International Journal of Theoretical Physics

, Volume 47, Issue 6, pp 1641–1653 | Cite as

Quantum Computation in a Ising Spin Chain Taking into Account Second Neighbor Couplings

  • G. V. López
  • T. Gorin
  • L. Lara


We consider the realization of a quantum computer in a chain of nuclear spins coupled by an Ising interaction. Quantum algorithms can be performed with the help of appropriate radio-frequency pulses. In addition to the standard nearest-neighbor Ising coupling, we also allow for a second neighbor coupling. It is shown, how to apply the 2π k method in this more general setting, where the additional coupling eventually allows to save a few pulses. We illustrate our results with two numerical simulations: the Shor prime factorization of the number 4 and the teleportation of a qubit along a chain of 3 qubits. In both cases, the optimal Rabi frequency (to suppress non-resonant effects) depends primarily on the strength of the second neighbor interaction.


Quantum computer Ising Quantum information Chain of spins Second neighbor Shor’s factorization Teleportation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bennett, C.H., Brassard, G., Crépeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993) CrossRefADSMathSciNetMATHGoogle Scholar
  2. 2.
    Berman, G.P., Doolen, G.D., Holm, D.D., Tsifrinovich, V.I.: Quantum computer on a class of one-dimensional Ising systems. Phys. Lett. A 193, 444–450 (1994) CrossRefADSGoogle Scholar
  3. 3.
    Berman, G.P., Doolen, G.D., López, G.V., Tsifrinovich, V.I.: Simulations of quantum-logic operations in a quantum computer with a large number of qubits. Phys. Rev. A 61, 062305 (2000) CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Berman, G.P., Borgonovi, F., Izrailev, F.M., Tsifrinovich, V.I.: Single-pulse preparation of the uniform superpositional state used in quantum algorithms. Phys. Lett. A 291, 232–236 (2001) CrossRefADSMathSciNetMATHGoogle Scholar
  5. 5.
    Berman, G.P., Kamenev, D.I., Doolen, G.D., López, G.V., Tsifrinovich, V.I.: Perturbation theory and numerical modeling of quantum logic operations with a large number of qubits. Contemp. Math. 305, 13–41 (2002) Google Scholar
  6. 6.
    Berman, G.P., López, G.V., Tsifrinovich, V.I.: Teleportation in a nuclear spin quantum computer. Phys. Rev. A 66, 042312 (2002) CrossRefADSGoogle Scholar
  7. 7.
    Berman, G.P., Kamenev, D.I., Kassman, R.B., Pineda, C., Tsifrinovich, V.I.: Method for implementation of universal quantum logic gates in a scalable Ising spin quantum computer. quant-ph/0212070 (2002) Google Scholar
  8. 8.
    Celardo, G.L., Pineda, C., Žnidarič, M.: Stability of the quantum Fourier transformation on the Ising quantum computer. Int. J. Quantum Inf. 3, 441–462 (2005) CrossRefMATHGoogle Scholar
  9. 9.
    García-Ripoll, J.J., Cirac, J.I.: Spin dynamics for bosons in an optical lattice. New J. Phys. 5(76), 1–13 (2003) Google Scholar
  10. 10.
    Gorin, T., Prosen, T., Seligman, T.H., Žnidarič, M.: Dynamics of Loschmidt echoes and fidelity decay. Phys. Rep. 435, 33–156 (2006) CrossRefADSGoogle Scholar
  11. 11.
    Jones, J.A.: NMR quantum computation: a critical evaluation. Fortschr. Phys. 48, 909–924 (2000) CrossRefGoogle Scholar
  12. 12.
    Lloyd, S.: A potentially realizable quantum computer. Science 261, 1569–1571 (1993) CrossRefADSGoogle Scholar
  13. 13.
    Lloyd, S.: Quantum-mechanical computers. Sci. Am. 273, 140–145 (1995) CrossRefGoogle Scholar
  14. 14.
    López, G.V., Quezada, J., Berman, G.P., Doolen, G.D., Tsifrinovich, V.I.: Numerical simulation of a quantum controlled-not gate implemented on four-spin molecules at room temperature. J. Opt. B: Quantum Semiclass. Opt. 5, 184–189 (2003) CrossRefGoogle Scholar
  15. 15.
    López, G.V., Lara, L.: Numerical simulation of a controlled-controlled-not (CCN) quantum gate in a chain of three interacting nuclear spins system. J. Phys. B: At. Mol. Opt. Phys. 39, 3897–3904 (2006) CrossRefADSGoogle Scholar
  16. 16.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) MATHGoogle Scholar
  17. 17.
    Peres, A.: Stability of quantum motion in chaotic and regular systems. Phys. Rev. A 30, 1610–1615 (1984) CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Schumacher, B.: Quantum coding. Phys. Rev. A 51, 2738–2747 (1995) CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. IEEE Computer Society, Los Alamitos (1994) CrossRefGoogle Scholar
  20. 20.
    Slichter, C.P.: Principles of Magnetic Resonance, 3rd edn. Springer, Berlin (1996) Google Scholar
  21. 21.
    Vandersypen, L.M.K., Steffen, M., Breyta, G., Yannoni, C.S., Sherwood, M.H., Chuang, I.L.: Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidad de GuadalajaraGuadalajaraMexico

Personalised recommendations