Quantum Principles and Mathematical Computability

  • Tien D. Kieu


Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical “algorithm” for one of the insoluble problems of mathematics, the Hilbert's tenth and equivalently the Turing halting problem. The key element of this algorithm is the computability and measurability of both the values of physical observables and of the quantum-mechanical probability distributions for these values.


quantum computation computability Hilbert's tenth problem Turing halting problem 


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  1. Bernstein, E. and Vazirani, U. (1997). Quantum complexity theory. SIAM Journal of Computing 26, 1411.CrossRefMathSciNetGoogle Scholar
  2. Calude, C. S. and Paun, G. (2001). Computing with Cells and Atoms, Taylor and Francis, London.Google Scholar
  3. Calude, C. S. and Pavlov, B. (2001). Coins, Quantum Measurements, and Turing's Barrier, Archive quant-ph/0112087.Google Scholar
  4. Chaitin, G. J. (1992). Algorithmic Information Theory, Cambridge University Press, Cambridge, UK.Google Scholar
  5. Davis, M. (1982). Computability and Unsolvability, Dover, New York.Google Scholar
  6. Deutsch, D. (1977). The Fabric of Reality, Penguin, New York.Google Scholar
  7. Deutsch, D., Ekert, A., and Lupacchini, R. (2000). Machines, logic, and quantum physics. Bulletin on Symbolic Logic 6, 265–283. See also a review of this paper by E. Knill on http://quickreviews. org/cgi/display.cgi?reviewID = knill.bsl.6.265Google Scholar
  8. Geroch, R. and Hartle, J. B. (1986). Journal of Foundations in Physics 16, 533–550.MathSciNetGoogle Scholar
  9. Kieu, T. D. (2003a). Quantum Algorithm for Hilbert's Tenth Problem. International Journal for Theoretical Physics 42, 1451–1468.MathSciNetGoogle Scholar
  10. Kieu, T. D. (2003b). Computing the Noncomputable, Contemporary Physics 44, 51–77.ADSGoogle Scholar
  11. Kieu, T. D. (2004). A Reformulation of the Hilbert's Tenth Problem Through Quantum Mechanics. Proc. Roy. Soc. A 460, 1535–1545.ADSMATHMathSciNetGoogle Scholar
  12. Kieu, T. D. (2005a). Hypercomputability of quantum adiabatic processes: Fact versus Prejudices. Archive quant-ph/0504101.Google Scholar
  13. Kieu T. D. (2005b). Mathematical computability questions for some classes of linear and non-linear differential equations originated from Hilbert's tenth problem. Archive math. GM/0507109.Google Scholar
  14. Lewis, H. R. and Papadimitriou, C. H. (1981). Elements of the Theory of Computation, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
  15. Matiyasevich, Y. V. (1993). Hilbert's Tenth Problem, MIT Press, Cambridge, MA.Google Scholar
  16. Omnes, R., (1994). The Interpretation of Quantum Mechanics, Princeton University Press, Princeton, NJ.Google Scholar
  17. Pour-El, M. and Richards, I. (1989). Computability in Analysis and Physics, Springer-Verlag, New York.Google Scholar
  18. Siegelmann, H. T. (1995). Computation beyond the Turing limit. Science 268, 545–548.ADSGoogle Scholar
  19. Stannett, M. (2001). Hypercomputation Is Experimentally Irrefutable, Tech. Report CS-01-04, Department of Computer Science, Sheffield University, Sheffield, UK.Google Scholar
  20. Tarski, A. (1951). A Decision Method for Elementary Algebra and Geometry, University of California Press, Los Angeles, CA.Google Scholar
  21. von Neumann, J. (1983). Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, NJ.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Centre for Atom Optics and Ultrafast SpectroscopySwinburne University of TechnologyHawthornAustralia

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