their properties are studied in the chapter. In particular, the equivalence criterion of quantum actions in terms of their generating operators has been found; the special class of quantum actions has been characterised in terms of generating operators too. This class is formed by quantum actions leading to the collapse of quantum states. Further in the chapter, the mathematical model of quantum information processing systems. It is defined as an ensemble of interacting quantum actions on the common memory. The term ”abstract quantum automata” is introduced to denote such model. At the end of the chapter models of some important quantum information processing systems are presented.

Keywords

Algorithmic solvability computational complexity quantum algorithm formal specification labelled transition system operational semantics Kraus’ family quantum action abstract quantum automaton 

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References

  1. 1.
    Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proc. 39th Ann. Symp. on Found. Comp. Sci., pp. 332–341. IEEE (1998)Google Scholar
  2. 2.
    Bouwmeester, D., Pan, J.-W., Mattle, K., Eible, M., Weinfurter, H., Zeilinger, A.: Experimental quantum teleportation. Nature 390, 575–579 (1997)CrossRefGoogle Scholar
  3. 3.
    Boschi, D., Branca, S., De Martini, F., Hardy, L., Popescu, S.: Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein–Podolsky–Rosen Channel. Phys. Rev. Lett. 80, 1121–1125 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Church, A.: An unsolvable problem of elementary number theory. Amer. J. Math. 58(2), 345–363 (1936)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cook, S.: The Complexity of Theorem Proving Procedures. In: Proc. 3rd Ann. ACM Symp. on Theory of Computing, pp. 151–158. ACM, New York (1971)Google Scholar
  6. 6.
    Deutsch, D.: Quantum theory, the Church-Turing principle and the universal quantum computer. Proc. Roy. Soc. Lond., Series A 400(1818), 97–117 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Deutsch, D., Jozsa, R.: Rapid Solution of Problems by Quantum Computation. Proc. Roy. Soc. Lond., Series A 439(1907), 553–558 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press (1958)Google Scholar
  9. 9.
    Feynman, R.P.: Simulating Physics with Computer. Int. J. Theor. Phys. 21, 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Publishing Company, Amsterdam (1982)zbMATHGoogle Scholar
  11. 11.
    Karp, R.M.: Complexity of Computation. SIAM-AMS Proceedings, vol. 7. AMS, Providence (1974)zbMATHGoogle Scholar
  12. 12.
    Manin, Y.I.: Computable and Uncomputable (Cybernetics), Sovetskoe radio, Moscow (1980) (in Russian)Google Scholar
  13. 13.
    Manin, Y.I.: Mathematics as metaphor: selected essays of Yuri I. Manin. AMS (2007)Google Scholar
  14. 14.
    Milner, R.: Communicating and Mobile System: the π-Calculus. Cambridge University Press, Cambridge (1999)Google Scholar
  15. 15.
    Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theoret. Comput. Sci. 237, 99–136 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Neumann von, J.: Mathematische Grundlagen Der Quantenmechanik. Verlag von Julius Springer, Berlin (1932)zbMATHGoogle Scholar
  17. 17.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. 10th Anniversary Edition. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  18. 18.
    OMG Unified Modelling Language (OMG UML), Superstructure. OMG, v 2.4.1 (2011), http://www.omg.org/spec/UML/2.4.1/Superstructure/PDF
  19. 19.
    Post, E.L.: Finite Combinatory Processes – Formulation 1. J. Symb. Logics. 1(3), 103–105 (1936)zbMATHCrossRefGoogle Scholar
  20. 20.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc. 2(42), 230–265 (1936)MathSciNetGoogle Scholar
  21. 21.
    Turing, A.M.: Computability and λ-Definability. J. Symb. Logics. 2(4), 153–163 (1937)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Mizal Alobaidi
    • 1
  • Andriy Batyiv
    • 2
  • Grygoriy Zholtkevych
    • 2
  1. 1.Faculty of Computer Science and MathematicsTikrit UniversityTikritIraq
  2. 2.School of Mathematics and MechanicsV.N. Karazin Kharkiv National UniversityKharkivUkraine

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