Abstract

We propose a potentially practical scheme for realization of an n-qubit (n>2) conditional phase flip (CPF) gate and implementation of Grover’s search algorithm in an ion-trap system. We demonstrate, both analytically and numerically, that our scheme could be achieved efficiently to find a marked state with high fidelity and high success probability. We also show the merits of the proposal that the increase of the ion number can improve the fidelity and success probability of the CPF gate. The required operations for a Grover search are very close to the capabilities of current ion-trap techniques.

© 2008 Optical Society of America

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  1. L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. 79, 325-328 (1997).
    [CrossRef]
  2. D. Daems and S. Guérin, “Adiabatic quantum search scheme with atoms in a cavity driven by lasers,” Phys. Rev. Lett. 99, 170503 (2007).
    [CrossRef] [PubMed]
  3. J. Roland and N. J. Cerf, “Quantum search by local adiabatic evolution,” Phys. Rev. A 65, 042308 (2002).
    [CrossRef]
  4. R. Schützhold and G. Schaller, “Adiabatic quantum algorithms as quantum phase transitions: First versus second order,” Phys. Rev. A 74, 060304(R) (2006).
    [CrossRef]
  5. M. Stewart Siu, “Adiabatic rotation, quantum search, and preparation of superposition states,” Phys. Rev. A 75, 062337 (2007).
    [CrossRef]
  6. A. Pérez and A. Romanelli, “Nonadiabatic quantum search algorithms,” Phys. Rev. A 76, 052318 (2007).
    [CrossRef]
  7. D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, “Sufficiency criterion for the validity of the adiabatic approximation,” Phys. Rev. Lett. 98, 150402 (2007).
    [CrossRef] [PubMed]
  8. Z. Wei and M. Ying, “Quantum adiabatic computation and adiabatic conditions,” Phys. Rev. A 76, 024304 (2007).
    [CrossRef]
  9. H. Azuma, “Higher-order perturbation theory for decoherence in Grover's algorithm,” Phys. Rev. A 72, 042305 (2005).
    [CrossRef]
  10. M. Tiersch and R. Schützhold, “Non-Markovian decoherence in the adiabatic quantum search algorithm,” Phys. Rev. A 75, 062313 (2007).
    [CrossRef]
  11. G.-L. Long, Y.-S. Li, W.-L. Zhang, and C.-C. Tu, “Dominant gate imperfection in Grover's quantum search algorithm,” Phys. Rev. A 61, 042305 (2000).
    [CrossRef]
  12. B. Pablo-Norman and M. Ruiz-Altaba, “Noise in Grover's quantum search algorithm,” Phys. Rev. A 61, 012301 (1999).
    [CrossRef]
  13. D. Shapira, S. Mozes, and O. Biham, “The effect of unitary noise on Grover's quantum search algorithm,” e-print arXiv:quant-ph/0307142.
  14. P. J. Salas, “Noise effect on Grover algorithm,” e-print arXiv:quant-ph/0801.1261.
  15. J. A. Jones, M. Mosca, and R. H. Hansen, “Implementation of a quantum search algorithm on a quantum computer,” Nature (London) 393, 344-346 (1998).
    [CrossRef]
  16. I. L. Chuang, N. Gershenfeld, and M. Kubinec, “Experimental implementation of fast quantum searching,” Phys. Rev. Lett. 80, 3408-3411 (1998).
    [CrossRef]
  17. P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
    [CrossRef]
  18. P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, “Grover's search algorithm: An optical approach,” J. Mod. Opt. 47, 257-266 (2000).
    [CrossRef]
  19. J. L. Dodd, T. C. Ralph, and G. J. Milburn, “Experimental requirements for Grover's algorithm in optical quantum computation,” Phys. Rev. A 68, 042328 (2003).
    [CrossRef]
  20. M. Feng, “Grover search with pairs of trapped ions,” Phys. Rev. A 63, 052308 (2001).
    [CrossRef]
  21. S. Fujiwara and S. Hasegawa, “General method for realizing the conditional phase-shift gate and a simulation of Grover's algorithm in an ion-trap system,” Phys. Rev. A 71, 012337 (2005).
    [CrossRef]
  22. K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, “Implementation of Grover's quantum search algorithm in a scalable system,” Phys. Rev. A 72, 050306(R) (2005).
    [CrossRef]
  23. A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
    [CrossRef]
  24. F. Yamaguchi, P. Milman, M. Brune, J. M. Raimond, and S. Haroche, “Quantum search with two-atom collisions in cavity QED,” Phys. Rev. A 66, 010302(R) (2002).
    [CrossRef]
  25. W. L. Yang, C. Y. Chen, and M. Feng, “Implementation of three-qubit Grover search in cavity quantum electrodynamics,” Phys. Rev. A 76, 054301 (2007).
    [CrossRef]
  26. Z. J. Deng, M. Feng, and K. L. Gao, “Simple scheme for the two-qubit Grover search in cavity QED,” Phys. Rev. A 72, 034306 (2005).
    [CrossRef]
  27. A. Joshi and M. Xiao, “Three-qubit quantum-gate operation in a cavity QED system,” Phys. Rev. A 74, 052318 (2006).
    [CrossRef]
  28. Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature (London) 398, 786-788 (1999).
    [CrossRef]
  29. D. Vion, A. Aassime, and A. Cottet, “Manipulating the quantum state of an electrical circuit,” Science 296, 886-889 (2002).
    [CrossRef] [PubMed]
  30. M. S. Anwar, D. Blazina, H. A. Carteret, S. B. Duckett, and J. A. Jones, “Implementing Grover's quantum search on a para-hydrogen based pure state NMR quantum computer,” Chem. Phys. Lett. 400, 94-97 (2004).
    [CrossRef]
  31. K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835-1838 (1999).
    [CrossRef]
  32. A. Sørensen and K. Mølmer, “Quantum computation with ions in thermal motion,” Phys. Rev. Lett. 82, 1971-1974 (1999).
    [CrossRef]
  33. M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
    [CrossRef] [PubMed]
  34. S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett. 75, 346-349 (1995).
    [CrossRef] [PubMed]
  35. N. Schuch and J. Siewert, “Programmable networks for quantum algorithms,” Phys. Rev. Lett. 91, 027902 (2003).
    [CrossRef] [PubMed]
  36. For example, R. L. de Matos Filho and W. Vogel, “Even and odd coherent states of the motion of a trapped ion,” Phys. Rev. Lett. 76, 608-611 (1996) and W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes-Cummings dynamics of a trapped ion,” Phys. Rev. A 52, 4214-4217 (1995). For clarity of description below, we first assume a weak radiation without inducing AC Stark shift. So we consider in Eq. the resonant excitation involving motional quanta exchange. Actually, with a strong radiation, we could also do our job by carrier transition, as discussed later.
    [CrossRef] [PubMed]
  37. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390-393 (1997).
    [CrossRef]
  38. M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
    [CrossRef] [PubMed]
  39. C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
    [CrossRef] [PubMed]
  40. D. Jonathan, M. B. Plenio, and P. L. Knight, “Fast quantum gates for cold trapped ions,” Phys. Rev. A 62, 042307 (2000).
    [CrossRef]
  41. M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
    [CrossRef] [PubMed]
  42. H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
    [CrossRef]
  43. D. Kielpinski, C. Monroe, and D. J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature (London) 417, 709-711 (2002).
    [CrossRef]
  44. D. Leibfried, E. Knill, C. Ospelkaus, and D. J. Wineland, “Transport quantum logic gates for trapped ions,” Phys. Rev. A 76, 032324 (2007).
    [CrossRef]
  45. D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
    [CrossRef]
  46. R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188-5191 (2001).
    [CrossRef] [PubMed]

