## Implementation of Deutsch-Jozsa algorithm and determination of value of function via Rydberg blockade |

Optics Express, Vol. 19, Issue 3, pp. 2037-2045 (2011)

http://dx.doi.org/10.1364/OE.19.002037

Acrobat PDF (1004 KB)

### Abstract

We propose an efficient scheme in which the Deutsch-Jozsa algorithm can be realized via Rydberg blockade interaction. Deutsch-Jozsa algorithm can fast determine whether function is constant or balanced, but this algorithm does not give the concrete value of function. Using the Rydberg blockade, value of function may be determined in our scheme. According to the quantitative calculation of Rydberg blockade, we discuss the experimental feasibility of our scheme.

© 2011 OSA

## 1. Introduction

1. P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Comput. **26**(5), 1484–1509 (1997). [CrossRef]

2. L. K. Grover, “Quantum Computers Can Search Rapidly by Using Almost Any Transformation,” Phys. Rev. Lett. **80**(19), 4329–4332 (1998). [CrossRef]

3. D. Deutsch and R. Jozsa, “Rapid solution of problems by quantum computation,” Proc. R. Soc. Lond. A **439**(1907), 553–558 (1992). [CrossRef]

*x*, where

4. I. L. Chuang, I. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature **393**(6681), 143–146 (1998). [CrossRef]

5. J. A. Jones and M. Mosca, “Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer,” J. Chem. Phys. **109**(5), 1648–1653 (1998). [CrossRef]

6. M. Mohseni, J. S. Lundeen, K. J. Resch, and A. M. Steinberg, “Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm,” Phys. Rev. Lett. **91**(18), 187903 (2003). [CrossRef] [PubMed]

7. F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. **105**(4), 040504 (2010). [CrossRef] [PubMed]

8. S.-B. Zheng, “Scheme for implementing the Deutsch-Jozsa algorithm in cavity QED,” Phys. Rev. A **70**(3), 034301 (2004). [CrossRef]

9. L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. **104**(1), 010503 (2010). [CrossRef] [PubMed]

16. Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Preparation of multiparty entangled states using pairwise perfectly efficient single-probe photon four-wave mixing,” Phys. Rev. A **69**(6), 063803 (2004). [CrossRef]

9. L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. **104**(1), 010503 (2010). [CrossRef] [PubMed]

10. T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. **104**(1), 010502 (2010). [CrossRef] [PubMed]

11. M. Saffman and K. Mølmer, “Efficient multiparticle entanglement via asymmetric Rydberg blockade,” Phys. Rev. Lett. **102**(24), 240502 (2009). [CrossRef] [PubMed]

12. H.-Z. Wu, Z.-B. Yang, and S.-B. Zheng, “Implementation of a multiqubit quantum phase gate in a neutral atomic ensemble via the asymmetric Rydberg blockade,” Phys. Rev. A **82**(3), 034307 (2010). [CrossRef]

17. E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. **5**(2), 110–114 (2009). [CrossRef]

9. L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. **104**(1), 010503 (2010). [CrossRef] [PubMed]

## 2. Implementation of two-qubit Deutsch-Jozsa Algorithm

17. E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. **5**(2), 110–114 (2009). [CrossRef]

*i*and

*a*stand for an input query qubit and an auxiliary qubit, respectively. Considering the transition between internal levels

*π*on the state

*π*pulse to couple states

19. C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Many-body theory of excitation dynamics in an ultracold Rydberg gas,” Phys. Rev. A **76**(1), 013413 (2007). [CrossRef]

*π*pulse leads to

## 3. Determination of value of function

10. T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. **104**(1), 010502 (2010). [CrossRef] [PubMed]

## 4. Conclusion

^{87}Rb atom. Rydberg blockade between two rubidium atoms localized in spatially separated trapping sites has been observed [17

17. E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. **5**(2), 110–114 (2009). [CrossRef]

**104**(1), 010503 (2010). [CrossRef] [PubMed]

20. T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. **100**(11), 113003 (2008). [CrossRef] [PubMed]

19. C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Many-body theory of excitation dynamics in an ultracold Rydberg gas,” Phys. Rev. A **76**(1), 013413 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Comput. |

2. | L. K. Grover, “Quantum Computers Can Search Rapidly by Using Almost Any Transformation,” Phys. Rev. Lett. |

3. | D. Deutsch and R. Jozsa, “Rapid solution of problems by quantum computation,” Proc. R. Soc. Lond. A |

4. | I. L. Chuang, I. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature |

5. | J. A. Jones and M. Mosca, “Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer,” J. Chem. Phys. |

6. | M. Mohseni, J. S. Lundeen, K. J. Resch, and A. M. Steinberg, “Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm,” Phys. Rev. Lett. |

7. | F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. |

8. | S.-B. Zheng, “Scheme for implementing the Deutsch-Jozsa algorithm in cavity QED,” Phys. Rev. A |

9. | L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. |

10. | T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. |

11. | M. Saffman and K. Mølmer, “Efficient multiparticle entanglement via asymmetric Rydberg blockade,” Phys. Rev. Lett. |

