Abstract

A new physical scheme for implementing an N-bit discrete quantum Fourier transform (DQFT) is proposed via superconducting (SC) qubits coupled to a single-mode SC cavity. Two-qubit and one-qubit gates as well as a new two-qubit gate are realized. Such gates are used for implementing the algorithm of N-bit DQFT. We propose and analyze a detailed experimental procedure for implementing the algorithm and compute the fidelity measure to quantify the success of this algorithm. Estimates show that the protocol can be successfully implemented within the present experimental limits.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  6. L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
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    [CrossRef]
  8. L. Fu, J. Luo, Li Xiao, and X. Zeng, “Experimental realization of discrete Fourier transformation on NMR quantum computer,” Appl. Magn. Reson. 19, 153–159 (2000).
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  14. A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
    [CrossRef]
  15. H.-F. Wang, X.-X. Jiang, S. Zhang, and K.-H. Yeon, “Efficient quantum circuit for implementing discrete quantum Fourier transform in solid-state qubits,” J. Phys. B 44, 115502 (2011).
    [CrossRef]
  16. J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42–47 (2005).
    [CrossRef]
  17. R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
    [CrossRef]
  18. Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
    [CrossRef]
  19. D. V. Averin, “Quantum computing and quantum measurement with mesoscopic Josephson junctions,” Fortsch. Phys. 48, 1055–1074 (2000).
    [CrossRef]
  20. A. Shnirman, G. Schön, and Z. Hermon, “Quantum manipulations of small Josephson junctions,” Phys. Rev. Lett. 79, 2371–2374 (1997).
    [CrossRef]
  21. J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68, 064509 (2003).
    [CrossRef]
  22. C.-P. Yang, Y.-X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
    [CrossRef]
  23. A.-S. F. Obada, H. A. Hessian, A.-B. A. Mohamed, and A. H. Homid, “Quantum logic gates generated by SC-charge qubits coupled to a resonator,” J. Phys. A 45, 485305(2012).
    [CrossRef]
  24. Y. Makhlin, G. Schön, and A. Shnirman, “Josephson-junction qubits with controlled couplings,” Nature 398, 305–307 (1999).
    [CrossRef]
  25. Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
    [CrossRef]
  26. Y.-X. Liu, L. F. Wei, and F. Nori, “Measuring the quality factor of a microwave cavity using superconducting qubit devices,” Phys. Rev. A 72, 033818 (2005).
    [CrossRef]
  27. D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120–126 (1998).
    [CrossRef]
  28. K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
    [CrossRef]
  29. 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]
  30. K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
    [CrossRef]
  31. Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature 398, 786–788 (1999).
    [CrossRef]

2012 (1)

A.-S. F. Obada, H. A. Hessian, A.-B. A. Mohamed, and A. H. Homid, “Quantum logic gates generated by SC-charge qubits coupled to a resonator,” J. Phys. A 45, 485305(2012).
[CrossRef]

2011 (2)

H.-F. Wang, A.-D. Zhu, S. Zhang, and K.-H. Yeon, “Simple implementation of discrete quantum Fourier transform via cavity quantum electrodynamics,” New J. Phys. 13, 013021 (2011).
[CrossRef]

H.-F. Wang, X.-X. Jiang, S. Zhang, and K.-H. Yeon, “Efficient quantum circuit for implementing discrete quantum Fourier transform in solid-state qubits,” J. Phys. B 44, 115502 (2011).
[CrossRef]

2010 (2)

H. F. Wang, X. Q. Shao, Y. F. Zhao, S. Zhang, and K. H. Yeon, “Protocol and quantum circuit for implementing the N-bit discrete quantum Fourier transform in cavity QED,” J. Phys. B 43, 065503 (2010).
[CrossRef]

C.-P. Yang, Y.-X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
[CrossRef]

2008 (1)

H. F. Wang, S. Zhang, and K. H. Yeon, “Implementing quantum discrete Fourier transform by using cavity quantum electrodynamics,” J. Korean Phys. Soc. 53, 1787–1790 (2008).