2007 (8)

D. Daems and S. Guérin, “Adiabatic quantum search scheme with atoms in a cavity driven by lasers,” Phys. Rev. Lett. 99, 170503 (2007).
[CrossRef] [PubMed]

M. Stewart Siu, “Adiabatic rotation, quantum search, and preparation of superposition states,” Phys. Rev. A 75, 062337 (2007).
[CrossRef]

A. Pérez and A. Romanelli, “Nonadiabatic quantum search algorithms,” Phys. Rev. A 76, 052318 (2007).
[CrossRef]

D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, “Sufficiency criterion for the validity of the adiabatic approximation,” Phys. Rev. Lett. 98, 150402 (2007).
[CrossRef] [PubMed]

Z. Wei and M. Ying, “Quantum adiabatic computation and adiabatic conditions,” Phys. Rev. A 76, 024304 (2007).
[CrossRef]

M. Tiersch and R. Schützhold, “Non-Markovian decoherence in the adiabatic quantum search algorithm,” Phys. Rev. A 75, 062313 (2007).
[CrossRef]

W. L. Yang, C. Y. Chen, and M. Feng, “Implementation of three-qubit Grover search in cavity quantum electrodynamics,” Phys. Rev. A 76, 054301 (2007).
[CrossRef]

D. Leibfried, E. Knill, C. Ospelkaus, and D. J. Wineland, “Transport quantum logic gates for trapped ions,” Phys. Rev. A 76, 032324 (2007).
[CrossRef]

2006 (2)

A. Joshi and M. Xiao, “Three-qubit quantum-gate operation in a cavity QED system,” Phys. Rev. A 74, 052318 (2006).
[CrossRef]

R. Schützhold and G. Schaller, “Adiabatic quantum algorithms as quantum phase transitions: First versus second order,” Phys. Rev. A 74, 060304(R) (2006).
[CrossRef]

2005 (6)

H. Azuma, “Higher-order perturbation theory for decoherence in Grover's algorithm,” Phys. Rev. A 72, 042305 (2005).
[CrossRef]

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Z. J. Deng, M. Feng, and K. L. Gao, “Simple scheme for the two-qubit Grover search in cavity QED,” Phys. Rev. A 72, 034306 (2005).
[CrossRef]

S. Fujiwara and S. Hasegawa, “General method for realizing the conditional phase-shift gate and a simulation of Grover's algorithm in an ion-trap system,” Phys. Rev. A 71, 012337 (2005).
[CrossRef]

K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, “Implementation of Grover's quantum search algorithm in a scalable system,” Phys. Rev. A 72, 050306(R) (2005).
[CrossRef]

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

2004 (4)

M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
[CrossRef] [PubMed]

M. S. Anwar, D. Blazina, H. A. Carteret, S. B. Duckett, and J. A. Jones, “Implementing Grover's quantum search on a para-hydrogen based pure state NMR quantum computer,” Chem. Phys. Lett. 400, 94-97 (2004).
[CrossRef]

M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
[CrossRef] [PubMed]

C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
[CrossRef] [PubMed]

2003 (3)

N. Schuch and J. Siewert, “Programmable networks for quantum algorithms,” Phys. Rev. Lett. 91, 027902 (2003).
[CrossRef] [PubMed]

J. L. Dodd, T. C. Ralph, and G. J. Milburn, “Experimental requirements for Grover's algorithm in optical quantum computation,” Phys. Rev. A 68, 042328 (2003).
[CrossRef]

D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
[CrossRef]

2002 (5)

D. Kielpinski, C. Monroe, and D. J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature (London) 417, 709-711 (2002).
[CrossRef]