12. | H.-Z. Wu, Z.-B. Yang, and S.-B. Zheng, “Implementation of a multiqubit quantum phase gate in a neutral atomic ensemble via the asymmetric Rydberg blockade,” Phys. Rev. A |

13. | B. Zhao, M. Müller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic Rydberg gates,” Phys. Rev. Lett. |

14. | D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. |

15. | M. Müller, I. Lesanovsky, H. Weimer, H. P. Büchler, and P. Zoller, “Mesoscopic Rydberg gate based on electromagnetically induced transparency,” Phys. Rev. Lett. |

16. | Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Preparation of multiparty entangled states using pairwise perfectly efficient single-probe photon four-wave mixing,” Phys. Rev. A |

17. | E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. |

18. | M. O. Scully, and M. S. Zubairy, |

19. | C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Many-body theory of excitation dynamics in an ultracold Rydberg gas,” Phys. Rev. A |

20. | T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: November 15, 2010

Revised Manuscript: December 18, 2010

Manuscript Accepted: January 9, 2011

Published: January 19, 2011

**Citation**

Aixi Chen, "Implementation of Deutsch-Jozsa algorithm and determination of value of function via Rydberg blockade," Opt. Express **19**, 2037-2045 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2037

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### References

- P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Comput. 26(5), 1484–1509 (1997). [CrossRef]
- L. K. Grover, “Quantum Computers Can Search Rapidly by Using Almost Any Transformation,” Phys. Rev. Lett. 80(19), 4329–4332 (1998). [CrossRef]
- D. Deutsch and R. Jozsa, “Rapid solution of problems by quantum computation,” Proc. R. Soc. Lond. A 439(1907), 553–558 (1992). [CrossRef]
- I. L. Chuang, I. M. K. Vandersypen, X. Zhou, D. W. Leung, and S. Lloyd, “Experimental realization of a quantum algorithm,” Nature 393(6681), 143–146 (1998). [CrossRef]
- J. A. Jones and M. Mosca, “Implementation of a quantum algorithm on a nuclear magnetic resonance quantum computer,” J. Chem. Phys. 109(5), 1648–1653 (1998). [CrossRef]
- M. Mohseni, J. S. Lundeen, K. J. Resch, and A. M. Steinberg, “Experimental application of decoherence-free subspaces in an optical quantum-computing algorithm,” Phys. Rev. Lett. 91(18), 187903 (2003). [CrossRef] [PubMed]
- F. Shi, X. Rong, N. Xu, Y. Wang, J. Wu, B. Chong, X. Peng, J. Kniepert, R. S. Schoenfeld, W. Harneit, M. Feng, and J. Du, “Room-temperature implementation of the Deutsch-Jozsa algorithm with a single electronic spin in diamond,” Phys. Rev. Lett. 105(4), 040504 (2010). [CrossRef] [PubMed]
- S.-B. Zheng, “Scheme for implementing the Deutsch-Jozsa algorithm in cavity QED,” Phys. Rev. A 70(3), 034301 (2004). [CrossRef]
- L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104(1), 010503 (2010). [CrossRef] [PubMed]
- T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104(1), 010502 (2010). [CrossRef] [PubMed]
- M. Saffman and K. Mølmer, “Efficient multiparticle entanglement via asymmetric Rydberg blockade,” Phys. Rev. Lett. 102(24), 240502 (2009). [CrossRef] [PubMed]
- H.-Z. Wu, Z.-B. Yang, and S.-B. Zheng, “Implementation of a multiqubit quantum phase gate in a neutral atomic ensemble via the asymmetric Rydberg blockade,” Phys. Rev. A 82(3), 034307 (2010). [CrossRef]
- B. Zhao, M. Müller, K. Hammerer, and P. Zoller, “Efficient quantum repeater based on deterministic Rydberg gates,” Phys. Rev. Lett. 81, 052329 (2010).
- D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000). [CrossRef] [PubMed]
- M. Müller, I. Lesanovsky, H. Weimer, H. P. Büchler, and P. Zoller, “Mesoscopic Rydberg gate based on electromagnetically induced transparency,” Phys. Rev. Lett. 102(17), 170502 (2009). [CrossRef] [PubMed]
- Y. Wu, M. G. Payne, E. W. Hagley, and L. Deng, “Preparation of multiparty entangled states using pairwise perfectly efficient single-probe photon four-wave mixing,” Phys. Rev. A 69(6), 063803 (2004). [CrossRef]
- E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5(2), 110–114 (2009). [CrossRef]
- M. O. Scully, and M. S. Zubairy, Quantum Optics (Cambridge: Cambridge University Press,1997), Chap.5.
- C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Many-body theory of excitation dynamics in an ultracold Rydberg gas,” Phys. Rev. A 76(1), 013413 (2007). [CrossRef]
- T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole-dipole atomic interactions,” Phys. Rev. Lett. 100(11), 113003 (2008). [CrossRef] [PubMed]

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