2005 (3)

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42–47 (2005).
[CrossRef]

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Y.-X. Liu, L. F. Wei, and F. Nori, “Measuring the quality factor of a microwave cavity using superconducting qubit devices,” Phys. Rev. A 72, 033818 (2005).
[CrossRef]

2003 (2)

J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68, 064509 (2003).
[CrossRef]

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

2002 (2)

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

M. O. Scully and M. S. Zubairy, “Cavity QED implementation of the discrete quantum Fourier transform,” Phys. Rev. A 65, 052324 (2002).
[CrossRef]

2001 (4)

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,” Nature 414, 883–887 (2001).
[CrossRef]

Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory, “Implementation of the quantum Fourier transform,” Phys. Rev. Lett. 86, 1889–1891 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

2000 (4)

D. V. Averin, “Quantum computing and quantum measurement with mesoscopic Josephson junctions,” Fortsch. Phys. 48, 1055–1074 (2000).
[CrossRef]

J. C. Howell and J. A. Yeazell, “Reducing the complexity of linear optics quantum circuits,” Phys. Rev. A 61, 052303 (2000).
[CrossRef]

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
[CrossRef]

L. Fu, J. Luo, Li Xiao, and X. Zeng, “Experimental realization of discrete Fourier transformation on NMR quantum computer,” Appl. Magn. Reson. 19, 153–159 (2000).
[CrossRef]

1999 (3)

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

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

Y. Makhlin, G. Schön, and A. Shnirman, “Josephson-junction qubits with controlled couplings,” Nature 398, 305–307 (1999).
[CrossRef]

1998 (2)

D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120–126 (1998).
[CrossRef]

M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight bounds on quantum searching,” Fortsch. Phys. 46, 493–505 (1998).
[CrossRef]

1997 (2)

A. Shnirman, G. Schön, and Z. Hermon, “Quantum manipulations of small Josephson junctions,” Phys. Rev. Lett. 79, 2371–2374 (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]

1995 (1)

B. Schumacher, “Quantum coding,” Phys. Rev. A 51, 2738–2747 (1995).
[CrossRef]

1992 (1)

D. Deutsch and R. Josza, “Rapid solution of problems by quantum computation,” Proc. R. Soc. London, Ser. A 439, 553–558 (1992).
[CrossRef]

Averin, D. V.

D. V. Averin, “Quantum computing and quantum measurement with mesoscopic Josephson junctions,” Fortsch. Phys. 48, 1055–1074 (2000).
[CrossRef]

Awschalom, D. D.

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

Birkl, G.

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

Bladh, K.

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

Boyer, M.

M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight bounds on quantum searching,” Fortsch. Phys. 46, 493–505 (1998).
[CrossRef]

Brassard, G.

M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight bounds on quantum searching,” Fortsch. Phys. 46, 493–505 (1998).
[CrossRef]

Breyta, G.

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,” Nature 414, 883–887 (2001).
[CrossRef]

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
[CrossRef]

Bruss, D.

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

Burkard, G.

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

Chuang, I. L.

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,” Nature 414, 883–887 (2001).
[CrossRef]

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
[CrossRef]

Cicak, K.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Cirac, J. I.

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]

Cleve, R.

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
[CrossRef]

Cooper, K. B.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Cory, D. G.

Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory, “Implementation of the quantum Fourier transform,” Phys. Rev. Lett. 86, 1889–1891 (2001).
[CrossRef]

Delsing, P.

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

Deutsch, D.

D. Deutsch and R. Josza, “Rapid solution of problems by quantum computation,” Proc. R. Soc. London, Ser. A 439, 553–558 (1992).
[CrossRef]

DiVincenzo, D. P.

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120–126 (1998).
[CrossRef]

Eckert, K.

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

Fortunato, E. M.

Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory, “Implementation of the quantum Fourier transform,” Phys. Rev. Lett. 86, 1889–1891 (2001).
[CrossRef]

Fu, L.

L. Fu, J. Luo, Li Xiao, and X. Zeng, “Experimental realization of discrete Fourier transformation on NMR quantum computer,” Appl. Magn. Reson. 19, 153–159 (2000).
[CrossRef]

Gunnarsson, D.

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

Hermon, Z.