J. Roland and N. J. Cerf, “Quantum search by local adiabatic evolution,” Phys. Rev. A 65, 042308 (2002).
[CrossRef]

M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
[CrossRef] [PubMed]

F. Yamaguchi, P. Milman, M. Brune, J. M. Raimond, and S. Haroche, “Quantum search with two-atom collisions in cavity QED,” Phys. Rev. A 66, 010302(R) (2002).
[CrossRef]

D. Vion, A. Aassime, and A. Cottet, “Manipulating the quantum state of an electrical circuit,” Science 296, 886-889 (2002).
[CrossRef] [PubMed]

2001 (2)

M. Feng, “Grover search with pairs of trapped ions,” Phys. Rev. A 63, 052308 (2001).
[CrossRef]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188-5191 (2001).
[CrossRef] [PubMed]

2000 (3)

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, “Grover's search algorithm: An optical approach,” J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

G.-L. Long, Y.-S. Li, W.-L. Zhang, and C.-C. Tu, “Dominant gate imperfection in Grover's quantum search algorithm,” Phys. Rev. A 61, 042305 (2000).
[CrossRef]

D. Jonathan, M. B. Plenio, and P. L. Knight, “Fast quantum gates for cold trapped ions,” Phys. Rev. A 62, 042307 (2000).
[CrossRef]

1999 (5)

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835-1838 (1999).
[CrossRef]

A. Sørensen and K. Mølmer, “Quantum computation with ions in thermal motion,” Phys. Rev. Lett. 82, 1971-1974 (1999).
[CrossRef]

Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature (London) 398, 786-788 (1999).
[CrossRef]

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
[CrossRef]

B. Pablo-Norman and M. Ruiz-Altaba, “Noise in Grover's quantum search algorithm,” Phys. Rev. A 61, 012301 (1999).
[CrossRef]

1998 (2)

J. A. Jones, M. Mosca, and R. H. Hansen, “Implementation of a quantum search algorithm on a quantum computer,” Nature (London) 393, 344-346 (1998).
[CrossRef]

I. L. Chuang, N. Gershenfeld, and M. Kubinec, “Experimental implementation of fast quantum searching,” Phys. Rev. Lett. 80, 3408-3411 (1998).
[CrossRef]

1997 (2)

L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. 79, 325-328 (1997).
[CrossRef]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390-393 (1997).
[CrossRef]

1996 (1)

For example, R. L. de Matos Filho and W. Vogel, “Even and odd coherent states of the motion of a trapped ion,” Phys. Rev. Lett. 76, 608-611 (1996) and W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes-Cummings dynamics of a trapped ion,” Phys. Rev. A 52, 4214-4217 (1995). For clarity of description below, we first assume a weak radiation without inducing AC Stark shift. So we consider in Eq. the resonant excitation involving motional quanta exchange. Actually, with a strong radiation, we could also do our job by carrier transition, as discussed later.
[CrossRef] [PubMed]

1995 (1)

S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett. 75, 346-349 (1995).
[CrossRef] [PubMed]

Aassime, A.

D. Vion, A. Aassime, and A. Cottet, “Manipulating the quantum state of an electrical circuit,” Science 296, 886-889 (2002).
[CrossRef] [PubMed]

Acton, M.

K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, “Implementation of Grover's quantum search algorithm in a scalable system,” Phys. Rev. A 72, 050306(R) (2005).
[CrossRef]

Anwar, M. S.

M. S. Anwar, D. Blazina, H. A. Carteret, S. B. Duckett, and J. A. Jones, “Implementing Grover's quantum search on a para-hydrogen based pure state NMR quantum computer,” Chem. Phys. Lett. 400, 94-97 (2004).
[CrossRef]

Aspelmeyer, M.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Azuma, H.

H. Azuma, “Higher-order perturbation theory for decoherence in Grover's algorithm,” Phys. Rev. A 72, 042305 (2005).
[CrossRef]

Barrett, M.

D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
[CrossRef]

Barrett, M. D.

M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
[CrossRef] [PubMed]

Becher, C.

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
[CrossRef] [PubMed]

Benhelm, J.

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
[CrossRef] [PubMed]

Ben-Kish, A.

D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
[CrossRef]

Bertet, P.

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
[CrossRef]

Biham, O.

D. Shapira, S. Mozes, and O. Biham, “The effect of unitary noise on Grover's quantum search algorithm,” e-print arXiv:quant-ph/0307142.

Blatt, R.

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
[CrossRef] [PubMed]

Blazina, D.

M. S. Anwar, D. Blazina, H. A. Carteret, S. B. Duckett, and J. A. Jones, “Implementing Grover's quantum search on a para-hydrogen based pure state NMR quantum computer,” Chem. Phys. Lett. 400, 94-97 (2004).
[CrossRef]

Bremner, M. J.

M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
[CrossRef] [PubMed]

Brickman, K.-A.

K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, “Implementation of Grover's quantum search algorithm in a scalable system,” Phys. Rev. A 72, 050306(R) (2005).
[CrossRef]

Briegel, H. J.

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188-5191 (2001).
[CrossRef] [PubMed]

Britton, J.

M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
[CrossRef] [PubMed]

D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
[CrossRef]

Brune, M.

