Abstract
We discuss the simulation of complex dynamical systems on a quantum computer. We show that a quantum computer can be used to efficiently extract relevant physical information. It is possible to simulate the dynamical localization of classical chaos and extract the localization length with quadratic speed up with respect to any known classical computation. We can also compute with algebraic speed up the diffusion coefficient and the diffusion exponent, both in the regimes of Brownian and anomalous diffusion. Finally, we show that it is possible to extract the fidelity of the quantum motion, which measures the stability of the system under perturbations, with exponential speed up. The so-called quantum sawtooth map model is used as a test bench to illustrate these results.
PACS: 03.67.Lx, 05.45.Mt
Similar content being viewed by others
REFERENCES
R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
S. Lloyd, Science 273, 1073 (1996).
G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Phys. Rev. A 64, 022319 (2001).
R. Schack, Phys. Rev. A 57, 1634 (1998).
B. Georgeot and D. L. Shepelyansky, Phys. Rev. Lett. 86, 2890 (2001).
G. Benenti, G. Casati, S. Montangero, and D. L. Shepelyansky, Phys. Rev. Lett. 87, 227901 (2001).
G. Benenti, G. Casati, S. Montangero, and D. L. Shepelyansky, Phys. Rev. A 67, 052312 (2003).
I. Dana, N. W. Murray, and I. C. Percival, Phys. Rev. Lett. 62, 233 (1989).
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
G. Casati, B. V. Chirikov, J. Ford, and F. M. Izrailev, Lecture Notes Phys. 93, 334 (1979); for a review see, e.g., F.M. Izrailev, Phys. Rep. 196, 299 (1990).
S. Fishman, D. R. Grempel, and R. E. Prange, Phys. Rev. Lett. 49, 509 (1982).
P. M. Koch and K. A. H. van Leeuwen, Phys. Rep. 255, 289 (1995), and references therein.
F. L. Moore, J. C. Robinson, C. F. Barucha, B. Sundaram, and M. G. Raizen, Phys. Rev. Lett. 75, 4598 (1995); H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen, ibid. 80, 4111 (1998); D. A. Steck, W. H. Oskay, and M. G. Raizen, ibid. 88, 120406 (2002).
A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, Phys. Rev.E 54, 3221 (1996).
B. V. Chirikov, F. M. Izrailev, and D.L. Shepelyansky, Sov. Sci. Rev. C 2, 209 (1981).
Y. S. Weinstein, S. Lloyd, J. Emerson, and D. G. Cory Phys. Rev. Lett. 89, 157902 (2002).
L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, Nature 414, 883 (2001).
S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. H¨affner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt, Nature 421, 48 (2003).
A. Peres, Phys. Rev. A 30, 1610 (1984).
R. A. Jalabert and H. M. Pastawski, Phys. Rev. Lett. 86, 2490 (2001).
Ph. Jacquod, P. G. Silvestrov, and C. W. J. Beenakker, Phys. Rev. E 64, 055203(R) (2001).
N. R. Cerruti and S. Tomsovic, Phys. Rev. Lett. 88, 054103 (2002).
G. Benenti and G. Casati, Phys. Rev. E 65, 066205 (2002).
T. Prosen and M. ? Znidari?c, J. Phys. A 35, 1455 (2002).
J. Emerson, Y. S. Weinstein, S. Lloyd, and D. Cory, Phys. Rev. Lett. 89, 284102 (2002).
F. M. Cucchietti, D. A. R. Dalvit, J. P. Paz, and W. H. Zurek, Phys. Rev. Lett. 91, 210403 (2003).
S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 79, 4790 (1997).
C. Miquel, J. P. Paz, M. Saraceno, E. Knill, R. Laflamme, and C. Negrevergne, Nature 418, 59 (2002).
E. Ott, T. M. Antonsen, Jr., and J. D. Hanson, Phys. Rev. Lett. 53, 2187 (1984).
P. H. Song and D. L. Shepelyansky, Phys. Rev. Lett. 86, 2162 (2001).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Benenti, G., Casati, G. & Montangero, S. Quantum Computing and Information Extraction for Dynamical Quantum Systems. Quantum Information Processing 3, 273–293 (2004). https://doi.org/10.1007/s11128-004-0415-2
Issue Date:
DOI: https://doi.org/10.1007/s11128-004-0415-2