A. Shnirman, G. Schön, and Z. Hermon, “Quantum manipulations of small Josephson junctions,” Phys. Rev. Lett. 79, 2371–2374 (1997).
[CrossRef]

Hessian, H. A.

A.-S. F. Obada, H. A. Hessian, A.-B. A. Mohamed, and A. H. Homid, “Quantum logic gates generated by SC-charge qubits coupled to a resonator,” J. Phys. A 45, 485305(2012).
[CrossRef]

Homid, A. H.

A.-S. F. Obada, H. A. Hessian, A.-B. A. Mohamed, and A. H. Homid, “Quantum logic gates generated by SC-charge qubits coupled to a resonator,” J. Phys. A 45, 485305(2012).
[CrossRef]

Howell, J. C.

J. C. Howell and J. A. Yeazell, “Reducing the complexity of linear optics quantum circuits,” Phys. Rev. A 61, 052303 (2000).
[CrossRef]

Hoyer, P.

M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight bounds on quantum searching,” Fortsch. Phys. 46, 493–505 (1998).
[CrossRef]

Imamoglu, A.

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

Jiang, X.-X.

H.-F. Wang, X.-X. Jiang, S. Zhang, and K.-H. Yeon, “Efficient quantum circuit for implementing discrete quantum Fourier transform in solid-state qubits,” J. Phys. B 44, 115502 (2011).
[CrossRef]

Josza, R.

D. Deutsch and R. Josza, “Rapid solution of problems by quantum computation,” Proc. R. Soc. London, Ser. A 439, 553–558 (1992).
[CrossRef]

Lehnert, K. W.

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

Lewenstein, M.

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

Liu, Y.-X.

C.-P. Yang, Y.-X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
[CrossRef]

Y.-X. Liu, L. F. Wei, and F. Nori, “Measuring the quality factor of a microwave cavity using superconducting qubit devices,” Phys. Rev. A 72, 033818 (2005).
[CrossRef]

Lloyd, S.

Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory, “Implementation of the quantum Fourier transform,” Phys. Rev. Lett. 86, 1889–1891 (2001).
[CrossRef]

Loss, D.

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120–126 (1998).
[CrossRef]

Luo, J.

L. Fu, J. Luo, Li Xiao, and X. Zeng, “Experimental realization of discrete Fourier transformation on NMR quantum computer,” Appl. Magn. Reson. 19, 153–159 (2000).
[CrossRef]

Makhlin, Y.

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Josephson-junction qubits with controlled couplings,” Nature 398, 305–307 (1999).
[CrossRef]

Martinis, J. M.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

McDermott, R.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Mohamed, A.-B. A.

A.-S. F. Obada, H. A. Hessian, A.-B. A. Mohamed, and A. H. Homid, “Quantum logic gates generated by SC-charge qubits coupled to a resonator,” J. Phys. A 45, 485305(2012).
[CrossRef]

Mompart, J.

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

Nakamura, Y.

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

Nori, F.

C.-P. Yang, Y.-X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
[CrossRef]

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42–47 (2005).
[CrossRef]

Y.-X. Liu, L. F. Wei, and F. Nori, “Measuring the quality factor of a microwave cavity using superconducting qubit devices,” Phys. Rev. A 72, 033818 (2005).
[CrossRef]

J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68, 064509 (2003).
[CrossRef]

Obada, A.-S. F.

A.-S. F. Obada, H. A. Hessian, A.-B. A. Mohamed, and A. H. Homid, “Quantum logic gates generated by SC-charge qubits coupled to a resonator,” J. Phys. A 45, 485305(2012).
[CrossRef]

Oh, S.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Osborn, K. D.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Pappas, D. P.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Pashkin, Y. A.

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

Poyatos, J. F.

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]

Pravia, M. A.

Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory, “Implementation of the quantum Fourier transform,” Phys. Rev. Lett. 86, 1889–1891 (2001).
[CrossRef]

Schliemann, J.

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

Schoelkopf, R. J.

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

Schön, G.