F. Yamaguchi, P. Milman, M. Brune, J. M. Raimond, and S. Haroche, “Quantum search with two-atom collisions in cavity QED,” Phys. Rev. A 66, 010302(R) (2002).
[CrossRef]

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H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
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M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
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H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
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K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, “Implementation of Grover's quantum search algorithm in a scalable system,” Phys. Rev. A 72, 050306(R) (2005).
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J. A. Jones, M. Mosca, and R. H. Hansen, “Implementation of a quantum search algorithm on a quantum computer,” Nature (London) 393, 344-346 (1998).
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F. Yamaguchi, P. Milman, M. Brune, J. M. Raimond, and S. Haroche, “Quantum search with two-atom collisions in cavity QED,” Phys. Rev. A 66, 010302(R) (2002).
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A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
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M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
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S. Fujiwara and S. Hasegawa, “General method for realizing the conditional phase-shift gate and a simulation of Grover's algorithm in an ion-trap system,” Phys. Rev. A 71, 012337 (2005).
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M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
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M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
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D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
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D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
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M. S. Anwar, D. Blazina, H. A. Carteret, S. B. Duckett, and J. A. Jones, “Implementing Grover's quantum search on a para-hydrogen based pure state NMR quantum computer,” Chem. Phys. Lett. 400, 94-97 (2004).
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J. A. Jones, M. Mosca, and R. H. Hansen, “Implementation of a quantum search algorithm on a quantum computer,” Nature (London) 393, 344-346 (1998).
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M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
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D. Kielpinski, C. Monroe, and D. J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature (London) 417, 709-711 (2002).
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D. Jonathan, M. B. Plenio, and P. L. Knight, “Fast quantum gates for cold trapped ions,” Phys. Rev. A 62, 042307 (2000).
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D. Leibfried, E. Knill, C. Ospelkaus, and D. J. Wineland, “Transport quantum logic gates for trapped ions,” Phys. Rev. A 76, 032324 (2007).
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M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
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H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
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I. L. Chuang, N. Gershenfeld, and M. Kubinec, “Experimental implementation of fast quantum searching,” Phys. Rev. Lett. 80, 3408-3411 (1998).
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C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
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M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
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D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
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K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, “Implementation of Grover's quantum search algorithm in a scalable system,” Phys. Rev. A 72, 050306(R) (2005).
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D. Leibfried, E. Knill, C. Ospelkaus, and D. J. Wineland, “Transport quantum logic gates for trapped ions,” Phys. Rev. A 76, 032324 (2007).
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M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
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D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
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D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
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M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
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M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
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D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
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J. L. Dodd, T. C. Ralph, and G. J. Milburn, “Experimental requirements for Grover's algorithm in optical quantum computation,” Phys. Rev. A 68, 042328 (2003).
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F. Yamaguchi, P. Milman, M. Brune, J. M. Raimond, and S. Haroche, “Quantum search with two-atom collisions in cavity QED,” Phys. Rev. A 66, 010302(R) (2002).
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P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, “Grover's search algorithm: An optical approach,” J. Mod. Opt. 47, 257-266 (2000).
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K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, “Implementation of Grover's quantum search algorithm in a scalable system,” Phys. Rev. A 72, 050306(R) (2005).
[CrossRef]

D. Kielpinski, C. Monroe, and D. J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature (London) 417, 709-711 (2002).
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M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
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J. A. Jones, M. Mosca, and R. H. Hansen, “Implementation of a quantum search algorithm on a quantum computer,” Nature (London) 393, 344-346 (1998).
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D. Shapira, S. Mozes, and O. Biham, “The effect of unitary noise on Grover's quantum search algorithm,” e-print arXiv:quant-ph/0307142.

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Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature (London) 398, 786-788 (1999).
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M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
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A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
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D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, “Sufficiency criterion for the validity of the adiabatic approximation,” Phys. Rev. Lett. 98, 150402 (2007).
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M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
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A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
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D. Leibfried, E. Knill, C. Ospelkaus, and D. J. Wineland, “Transport quantum logic gates for trapped ions,” Phys. Rev. A 76, 032324 (2007).
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M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
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Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature (London) 398, 786-788 (1999).
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D. Jonathan, M. B. Plenio, and P. L. Knight, “Fast quantum gates for cold trapped ions,” Phys. Rev. A 62, 042307 (2000).
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J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390-393 (1997).
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F. Yamaguchi, P. Milman, M. Brune, J. M. Raimond, and S. Haroche, “Quantum search with two-atom collisions in cavity QED,” Phys. Rev. A 66, 010302(R) (2002).
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A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
[CrossRef]

Ralph, T. C.

J. L. Dodd, T. C. Ralph, and G. J. Milburn, “Experimental requirements for Grover's algorithm in optical quantum computation,” Phys. Rev. A 68, 042328 (2003).
[CrossRef]

Ramos, A.

M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
[CrossRef] [PubMed]

Rapol, U. D.

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

Rauschenbeutel, A.

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
[CrossRef]

Raussendorf, R.

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188-5191 (2001).
[CrossRef] [PubMed]

Resch, K. J.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Riebe, M.

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
[CrossRef] [PubMed]

Roland, J.

J. Roland and N. J. Cerf, “Quantum search by local adiabatic evolution,” Phys. Rev. A 65, 042308 (2002).
[CrossRef]

Romanelli, A.

A. Pérez and A. Romanelli, “Nonadiabatic quantum search algorithms,” Phys. Rev. A 76, 052318 (2007).
[CrossRef]

Roos, C. F.

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
[CrossRef] [PubMed]

Rosenband, T.

D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
[CrossRef]

Rowe, M.

D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
[CrossRef]

Rudolph, T.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Ruiz-Altaba, M.

B. Pablo-Norman and M. Ruiz-Altaba, “Noise in Grover's quantum search algorithm,” Phys. Rev. A 61, 012301 (1999).
[CrossRef]

Salas, P. J.

P. J. Salas, “Noise effect on Grover algorithm,” e-print arXiv:quant-ph/0801.1261.

Schaetz, T.

M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
[CrossRef] [PubMed]

Schaller, G.

