International Journal of Theoretical Physics

, Volume 42, Issue 7, pp 1461–1478 | Cite as

Quantum Algorithm for Hilbert's Tenth Problem

  • Tien D Kieu


We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain Hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the Church—Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles.

quantum algorithms computability quantum adiabatic computation hypercomputation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Benioff, P. (1980). The computer as a physical system. Journal of Statistical Physics 22, 563-591.Google Scholar
  2. Bernstein, E. and Vazirani, U. (1997). Quantum complexity theory. SIAM Journal of Computing 26, 1411.Google Scholar
  3. Braunstein, S. (1998). Error correction for continuous variables. Physical Review Letters 80, 4084.Google Scholar
  4. Calude, C. S. and Pavlov, B. (2001). Coins, quantum measurements and Turing's barrier. Preprint quant-ph/0112087.Google Scholar
  5. Childs, A. M., Farhi, E., and Preskill, J. (2001). Robusteness of adiabatic quantum computation. Preprint quant-ph/0108048.Google Scholar
  6. Davis, M. (1982). Computability and Unsolvability, Dover, New York.Google Scholar
  7. Etesi, G. and Németi, I. (2001). Non-Turing computations via Malament-Hogarth space-times, gr-qc/0104023.Google Scholar
  8. Farhi, E., Goldstone, J., Gutmann, S., and Sipser, M. (2000). Quantum computation by adiabatic evolution. Preprint quant-ph/0001106.Google Scholar
  9. Feynman, R. P. (1982). Simulating physics with computers. International Journal of Theoretical Physics 21, 467.Google Scholar
  10. Grover, L. K. (1997). Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters 79, 325-328.Google Scholar
  11. Itzykson, C. and Zuber, J.-B. (1985). Quantum Field Theory, McGraw-Hill, New York.Google Scholar
  12. Kadowaki, T. and Nishimori, H. (1998). Quantum annealing in the transverse Ising model. Physical Review E 58, 5355.Google Scholar
  13. Kieu, T. D. (2001a). A reformulation of the Hilbert's tenth problem through Quantum Mechanics. Preprint quant-ph/0111063.Google Scholar
  14. Kieu, T. D. (2001b). Gödel's Indompleteness, Chaitin's ω and Quantum Physics, quant-ph/0111062.Google Scholar
  15. Kieu, T. D. (2002). Computing the noncomputable. Contemporary Physics 44, 51-71.Google Scholar
  16. Kieu, T. D. (2003). Numerical simulations of a quantum algorithm for Hilbert's tenth problem. Preprint quant-ph/0304114.Google Scholar
  17. Lloyd, S. and Braunstein, S. L. (1999). Quantum computation over continuous variables. Physical Review Letters 82, 1784.Google Scholar
  18. Matiyasevich, Y. V. (1993). Hilbert's Tenth Problem, MIT Press, Cambridge, MA.Google Scholar
  19. Nielsen, M. A. (1997). Computable functions, quantum measurements, and quantum dynamics, Physical Review Letters 79, 2915-2918.Google Scholar
  20. Nielsen, M. and Chuang, I. L. (2000). Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK.Google Scholar
  21. Ord, T. and Kieu, T. D. (2003). The diagonal method and hypercomputation. Preprint math-LO/0307020.Google Scholar
  22. Ozawa, M. (1998). Measurability and computability. Preprint quant-ph/9809048.Google Scholar
  23. Renyi, A. (1970). Probability Theory, North-Holland, New York.Google Scholar
  24. Rogers, H., Jr. (1987). Theory of Recursive Functions and Effective Computability, MIT Press, Cambridge, MA.Google Scholar
  25. Ruskai, M. B. (2002). Comments on adiabatic quantum algorithms. Preprint quant-ph/0203127.Google Scholar
  26. Shor, P. W. (1997). Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal of Computing 26, 1484-1509.Google Scholar
  27. Wooters, W. K. and Zurek, W. H. (1982). A single quantum cannot be cloned. Nature 299, 802-803.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Tien D Kieu
    • 1
  1. 1.Centre for Atom Optics and Ultrafast SpectroscopySwinburne University of TechnologyHawthornAustralia

Personalised recommendations