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Josephson-junction qubits with controlled couplings,” Nature 398, 305–307 (1999).
[CrossRef]

A. Shnirman, G. Schön, and Z. Hermon, “Quantum manipulations of small Josephson junctions,” Phys. Rev. Lett. 79, 2371–2374 (1997).
[CrossRef]

Schumacher, B.

B. Schumacher, “Quantum coding,” Phys. Rev. A 51, 2738–2747 (1995).
[CrossRef]

Schuster, D. I.

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

Scully, M. O.

M. O. Scully and M. S. Zubairy, “Cavity QED implementation of the discrete quantum Fourier transform,” Phys. Rev. A 65, 052324 (2002).
[CrossRef]

Shao, X. Q.

H. F. Wang, X. Q. Shao, Y. F. Zhao, S. Zhang, and K. H. Yeon, “Protocol and quantum circuit for implementing the N-bit discrete quantum Fourier transform in cavity QED,” J. Phys. B 43, 065503 (2010).
[CrossRef]

Sherwin, M.

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

Sherwood, M. H.

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,” Nature 414, 883–887 (2001).
[CrossRef]

Shnirman, A.

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Josephson-junction qubits with controlled couplings,” Nature 398, 305–307 (1999).
[CrossRef]

A. Shnirman, G. Schön, and Z. Hermon, “Quantum manipulations of small Josephson junctions,” Phys. Rev. Lett. 79, 2371–2374 (1997).
[CrossRef]

Shor, P. W.

P. W. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in 35th Annual Symposium on Foundations of Computer Science, S. Goldwasser, ed. (IEEE, 1994), pp. 124–134.

Simmonds, R. W.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Small, A.

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

Spietz, L. F.

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

Steffen, M.

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,” Nature 414, 883–887 (2001).
[CrossRef]

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
[CrossRef]

Tapp, A.

M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight bounds on quantum searching,” Fortsch. Phys. 46, 493–505 (1998).
[CrossRef]

Tsai, J. S.

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

Vandersypen, L. M. K.

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,” Nature 414, 883–887 (2001).
[CrossRef]

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
[CrossRef]

Wang, H. F.

H. F. Wang, X. Q. Shao, Y. F. Zhao, S. Zhang, and K. H. Yeon, “Protocol and quantum circuit for implementing the N-bit discrete quantum Fourier transform in cavity QED,” J. Phys. B 43, 065503 (2010).
[CrossRef]

H. F. Wang, S. Zhang, and K. H. Yeon, “Implementing quantum discrete Fourier transform by using cavity quantum electrodynamics,” J. Korean Phys. Soc. 53, 1787–1790 (2008).

Wang, H.-F.

H.-F. Wang, X.-X. Jiang, S. Zhang, and K.-H. Yeon, “Efficient quantum circuit for implementing discrete quantum Fourier transform in solid-state qubits,” J. Phys. B 44, 115502 (2011).
[CrossRef]

H.-F. Wang, A.-D. Zhu, S. Zhang, and K.-H. Yeon, “Simple implementation of discrete quantum Fourier transform via cavity quantum electrodynamics,” New J. Phys. 13, 013021 (2011).
[CrossRef]

Wei, L. F.

Y.-X. Liu, L. F. Wei, and F. Nori, “Measuring the quality factor of a microwave cavity using superconducting qubit devices,” Phys. Rev. A 72, 033818 (2005).
[CrossRef]

Weinstein, Y. S.

Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory, “Implementation of the quantum Fourier transform,” Phys. Rev. Lett. 86, 1889–1891 (2001).
[CrossRef]

Xiao, Li

L. Fu, J. Luo, Li Xiao, and X. Zeng, “Experimental realization of discrete Fourier transformation on NMR quantum computer,” Appl. Magn. Reson. 19, 153–159 (2000).
[CrossRef]

Yang, C.-P.

C.-P. Yang, Y.-X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
[CrossRef]

Yannoni, C. S.

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,” Nature 414, 883–887 (2001).
[CrossRef]

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
[CrossRef]

Yeazell, J. A.

J. C. Howell and J. A. Yeazell, “Reducing the complexity of linear optics quantum circuits,” Phys. Rev. A 61, 052303 (2000).
[CrossRef]

Yeon, K. H.