R. Schützhold and G. Schaller, “Adiabatic quantum algorithms as quantum phase transitions: First versus second order,” Phys. Rev. A 74, 060304(R) (2006).
[CrossRef]

Schenck, E.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Schmidt, P. O.

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

Schmidt-Kaler, F.

C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
[CrossRef] [PubMed]

Schuch, N.

N. Schuch and J. Siewert, “Programmable networks for quantum algorithms,” Phys. Rev. Lett. 91, 027902 (2003).
[CrossRef] [PubMed]

Schützhold, R.

M. Tiersch and R. Schützhold, “Non-Markovian decoherence in the adiabatic quantum search algorithm,” Phys. Rev. A 75, 062313 (2007).
[CrossRef]

R. Schützhold and G. Schaller, “Adiabatic quantum algorithms as quantum phase transitions: First versus second order,” Phys. Rev. A 74, 060304(R) (2006).
[CrossRef]

Schwindt, P. D. D.

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, “Grover's search algorithm: An optical approach,” J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

Shapira, D.

D. Shapira, S. Mozes, and O. Biham, “The effect of unitary noise on Grover's quantum search algorithm,” e-print arXiv:quant-ph/0307142.

Siewert, J.

N. Schuch and J. Siewert, “Programmable networks for quantum algorithms,” Phys. Rev. Lett. 91, 027902 (2003).
[CrossRef] [PubMed]

Singh, K.

D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, “Sufficiency criterion for the validity of the adiabatic approximation,” Phys. Rev. Lett. 98, 150402 (2007).
[CrossRef] [PubMed]

Sørensen, A.

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835-1838 (1999).
[CrossRef]

A. Sørensen and K. Mølmer, “Quantum computation with ions in thermal motion,” Phys. Rev. Lett. 82, 1971-1974 (1999).
[CrossRef]

Stacey, D. N.

M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
[CrossRef] [PubMed]

Stacey, J.-P.

M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
[CrossRef] [PubMed]

Steane, A. M.

M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
[CrossRef] [PubMed]

Stewart Siu, M.

M. Stewart Siu, “Adiabatic rotation, quantum search, and preparation of superposition states,” Phys. Rev. A 75, 062337 (2007).
[CrossRef]

Tiersch, M.

M. Tiersch and R. Schützhold, “Non-Markovian decoherence in the adiabatic quantum search algorithm,” Phys. Rev. A 75, 062313 (2007).
[CrossRef]

Tong, D. M.

D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, “Sufficiency criterion for the validity of the adiabatic approximation,” Phys. Rev. Lett. 98, 150402 (2007).
[CrossRef] [PubMed]

Tsai, J. S.

Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature (London) 398, 786-788 (1999).
[CrossRef]

Tu, C.-C.

G.-L. Long, Y.-S. Li, W.-L. Zhang, and C.-C. Tu, “Dominant gate imperfection in Grover's quantum search algorithm,” Phys. Rev. A 61, 042305 (2000).
[CrossRef]

Vedral, V.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Vion, D.

D. Vion, A. Aassime, and A. Cottet, “Manipulating the quantum state of an electrical circuit,” Science 296, 886-889 (2002).
[CrossRef] [PubMed]

Vogel, W.

For example, R. L. de Matos Filho and W. Vogel, “Even and odd coherent states of the motion of a trapped ion,” Phys. Rev. Lett. 76, 608-611 (1996) and W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes-Cummings dynamics of a trapped ion,” Phys. Rev. A 52, 4214-4217 (1995). For clarity of description below, we first assume a weak radiation without inducing AC Stark shift. So we consider in Eq. the resonant excitation involving motional quanta exchange. Actually, with a strong radiation, we could also do our job by carrier transition, as discussed later.
[CrossRef] [PubMed]

Walther, P.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Webster, S. C.

M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
[CrossRef] [PubMed]

Wei, Z.

Z. Wei and M. Ying, “Quantum adiabatic computation and adiabatic conditions,” Phys. Rev. A 76, 024304 (2007).
[CrossRef]

Weinfurter, H.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

White, A. G.

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, “Grover's search algorithm: An optical approach,” J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

Wineland, D. J.

D. Leibfried, E. Knill, C. Ospelkaus, and D. J. Wineland, “Transport quantum logic gates for trapped ions,” Phys. Rev. A 76, 032324 (2007).
[CrossRef]

M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
[CrossRef] [PubMed]

D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
[CrossRef]

D. Kielpinski, C. Monroe, and D. J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature (London) 417, 709-711 (2002).
[CrossRef]

Xiao, M.

A. Joshi and M. Xiao, “Three-qubit quantum-gate operation in a cavity QED system,” Phys. Rev. A 74, 052318 (2006).
[CrossRef]

Yamaguchi, F.

F. Yamaguchi, P. Milman, M. Brune, J. M. Raimond, and S. Haroche, “Quantum search with two-atom collisions in cavity QED,” Phys. Rev. A 66, 010302(R) (2002).
[CrossRef]

Yang, W. L.

W. L. Yang, C. Y. Chen, and M. Feng, “Implementation of three-qubit Grover search in cavity quantum electrodynamics,” Phys. Rev. A 76, 054301 (2007).
[CrossRef]

Ying, M.

Z. Wei and M. Ying, “Quantum adiabatic computation and adiabatic conditions,” Phys. Rev. A 76, 024304 (2007).
[CrossRef]

Zeilinger, A.

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Zhang, W.-L.

