Quantum Algorithm for Hilbert's Tenth Problem Article

DOI :
10.1023/A:1025780028846

Cite this article as: Kieu, T.D. International Journal of Theoretical Physics (2003) 42: 1461. doi:10.1023/A:1025780028846
Abstract 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

REFERENCES Benioff, P. (1980). The computer as a physical system.

Journal of Statistical Physics
22 , 563-591.

Google Scholar Bernstein, E. and Vazirani, U. (1997). Quantum complexity theory.

SIAM Journal of Computing
26 , 1411.

Google Scholar Braunstein, S. (1998). Error correction for continuous variables.

Physical Review Letters
80 , 4084.

Google Scholar Calude, C. S. and Pavlov, B. (2001). Coins, quantum measurements and Turing's barrier. Preprint quant-ph/0112087.

Childs, A. M., Farhi, E., and Preskill, J. (2001). Robusteness of adiabatic quantum computation. Preprint quant-ph/0108048.

Davis, M. (1982).

Computability and Unsolvability , Dover, New York.

Google Scholar Etesi, G. and Németi, I. (2001). Non-Turing computations via Malament-Hogarth space-times, gr-qc/0104023.

Farhi, E., Goldstone, J., Gutmann, S., and Sipser, M. (2000). Quantum computation by adiabatic evolution. Preprint quant-ph/0001106.

Feynman, R. P. (1982). Simulating physics with computers.

International Journal of Theoretical Physics
21 , 467.

Google Scholar Grover, L. K. (1997). Quantum mechanics helps in searching for a needle in a haystack.

Physical Review Letters
79 , 325-328.

Google Scholar Itzykson, C. and Zuber, J.-B. (1985).

Quantum Field Theory , McGraw-Hill, New York.

Google Scholar Kadowaki, T. and Nishimori, H. (1998). Quantum annealing in the transverse Ising model.

Physical Review E
58 , 5355.

Google Scholar Kieu, T. D. (2001a). A reformulation of the Hilbert's tenth problem through Quantum Mechanics. Preprint quant-ph/0111063.

Kieu, T. D. (2001b). Gödel's Indompleteness, Chaitin's ω and Quantum Physics, quant-ph/0111062.

Kieu, T. D. (2002). Computing the noncomputable.

Contemporary Physics
44 , 51-71.

Google Scholar Kieu, T. D. (2003). Numerical simulations of a quantum algorithm for Hilbert's tenth problem. Preprint quant-ph/0304114.

Lloyd, S. and Braunstein, S. L. (1999). Quantum computation over continuous variables.

Physical Review Letters
82 , 1784.

Google Scholar Matiyasevich, Y. V. (1993).

Hilbert's Tenth Problem , MIT Press, Cambridge, MA.

Google Scholar Nielsen, M. A. (1997). Computable functions, quantum measurements, and quantum dynamics,

Physical Review Letters
79 , 2915-2918.

Google Scholar Nielsen, M. and Chuang, I. L. (2000).

Quantum Computation and Quantum Information , Cambridge University Press, Cambridge, UK.

Google Scholar Ord, T. and Kieu, T. D. (2003). The diagonal method and hypercomputation. Preprint math-LO/0307020.

Ozawa, M. (1998). Measurability and computability. Preprint quant-ph/9809048.

Renyi, A. (1970).

Probability Theory , North-Holland, New York.

Google Scholar Rogers, H., Jr. (1987).

Theory of Recursive Functions and Effective Computability , MIT Press, Cambridge, MA.

Google Scholar Ruskai, M. B. (2002). Comments on adiabatic quantum algorithms. Preprint quant-ph/0203127.

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 Wooters, W. K. and Zurek, W. H. (1982). A single quantum cannot be cloned.

Nature
299 , 802-803.

Google Scholar © Plenum Publishing Corporation 2003

Authors and Affiliations 1. Centre for Atom Optics and Ultrafast Spectroscopy Swinburne University of Technology Hawthorn Australia