H. F. Wang, X. Q. Shao, Y. F. Zhao, S. Zhang, and K. H. Yeon, “Protocol and quantum circuit for implementing the N-bit discrete quantum Fourier transform in cavity QED,” J. Phys. B 43, 065503 (2010).
[CrossRef]

H. F. Wang, S. Zhang, and K. H. Yeon, “Implementing quantum discrete Fourier transform by using cavity quantum electrodynamics,” J. Korean Phys. Soc. 53, 1787–1790 (2008).

Yeon, K.-H.

H.-F. Wang, A.-D. Zhu, S. Zhang, and K.-H. Yeon, “Simple implementation of discrete quantum Fourier transform via cavity quantum electrodynamics,” New J. Phys. 13, 013021 (2011).
[CrossRef]

H.-F. Wang, X.-X. Jiang, S. Zhang, and K.-H. Yeon, “Efficient quantum circuit for implementing discrete quantum Fourier transform in solid-state qubits,” J. Phys. B 44, 115502 (2011).
[CrossRef]

Yi, X. X.

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

You, J. Q.

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42–47 (2005).
[CrossRef]

J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68, 064509 (2003).
[CrossRef]

Zeng, X.

L. Fu, J. Luo, Li Xiao, and X. Zeng, “Experimental realization of discrete Fourier transformation on NMR quantum computer,” Appl. Magn. Reson. 19, 153–159 (2000).
[CrossRef]

Zhang, S.

H.-F. Wang, X.-X. Jiang, S. Zhang, and K.-H. Yeon, “Efficient quantum circuit for implementing discrete quantum Fourier transform in solid-state qubits,” J. Phys. B 44, 115502 (2011).
[CrossRef]

H.-F. Wang, A.-D. Zhu, S. Zhang, and K.-H. Yeon, “Simple implementation of discrete quantum Fourier transform via cavity quantum electrodynamics,” New J. Phys. 13, 013021 (2011).
[CrossRef]

H. F. Wang, X. Q. Shao, Y. F. Zhao, S. Zhang, and K. H. Yeon, “Protocol and quantum circuit for implementing the N-bit discrete quantum Fourier transform in cavity QED,” J. Phys. B 43, 065503 (2010).
[CrossRef]

H. F. Wang, S. Zhang, and K. H. Yeon, “Implementing quantum discrete Fourier transform by using cavity quantum electrodynamics,” J. Korean Phys. Soc. 53, 1787–1790 (2008).

Zhao, Y. F.

H. F. Wang, X. Q. Shao, Y. F. Zhao, S. Zhang, and K. H. Yeon, “Protocol and quantum circuit for implementing the N-bit discrete quantum Fourier transform in cavity QED,” J. Phys. B 43, 065503 (2010).
[CrossRef]

Zhu, A.-D.

H.-F. Wang, A.-D. Zhu, S. Zhang, and K.-H. Yeon, “Simple implementation of discrete quantum Fourier transform via cavity quantum electrodynamics,” New J. Phys. 13, 013021 (2011).
[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]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, “Cavity QED implementation of the discrete quantum Fourier transform,” Phys. Rev. A 65, 052324 (2002).
[CrossRef]

Appl. Magn. Reson. (1)

L. Fu, J. Luo, Li Xiao, and X. Zeng, “Experimental realization of discrete Fourier transformation on NMR quantum computer,” Appl. Magn. Reson. 19, 153–159 (2000).
[CrossRef]

Fortsch. Phys. (2)

D. V. Averin, “Quantum computing and quantum measurement with mesoscopic Josephson junctions,” Fortsch. Phys. 48, 1055–1074 (2000).
[CrossRef]

M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight bounds on quantum searching,” Fortsch. Phys. 46, 493–505 (1998).
[CrossRef]

J. Korean Phys. Soc. (1)

H. F. Wang, S. Zhang, and K. H. Yeon, “Implementing quantum discrete Fourier transform by using cavity quantum electrodynamics,” J. Korean Phys. Soc. 53, 1787–1790 (2008).