G.-L. Long, Y.-S. Li, W.-L. Zhang, and C.-C. Tu, “Dominant gate imperfection in Grover's quantum search algorithm,” Phys. Rev. A 61, 042305 (2000).
[CrossRef]

Zoller, P.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390-393 (1997).
[CrossRef]

Chem. Phys. Lett. (1)

M. S. Anwar, D. Blazina, H. A. Carteret, S. B. Duckett, and J. A. Jones, “Implementing Grover's quantum search on a para-hydrogen based pure state NMR quantum computer,” Chem. Phys. Lett. 400, 94-97 (2004).
[CrossRef]

J. Mod. Opt. (1)

P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, “Grover's search algorithm: An optical approach,” J. Mod. Opt. 47, 257-266 (2000).
[CrossRef]

J. Phys. B (1)

D. Leibfried, B. DeMarco, V. Meyer, M. Rowe, A. Ben-Kish, M. Barrett, J. Britton, J. Hughes, W. M. Itano, B. M. Jelenkovic, C. Langer, D. Lucas, T. Rosenband, and D. J. Wineland, “Towards quantum information with trapped ions at NIST,” J. Phys. B 36, 599-612 (2003).
[CrossRef]

Nature (1)

M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, “Deterministic quantum teleportation of atomic qubits,” Nature 429, 737-739 (2004).
[CrossRef] [PubMed]

Nature (London) (5)

H. Häffner, W. Hänsel, C. F. Roos, J. Benhelm, D. Chek-alkar, M. Chwalla, T. Körber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, “Scalable multiparticle entanglement of trapped ions,” Nature (London) 438, 643-646 (2005).
[CrossRef]

D. Kielpinski, C. Monroe, and D. J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature (London) 417, 709-711 (2002).
[CrossRef]

J. A. Jones, M. Mosca, and R. H. Hansen, “Implementation of a quantum search algorithm on a quantum computer,” Nature (London) 393, 344-346 (1998).
[CrossRef]

Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature (London) 398, 786-788 (1999).
[CrossRef]

P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, “Experimental one-way quantum computing,” Nature (London) 434, 169-176 (2005).
[CrossRef]

Phys. Rev. A (19)

D. Jonathan, M. B. Plenio, and P. L. Knight, “Fast quantum gates for cold trapped ions,” Phys. Rev. A 62, 042307 (2000).
[CrossRef]

F. Yamaguchi, P. Milman, M. Brune, J. M. Raimond, and S. Haroche, “Quantum search with two-atom collisions in cavity QED,” Phys. Rev. A 66, 010302(R) (2002).
[CrossRef]

W. L. Yang, C. Y. Chen, and M. Feng, “Implementation of three-qubit Grover search in cavity quantum electrodynamics,” Phys. Rev. A 76, 054301 (2007).
[CrossRef]

Z. J. Deng, M. Feng, and K. L. Gao, “Simple scheme for the two-qubit Grover search in cavity QED,” Phys. Rev. A 72, 034306 (2005).
[CrossRef]

A. Joshi and M. Xiao, “Three-qubit quantum-gate operation in a cavity QED system,” Phys. Rev. A 74, 052318 (2006).
[CrossRef]

J. L. Dodd, T. C. Ralph, and G. J. Milburn, “Experimental requirements for Grover's algorithm in optical quantum computation,” Phys. Rev. A 68, 042328 (2003).
[CrossRef]

M. Feng, “Grover search with pairs of trapped ions,” Phys. Rev. A 63, 052308 (2001).
[CrossRef]

S. Fujiwara and S. Hasegawa, “General method for realizing the conditional phase-shift gate and a simulation of Grover's algorithm in an ion-trap system,” Phys. Rev. A 71, 012337 (2005).
[CrossRef]

K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, “Implementation of Grover's quantum search algorithm in a scalable system,” Phys. Rev. A 72, 050306(R) (2005).
[CrossRef]

J. Roland and N. J. Cerf, “Quantum search by local adiabatic evolution,” Phys. Rev. A 65, 042308 (2002).
[CrossRef]

R. Schützhold and G. Schaller, “Adiabatic quantum algorithms as quantum phase transitions: First versus second order,” Phys. Rev. A 74, 060304(R) (2006).
[CrossRef]

M. Stewart Siu, “Adiabatic rotation, quantum search, and preparation of superposition states,” Phys. Rev. A 75, 062337 (2007).
[CrossRef]

A. Pérez and A. Romanelli, “Nonadiabatic quantum search algorithms,” Phys. Rev. A 76, 052318 (2007).
[CrossRef]

Z. Wei and M. Ying, “Quantum adiabatic computation and adiabatic conditions,” Phys. Rev. A 76, 024304 (2007).
[CrossRef]

H. Azuma, “Higher-order perturbation theory for decoherence in Grover's algorithm,” Phys. Rev. A 72, 042305 (2005).
[CrossRef]

M. Tiersch and R. Schützhold, “Non-Markovian decoherence in the adiabatic quantum search algorithm,” Phys. Rev. A 75, 062313 (2007).
[CrossRef]

G.-L. Long, Y.-S. Li, W.-L. Zhang, and C.-C. Tu, “Dominant gate imperfection in Grover's quantum search algorithm,” Phys. Rev. A 61, 042305 (2000).
[CrossRef]

B. Pablo-Norman and M. Ruiz-Altaba, “Noise in Grover's quantum search algorithm,” Phys. Rev. A 61, 012301 (1999).
[CrossRef]

D. Leibfried, E. Knill, C. Ospelkaus, and D. J. Wineland, “Transport quantum logic gates for trapped ions,” Phys. Rev. A 76, 032324 (2007).
[CrossRef]

Phys. Rev. Lett. (14)

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86, 5188-5191 (2001).
[CrossRef] [PubMed]

D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, “Sufficiency criterion for the validity of the adiabatic approximation,” Phys. Rev. Lett. 98, 150402 (2007).
[CrossRef] [PubMed]

