International Journal of Theoretical Physics

, Volume 44, Issue 11, pp 2059–2071 | Cite as

Computational Power of Infinite Quantum Parallelism

  • Martin ZieglerEmail author


Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing.

We compare this new one with other attempts towards hypercomputation by separating (1) its %principal computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic.

Key Words

Hypercomputation quantum mechanics recursion theory infinite parallelism 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adamyan, V. A., Calude, C. S., and Pavlov, B. S. (2004). Transcending the limits of Turing computability. T. Hida, K. Saito, S. Si, (ed), Quantum Information Complexity. Proceedings of Meijo Winter School 2003, World Scientific, Singapore,pp. 119–137. %! 0304128!Google Scholar
  2. Atallah, M. J. (ed.) (1999). Algorithms and Theory of Computation Handbook, CRC Press, Boca Raton, Florida.Google Scholar
  3. Berlekamp, E. R., Conway, J. H., and Guy, R. K. (2004). Winning Ways for Your Mathematical Plays, vol. 4, 2nd Edn., Academic Press, New York.Google Scholar
  4. Beggs, E. J. and Tucker, J. V. (2004). Computations via Experiments with Kinematic Systems, Technical Report 5–2004, Department of Computer Science, University of Wales Swansea.Google Scholar
  5. Benioff, P. (1980). The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing Machines. Journal of Statistical Physics 22, 563– 591.CrossRefMathSciNetADSGoogle Scholar
  6. Bennett, C. H. and Landauer, R. (1985). The fundamental physical limits of computation. Scientific American 253(1), 48–56.CrossRefGoogle Scholar
  7. Blass, A. (1984). Existence of bases implies the axiom of choice. Axiomatic Set Theory, Contemporary Mathematics 31, 31–33.zbMATHMathSciNetGoogle Scholar
  8. Blum, L., Shub, M., and Smale, S. (1989). On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions, and universal machines. Bulletin of the American Mathematical Society 21, 1–46.MathSciNetCrossRefzbMATHGoogle Scholar
  9. Burgin, M. and Klinger, A. (eds.) (2004). Super-recursive algorithms and hypercomputation. vol. 317. In Theoretical Computer Science, Elsevier, Amsterdam.Google Scholar
  10. Calude, C. S., Dinneen, M. J., and Svozil, K. (2001). Reflections on quantum computing. In Complexity, Vol. 6(1), Wiley, New York.Google Scholar
  11. Calude, C. S., Dinneen, M. J., and Peper, F. (eds.) (2002). Unconventional Models of Computation, Vol. 2509. In Lecture Notes in Computer Science, Springer, Heidelberg.Google Scholar
  12. Calude, C. S. and Pavlov, B. (2002). Coins, quantum measurements, and Turing's barrier. In Quantum Information Processing, Vol. 1, Plenum, New York, pp. 107–127.Google Scholar
  13. Copeland, J. (1997). The broad conception of computation. American Behavioural Scientist 40, 690–716.Google Scholar
  14. Copeland, J. (2002). Hypercomputation. Minds and Machines 12, 461–502.CrossRefzbMATHGoogle Scholar
  15. Copeland, J. and Proudfoot, D. (1999). Alan Turing's forgotten ideas in computer science. Scientific American 280(4), 98–103.CrossRefGoogle Scholar
  16. Du, D.-Z. and Ker-I Ko (2000). Theory of Computational Complexity, Wiley, New York.Google Scholar
  17. Eberbach, E. and Wegner, P. (2003). Beyond Turing Machines. Bulletin of the European Association for Theoretical Computer Science 81, 279–304.MathSciNetzbMATHGoogle Scholar
  18. van Emde Boas, P., Spaan, E., and Torenvliet, L. (1989). Nondeterminism, fairness and a fundamental analogy. The Bulletin of the European Association for Theoretical Computer Science 37, 186– 193.zbMATHGoogle Scholar
  19. Etesi, G. and Németi, I. (2002). Non-Turing computations via Malament–Hogarth Space–Times. International Journal of Theoretical Physics 41(2), 341–370.CrossRefMathSciNetzbMATHGoogle Scholar
  20. M. Fürer: “The tight deterministic time hierarchy”, % pp.8-16 in Proc. 14th ACM Symposium on Theory of Computing % (1982).Google Scholar
  21. Geroch, R. and Hartle, J. B. (1986). Computability and physical theories. Foundations of Physics 16(6), 533–550.CrossRefMathSciNetADSGoogle Scholar
  22. del, K. (1940). The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis With the Axioms of Set Theory, Princeton University Press, Princeton.Google Scholar
  23. Gruska, J. (1999). Quantum Computing, McGraw-Hill, New York.Google Scholar
  24. Hamkins, J. D. and Lewis, A. (2000). Infinite time Turing machines. Journal of Symbolic Logic 65(2), 567–604.MathSciNetzbMATHGoogle Scholar
  25. Hogarth, M. L. (1992). Does general relativity allow an observer to view an eternity in a finite time?. Foundations of Physics Letters 5(2), 173–181.CrossRefMathSciNetADSGoogle Scholar
  26. Hopcroft, J. E., Motwani, R., and Ullman, J. D. (2001). Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, MA.Google Scholar
  27. Kieu, T. (2003). Computing the non-computable. Contemporary Physics 44(1), 51–71.CrossRefADSGoogle Scholar
  28. Kieu, T. (2003). Quantum algorithm for Hilbert's Tenth Problem. International Journal of Theoretical Physics 42, 1461–1478.CrossRefzbMATHMathSciNetGoogle Scholar
  29. Matiyasevich, Y. V. (1970). Enumerable sets are diophantine. Soviet Mathematics. Doklady 11, 354–357.zbMATHGoogle Scholar
  30. Ord, T. (2002). Hypercomputation: Computing more than the Turing machine, Honours Thesis, University of Melbourne, Melbourne; available from \verb!!Google Scholar
  31. Odifreddi, P. (1989). Classical Recursion Theory, North-Holland, Amsterdam.Google Scholar
  32. Shagrir, O. and Pitowsky, I. (2003). Physical hypercomputation and the Church–Turing Hypothesis. Minds and Machines 13, 87–101.CrossRefzbMATHGoogle Scholar
  33. Soare, R. I. (1987). Recursively Enumerable Sets and Degrees, Springer, Heidelberg.Google Scholar
  34. Turing, A. M. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society 42(2), 230–265.zbMATHGoogle Scholar
  35. Turing, A. M. (1939). Systems of logic based on ordinals. Proceedings of the London Mathematical Society 45, 161–228. %zbMATHGoogle Scholar
  36. Yao, A. C.-C. (2003). Classical physics and the Church–Turing Thesis. Journal of the Association for Computing Machinery 50(1), 100–105. Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.University of PaderbornPaderbornGermany

Personalised recommendations