J. Phys. A (1)

A.-S. F. Obada, H. A. Hessian, A.-B. A. Mohamed, and A. H. Homid, “Quantum logic gates generated by SC-charge qubits coupled to a resonator,” J. Phys. A 45, 485305(2012).
[CrossRef]

J. Phys. B (2)

H. F. Wang, X. Q. Shao, Y. F. Zhao, S. Zhang, and K. H. Yeon, “Protocol and quantum circuit for implementing the N-bit discrete quantum Fourier transform in cavity QED,” J. Phys. B 43, 065503 (2010).
[CrossRef]

H.-F. Wang, X.-X. Jiang, S. Zhang, and K.-H. Yeon, “Efficient quantum circuit for implementing discrete quantum Fourier transform in solid-state qubits,” J. Phys. B 44, 115502 (2011).
[CrossRef]

Nature (3)

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, M. H. Sherwood, and I. L. Chuang, “Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance,” Nature 414, 883–887 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Josephson-junction qubits with controlled couplings,” Nature 398, 305–307 (1999).
[CrossRef]

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

New J. Phys. (1)

H.-F. Wang, A.-D. Zhu, S. Zhang, and K.-H. Yeon, “Simple implementation of discrete quantum Fourier transform via cavity quantum electrodynamics,” New J. Phys. 13, 013021 (2011).
[CrossRef]

Phys. Rev. A (7)

J. C. Howell and J. A. Yeazell, “Reducing the complexity of linear optics quantum circuits,” Phys. Rev. A 61, 052303 (2000).
[CrossRef]

B. Schumacher, “Quantum coding,” Phys. Rev. A 51, 2738–2747 (1995).
[CrossRef]

M. O. Scully and M. S. Zubairy, “Cavity QED implementation of the discrete quantum Fourier transform,” Phys. Rev. A 65, 052324 (2002).
[CrossRef]

C.-P. Yang, Y.-X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
[CrossRef]

Y.-X. Liu, L. F. Wei, and F. Nori, “Measuring the quality factor of a microwave cavity using superconducting qubit devices,” Phys. Rev. A 72, 033818 (2005).
[CrossRef]

D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120–126 (1998).
[CrossRef]

K. Eckert, J. Mompart, X. X. Yi, J. Schliemann, D. Bruss, G. Birkl, and M. Lewenstein, “Quantum computing in optical microtraps based on the motional states of neutral atoms,” Phys. Rev. A 66, 042317 (2002).
[CrossRef]

Phys. Rev. B (1)

J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68, 064509 (2003).
[CrossRef]

Phys. Rev. Lett. (6)

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]

K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing, and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic circuit,” Phys. Rev. Lett. 90, 027002 (2003).
[CrossRef]

A. Shnirman, G. Schön, and Z. Hermon, “Quantum manipulations of small Josephson junctions,” Phys. Rev. Lett. 79, 2371–2374 (1997).
[CrossRef]

Y. S. Weinstein, M. A. Pravia, E. M. Fortunato, S. Lloyd, and D. G. Cory, “Implementation of the quantum Fourier transform,” Phys. Rev. Lett. 86, 1889–1891 (2001).
[CrossRef]

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental realization of an order-finding algorithm with an NMR quantum computer,” Phys. Rev. Lett. 85, 5452–5455 (2000).
[CrossRef]

A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, and A. Small, “Quantum information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204–4207 (1999).
[CrossRef]

Phys. Today (1)

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42–47 (2005).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

D. Deutsch and R. Josza, “Rapid solution of problems by quantum computation,” Proc. R. Soc. London, Ser. A 439, 553–558 (1992).
[CrossRef]

Rev. Mod. Phys. (2)

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Y. Makhlin, G. Schön, and A. Shnirman, “Quantum-state engineering with Josephson-junction devices,” Rev. Mod. Phys. 73, 357–400 (2001).
[CrossRef]

Science (1)

R. McDermott, R. W. Simmonds, M. Steffen, K. B. Cooper, K. Cicak, K. D. Osborn, S. Oh, D. P. Pappas, and J. M. Martinis, “Simultaneous state measurement of coupled Josephson phase qubits,” Science 307, 1299–1302 (2005).
[CrossRef]

Other (1)

P. W. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in 35th Annual Symposium on Foundations of Computer Science, S. Goldwasser, ed. (IEEE, 1994), pp. 124–134.