A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J. M. Raimond, and S. Haroche, “Coherent operation of a tunable quantum phase gate in cavity QED,” Phys. Rev. Lett. 83, 5166-5169 (1999).
[CrossRef]

I. L. Chuang, N. Gershenfeld, and M. Kubinec, “Experimental implementation of fast quantum searching,” Phys. Rev. Lett. 80, 3408-3411 (1998).
[CrossRef]

L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. 79, 325-328 (1997).
[CrossRef]

D. Daems and S. Guérin, “Adiabatic quantum search scheme with atoms in a cavity driven by lasers,” Phys. Rev. Lett. 99, 170503 (2007).
[CrossRef] [PubMed]

K. Mølmer and A. Sørensen, “Multiparticle entanglement of hot trapped ions,” Phys. Rev. Lett. 82, 1835-1838 (1999).
[CrossRef]

A. Sørensen and K. Mølmer, “Quantum computation with ions in thermal motion,” Phys. Rev. Lett. 82, 1971-1974 (1999).
[CrossRef]

M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, “Practical scheme for quantum computation with any two-qubit entangling gate,” Phys. Rev. Lett. 89, 247902 (2002).
[CrossRef] [PubMed]

S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett. 75, 346-349 (1995).
[CrossRef] [PubMed]

N. Schuch and J. Siewert, “Programmable networks for quantum algorithms,” Phys. Rev. Lett. 91, 027902 (2003).
[CrossRef] [PubMed]

For example, R. L. de Matos Filho and W. Vogel, “Even and odd coherent states of the motion of a trapped ion,” Phys. Rev. Lett. 76, 608-611 (1996) and W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes-Cummings dynamics of a trapped ion,” Phys. Rev. A 52, 4214-4217 (1995). For clarity of description below, we first assume a weak radiation without inducing AC Stark shift. So we consider in Eq. the resonant excitation involving motional quanta exchange. Actually, with a strong radiation, we could also do our job by carrier transition, as discussed later.
[CrossRef] [PubMed]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390-393 (1997).
[CrossRef]

M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. M. Lucas, D. N. Stacey, and A. M. Steane, “High-efficiency detection of a single quantum of angular momentum by suppression of optical pumping,” Phys. Rev. Lett. 93, 153601 (2004).
[CrossRef] [PubMed]

Science (2)

C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and measurement of three-qubit entangled states,” Science 304, 1478-1480 (2004).
[CrossRef] [PubMed]

D. Vion, A. Aassime, and A. Cottet, “Manipulating the quantum state of an electrical circuit,” Science 296, 886-889 (2002).
[CrossRef] [PubMed]

Other (2)

D. Shapira, S. Mozes, and O. Biham, “The effect of unitary noise on Grover's quantum search algorithm,” e-print arXiv:quant-ph/0307142.

P. J. Salas, “Noise effect on Grover algorithm,” e-print arXiv:quant-ph/0801.1261.

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Figures (5)

Fig. 1
Fig. 1

Schematic setup for implementing n-qubit CPF gate and Grover search in a linear trap, where the inset shows the ionic-level configuration, with bold lines for the states encoding qubits and the arrows for the coupling of the lasers to the ions.

Fig. 2
Fig. 2

Infidelity versus the Rabi frequency ratio of the last ion to the others, where the solid, dashed-dotted, and dashed curves represent the case of n = 3 , 6 and 9, respectively.

Fig. 3
Fig. 3

Success probability versus the Rabi frequency ratio of the last ion to the others, where the solid, dashed-dotted, and dashed curves represent the case of n = 3 , 6 and 9, respectively.

Fig. 4
Fig. 4

Quantum circuit of one iteration of the four-qubit Grover search for the marked state f 1 g 2 g 3 e 4 , where W, J 1111 ( 4 ) and S x ( σ x ) are the Hadamard gate, four-qubit controlled phase gate, and single-qubit NOT gate, respectively. The state of ions is initially prepared in the average superposition state Ψ I = 1 4 ( g 1 + f 1 ) ( g 2 + f 2 ) ( g 3 + f 3 ) ( g 4 + e 4 ) . The operations in the dashed boxes could be reduced to the transforms R i and R i with i ( = 1 , 2 , 3 , 4 ) denoting the i th ion, which is helpful for experimental implementation. To maximize the search propability, we should implement the circuit several times.

Fig. 5
Fig. 5

The search probability for the marked states f 1 g 2 g 3 e 4 and f 1 g 2 g 3 e 4 e 5 versus the number of the iterations in the case of n = 4 and 5.

Tables (2)

Tables Icon

Table 1 Values of C s ( n )

Tables Icon

Table 2 Required Time t 0 for CPF Gate and the Values of ν and Ω m ( η = 0.1 )

Equations (40)