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Figures (3)

Fig. 1.
Fig. 1.

Quantum schematic circuit for implementing the 2 bit DQFT. This quantum circuit contains one- and two-qubit gates: here P^ is the two-qubit controlled phase gate at different phases. An experimental setup of a charge qubits-field system for implementing the 2 bit DQFT runs as follows: the superconductors are simultaneously interacting with the SC cavity mode, where the mode in the vacuum state. When the classical magnetic field is switched to ϕc=0, the interaction between the charge qubit and the field is switched off and results in one-qubit gates with different pulses of classical field at different positions on a two superconductors. Also, if the classical magnetic field is switched to ϕc=(1/2)ϕ0, the interaction between the charge qubits and the field is switched on and results in a two-qubit gate, where the interaction time is switched to t1=2π/μ to realize the controlled phase gate at different phases and is switched to t2=π/4μ to realize iSWC^Z gate.

Fig. 2.
Fig. 2.

Quantum circuit diagram and experimental setup for implementing the N-bit DQFT.

Fig. 3.
Fig. 3.

F for implementing 2 bit and 3 bit DQFT with the deviation amount Γ.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

|0=|000,|1=|0001,|2λ1=|111.
|m1m2mλ12λ2n=02λ1e2πimn/2λ|n=12λ2(|0+e2πi0·mλ|1)(|0+e2πi0·mλ1mλ|1)(|0+e2πi0·m1m2mλ1mλ|1),
H^(t)=ωb^b^+Ezi=1Nσ^ziEJcos(πϕcϕ0)i=1Nσ^xi(t)+πEJϕ0sin(πϕcϕ0)i=1N[ηib^(t)+h.c]σ^xi(t),
H^N=EJi=1Nσ^xi.
V^(θ,t)=(cos(EJt)ieiθsin(EJt)ieiθsin(EJt)cos(EJt)).
V^(θ,3π2EJ)V^(θ2,π2EJ)R^z(θ)=(eiθ200eiθ2),
eiθ2V^(θ,3π2EJ)V^(θ2,π2EJ)S^(θ)=(100eiθ).
eiπ2V^(π2,π4EJ)V^(2π,3π2EJ)H^=12(1111).
H^I(t)=πEJϕ0i=1N{ηiB^(t)ei(ωdω)t+h.c}{s^+i(t)eiθei(2Ez2ωd)t+h.c}.
H^(t)=CT^H^I(t)CT^=πEJϕ0i=1N{ηiB^(t)eiδt+h.c}σ^zi,
H^eff=(πEJϕo2δ)2i,j=1N{(ηiηj*+ηi*ηj)B^B^+ηiηj*}σ^ziσ^zj.
W^(t)=(eiϕ1(t)0000eiϕ2(t)0000eiϕ2(t)0000eiϕ1(t)),
W^(ϕ)=(eiϕ00001000010000eiϕ),ϕ={2π3|η|2ϕo2}.
f=12R^zf(ϕ)W^(ϕ)P^(β)=(100001000010000eiβ),R^zf(ϕ)=(eiϕ200eiϕ2),
H^N(t)=ωb^b^+Ezi=1Nσ^zi+πEJϕ0i=1N{ηib^(t)σ^+i(t)eiθ+h.c}.
H^eff=(2πEJϕoδ)2i,j=1N{ηiηj*(σ^+iσ^j)b^b^+ηjηi*(σ^ziδijσ^+jσ^i)b^b^}.
U^(t)=(eia1t0000eia3t(ϒ1a5*a4ϒ2)ia6a4eia3tϒ200ia6*a4eia3tϒ2eia3t(ϒ1a5a4ϒ2)0000eia2t),
U^(t)=(10000eiμtcos(μt)ieiμtsin(μt)00ieiμtsin(μt)eiμtcos(μt)0000e2iμt),μ={2πEJ|η|ϕ0δ}2.
iSWC^Z=(1000012(1i)12(1+i)0012(1+i)12(1i)0000i).
F^|Ψ=12N2n=02N1m=02N1cme2πimn/2N|n.
c0|g1|g2|gN1|gN+c1|g1|g2|gN1|eN++c2N1|e1|e2|eN1|eN,
|χ=c00|01|02+c01|01|12+c10|11|02+c11|11|12,
F^|χ=12{(c00+c01+c10+c11)|01|02+(c00+ic01c10ic11)|01|12+(c00c01+c10c11)|11|02+(c00ic01c10+ic11)|11|12}.
|φ=c0|g1|g2+c1|g1|e2+c2|e1|g2+c3|e1|e2.
|φ|φ1=12{eiπ4(c0+c1)(|g1|g2+|g1|e2)+eiπ4(c0c1)(|g1|g2|g1|e2)+eiπ4(c2+c3)(|e1|g2+|e1|e2)+eiπ4(c2c3)(|e1|g2|e1|e2)}.
|φ1|φ2=12{eiπ4(c0+c1)(|g1|g2+|g1|e2)+eiπ4(c0c1)(|g1|g2|g1|e2)+eiπ4(c2+c3)(|e1|g2|e1|e2)+eiπ4(c2c3)(|e1|g2+|e1|e2)}.
|φ2|φ3=12{eiπ4(c0+c1)|g1|g2eiπ4(c0c1)|g1|e2eiπ4(c2+c3)|e1|e2+eiπ4(c2c3)|e1|g2}.
|φ3|φ4=eiπ42{(c0+c1)|g1|g2+i(c2+c3)|e1|e2+12{(1+i)(c0c1)(1i)(c2c3)}|g1|e2+12{(1i)(c0c1)(1+i)(c2c3)}|e1|g2}.
|φ4|φ5=eiπ42{(c0+c1)|g1|g2i(c2+c3)|e1|e2+12{(1i)(c0c1)+(1+i)(c2c3)}|g1|e212{(1+i)(c0c1)+(1i)(c2c3)}|e1|g2}.
|φ5|φ6=eiπ42{(c0+c1)|g1|g2(c2+c3)|e1|e2+(c2c3)|g1|e2(c0c1)|e1|g2}.
|φ6|φ7=eiπ42{(c0+c1)|g1|g2+i(c2+c3)|e1|e2+(c2c3)|g1|e2(c0c1)|e1|g2}.
|φ7|φ8=eiπ42{i2c0|g1|g2+i2c1|e1|g2+c22(|g1|e2+i|g1|e2|e1|e2+i|e1|e2)+c32(|g1|e2i|g1|e2|e1|e2i|e1|e2)}.
|φ8|φ9=eiπ42{i2c0|g1|g2+i2c1|e1|g2+c22(|g1|e2+i|g1|e2+|e1|e2i|e1|e2)+c32(|g1|e2i|g1|e2+|e1|e2+i|e1|e2)}.
|φ9|φ10=eiπ42{i(c0+c1)|g1|g2+i(c0c1)|e1|g2+c2(i|g1|e2|e1|e2)+c3(i|g1|e2+|e1|e2)}.
|φ10|φ11=eiπ42{i(c0+c1)|g1|g2+i(c0c1)|e1|g2+c2(i|g1|e2+i|e1|e2)+c3(i|g1|e2i|e1|e2)}.
|φ11|φ12=ei3π42{(c0+c2)|g1|g2+(c1+c3)|e1|g2+(c0c2)|g1|e2+(c1c3)|e1|e2}.
|φ12|φ13=ei3π42{(c0+c2)|g1|g2+(c1+c3)|e1|g2+(c0c2)|g1|e2+i(c1c3)|e1|e2}.
|φ13|φ14=ei3π42{(c0+c1+c2+c3)|g1|g2+(c0+ic1c2ic3)|g1|e2+(c0c1+c2c3)|e1|g2+(c0ic1c2+ic3)|e1|e2}.
t1=t1+4πΓμ,t2=t2+4πΓμ,
F=Φin|U^Dρ^outU^D|Φin¯=|Φ(t)|Φideal|2¯,
F=(12π)N02πdϑ102πdϑ202πdϑN|Φ(t)|Φin|2.

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