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H = j = 1 n { ν a + a + ω 0 σ z , j + [ λ E + ( r , t ) σ j + + H.c. ] } ,
H I = j = 1 n ϰ = 0 Ω j ( t ) e η 2 2 e i ϕ j σ j + ( i η ) 2 ϰ + 1 ϰ ! ( ϰ + 1 ) ! ( a + ) ϰ a ϰ + 1 + H.c. ,
Ω j ( t ) = Ω max j exp { ( t t 0 ) 2 2 τ j 2 } ,
H 1 = j = 1 n i η Ω j ( t ) ( a + σ j e i ϕ j a σ j + e i ϕ j ) .
H 2 = j = 1 n η Ω j ( t ) ( a + σ j + a σ j + ) .
j = 1 , j k n 1 g k f j e n 0 exp ( i H 2 d t ) j , j k n 1 g k f j e n 0 = [ ϑ n 2 ϑ k 2 cos ( η ϑ k ) + ϑ k 2 ϑ n 2 ϑ k 2 ] × j , j k n 1 g k f j e n 0 + ϑ n ϑ k ϑ k 2 [ cos ( η ϑ k ) 1 ] e k g n j , j k n 1 f j 0 i ϑ n ϑ k sin ( η ϑ k ) g k g n j , j k n 1 f j 1 ,
ϑ j = 0 2 t 0 Ω max j exp { ( t t 0 ) 2 τ j 2 } d t = t 0 t 0 Ω max j exp ( t 2 2 τ j 2 ) d t = Ω max j 2 π τ j erf [ t 0 2 τ j ] Ω max j 2 π τ j ,
j , l = 1 , j l n 1 g j f l e n 0 exp ( i H 2 d t ) j , l = 1 , j l n 1 g j f l e n 0 = [ ϑ n 2 ϑ 2 cos ( η ϑ ) + ϑ 2 ϑ n 2 ϑ 2 ] × j , l = 1 , j l n 1 g j f l e n 0 + ϑ n ϑ 2 [ cos ( η ϑ ) 1 ] k = 1 s ϑ k e k g n j , l = 1 , j l n 1 g j f l 0 i ϑ j ϑ sin ( η ϑ ) j , l = 1 , j l n 1 g j f l g n 1 ,
j = 1 n 1 f j e n 0 exp ( i H 2 d t ) j = 1 n 1 f j e n 0 = [ cos ( η ϑ n ) e n 0 i sin ( η ϑ n ) g n 1 ] j = 1 n 1 f j .
j = 1 n 1 f j e n 0 j = 1 n 1 f j e n 0 .
ϑ 1 = ϑ 2 = = ϑ n 1 ϑ n .
m = ϑ n ϑ i = Ω max n τ n Ω max i τ i 1 ,
j = 1 , j k n 1 g k f j e n 0 β × j = 1 , j k n 1 g k f j e n 0
j , l = 1 , j l n 1 g j f l e n 0 α s × j , l = 1 , j l n 1 g j f l e n 0 ,
U CPF ( 4 ) = diag { 1 , 1 , α 3 , α 2 , 1 , 1 , α 2 , β , 1 , 1 , α 2 , β , 1 , 1 , β , 1 } ,
Infideity = 1 F = 1 [ s = 2 n 1 C s ( n ) α s + ( n 1 ) β + 2 n 1 + 1 ] 2 2 n [ s = 2 n 1 C s ( n ) α s 2 + ( n 1 ) β 2 + 2 n 1 + 1 ]
P = [ s = 2 n 1 C s ( n ) α s 2 + ( n 1 ) β 2 + 2 n 1 + 1 ] 2 n ,
C s ( n ) + C s + 1 ( n ) = C s + 1 ( n + 1 )
s = 2 n 1 C s ( n ) = 2 n 1 n .
Q ( n ) = W n J 00 0 ( n ) W n J ρ = W n J g 1 g 2 g n ( n ) W n J ρ = D ̂ ( n ) J ρ ,
W n = i = 1 n W i = ( 1 2 ) n [ 1 1 1 1 ] [ 1 1 1 1 ] .
J 00 0 ( n ) = σ x , n S x , n 1 S x , 1 J 11 1 ( n ) S x , 1 S x , n 1 σ x , n
D ̂ ( n ) = W n J 00 0 ( n ) W n ,
J g 1 g 2 g 3 g 4 = J 0000 = σ x , 4 S x , 3 S x , 2 S x , 1 J 1111 S x , 1 S x , 2 S x , 3 σ x , 4 ,
J g 1 g 2 g 3 e 4 = J 0001 = S x , 3 S x , 2 S x , 1 J 1111 S x , 1 S x , 2 S x , 3 ,
J g 1 g 2 f 3 g 4 = J 0010 = σ x , 4 S x , 2 S x , 1 J 1111 S x , 1 S x , 2 σ x , 4 ,
J g 1 g 2 f 3 e 4 = J 0011 = S x , 2 S x , 1 J 1111 S x , 1 S x , 2 ,
J g 1 f 2 g 3 g 4 = J 0100 = σ x , 4 S x , 3 S x , 1 J 1111 S x , 1 S x , 3 σ x , 4 ,
J g 1 f 2 g 3 e 4 = J 0101 = S x , 3 S x , 1 J 1111 S x , 1 S x , 3 ,
J g 1 f 2 f 3 g 4 = J 0110 = σ x , 4 S x , 1 J 1111 S x , 1 σ x , 4 ,
J g 1 f 2 f 3 e 4 = J 0111 = S x , 1 J 1111 S x , 1
J f 1 g 2 g 3 g 4 = J 1000 = σ x , 4 S x , 3 S x , 2 J 1111 S x , 2 S x , 3 σ x , 4 ,
J f 1 g 2 g 3 e 4 = J 1001 = S x , 3 S x , 2 J 1111 S x , 2 S x , 3 ,
J f 1 g 2 f 3 g 4 = J 1010 = σ x , 4 S x , 2 J 1111 S x , 2 σ x , 4 ,
J f 1 g 2 f 3 e 4 = J 1011 = S x , 2 J 1111 S x , 2 ,
J f 1 f 2 g 3 g 4 = J 1100 = σ x , 4 S x , 3 J 1111 S x , 3 σ x , 4 ,
J f 1 f 2 g 3 e 4 = J 1101 = S x , 3 J 1111 S x , 3 ,
J f 1 f 2 f 3 g 4 = J 1110 = σ x , 4 J 1111 σ x , 4 .
R 1 = R 4 = [ 1 1 1 1 ] , R 2 = R 3 = [ 1 1 1 1 ] ,
R 1 = R 2 = R 3 = R 4 = [ 1 1 1 1 ] .

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