Abstract

In a quantum computer, qubits are often stored in identical two-level systems separated by a distance shorter than the characteristic wavelength of the reservoirs that are responsible for decoherence. In this case the collective qubit–reservoir interaction, rather than the individual qubit–reservoir interaction, may determine the decoherence properties. We study the collective decoherence behavior in between each step in certain quantum algorithms and propose a simple alternative of implementing quantum algorithms using a quantum trajectory that is close to a decoherence–free subspace that avoids unstable Dicke’s superradiant states and Schrödinger’s cat state.

© 2007 Optical Society of America

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  1. T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
    [CrossRef]
  2. D. Loss and D. P. Divincenzo, "Quantum computation with quantum dots," Phys. Rev. A 57, 120-126 (1998).
    [CrossRef]
  3. B. E. Kane, "A silicon-based nuclear spin quantum computer," Nature 292, 133-137 (1998).
    [CrossRef]
  4. F. Yamaguchi and Y. Yamamoto, "Crystal lattice quantum computer," Appl. Phys. A Solids Surf. 68, 1-8 (1999).
    [CrossRef]
  5. T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
    [CrossRef]
  6. L.-M. Duan and G.-C. Guo, "Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment," Phys. Rev. A 57, 737-741 (1998).
    [CrossRef]
  7. P. Zanardi and M. Rasetti, "Noiseless quantum codes," Phys. Rev. Lett. 79, 3306-3309 (1997).
    [CrossRef]
  8. P. Zanardi, "Dissipation and decoherence in a quantum register," Phys. Rev. A 57, 3276-3284 (1998).
    [CrossRef]
  9. E. Knill, R. Laflamme, and L. Viola, "Theory of quantum error correction for general noise," Phys. Rev. Lett. 84, 2525-2528 (2000).
    [CrossRef] [PubMed]
  10. D. A. Lidar, I. L. Chuang, and K. B. Whaley, "Decoherence-free subspaces for quantum computation," Phys. Rev. Lett. 81, 2594-2597 (1998).
    [CrossRef]
  11. Daniel A. Lidar and L.-A. Wu, "Decoherence-free subspaces and subsystems," arXiv.org e-Print archive, quant-ph/0301032, January 9, 2003, http://arxiv.org/abs/quant-ph/0301032.
  12. R. H. Dicke, "Coherence in spontaneous radiation processes," Phys. Rev. 93, 99-110 (1954).
    [CrossRef]
  13. D. Deutsch and R. Jozsa, "Rapid solutions of problems by quantum computation," Proc. R. Soc. London Ser. A 439, 553-558 (1992).
    [CrossRef]
  14. L. K. Grover, "Quantum computers can search rapidly by using almost any transformation," Phys. Rev. Lett. 80, 4329-4332 (1998).
    [CrossRef]
  15. For example, see J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1993).
  16. P. W. Shor, "Algorithms for quantum computation: discrete log and factoring," in Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE, 1994), pp. 124-134.
    [CrossRef]
  17. A. Yu. Kitaev, "Quantum measurements and the Abelian stabilizer problem," arXiv.org e-Print archive, quant-ph/9511026, November 20, 1995, http://arxiv.org/abs/quant-ph/9511026.
  18. C. P. Master, S. Utsunomiya, and Y. Yamamoto, "Algorithm-based analysis of collective decoherence in quantum search," Prog. Inform. 3, 5-18 (2006).
    [CrossRef]
  19. C. P. Master, "Quantum computing under real-world constraints: efficiency of an ensemble quantum algorithm and fighting decoherence by gate design," Ph.D. dissertation (Stanford University, 2006).
  20. C. Cohen-Tannoudji, Quantum Mechanics: Vol 2 (Wiley-Interscience, 1977).

2006

T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
[CrossRef]

C. P. Master, S. Utsunomiya, and Y. Yamamoto, "Algorithm-based analysis of collective decoherence in quantum search," Prog. Inform. 3, 5-18 (2006).
[CrossRef]

2002

T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
[CrossRef]

2000

E. Knill, R. Laflamme, and L. Viola, "Theory of quantum error correction for general noise," Phys. Rev. Lett. 84, 2525-2528 (2000).
[CrossRef] [PubMed]

1999

F. Yamaguchi and Y. Yamamoto, "Crystal lattice quantum computer," Appl. Phys. A Solids Surf. 68, 1-8 (1999).
[CrossRef]

1998

P. Zanardi, "Dissipation and decoherence in a quantum register," Phys. Rev. A 57, 3276-3284 (1998).
[CrossRef]

D. A. Lidar, I. L. Chuang, and K. B. Whaley, "Decoherence-free subspaces for quantum computation," Phys. Rev. Lett. 81, 2594-2597 (1998).
[CrossRef]

L.-M. Duan and G.-C. Guo, "Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment," Phys. Rev. A 57, 737-741 (1998).
[CrossRef]

D. Loss and D. P. Divincenzo, "Quantum computation with quantum dots," Phys. Rev. A 57, 120-126 (1998).
[CrossRef]

B. E. Kane, "A silicon-based nuclear spin quantum computer," Nature 292, 133-137 (1998).
[CrossRef]

L. K. Grover, "Quantum computers can search rapidly by using almost any transformation," Phys. Rev. Lett. 80, 4329-4332 (1998).
[CrossRef]

1997

P. Zanardi and M. Rasetti, "Noiseless quantum codes," Phys. Rev. Lett. 79, 3306-3309 (1997).
[CrossRef]

1992

D. Deutsch and R. Jozsa, "Rapid solutions of problems by quantum computation," Proc. R. Soc. London Ser. A 439, 553-558 (1992).
[CrossRef]

1954

R. H. Dicke, "Coherence in spontaneous radiation processes," Phys. Rev. 93, 99-110 (1954).
[CrossRef]

Abe, E.

T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
[CrossRef]

Braunstein, S. L.

T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
[CrossRef]

Chuang, I. L.

D. A. Lidar, I. L. Chuang, and K. B. Whaley, "Decoherence-free subspaces for quantum computation," Phys. Rev. Lett. 81, 2594-2597 (1998).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, Quantum Mechanics: Vol 2 (Wiley-Interscience, 1977).

Deutsch, D.

D. Deutsch and R. Jozsa, "Rapid solutions of problems by quantum computation," Proc. R. Soc. London Ser. A 439, 553-558 (1992).
[CrossRef]

Dicke, R. H.

R. H. Dicke, "Coherence in spontaneous radiation processes," Phys. Rev. 93, 99-110 (1954).
[CrossRef]

Divincenzo, D. P.

D. Loss and D. P. Divincenzo, "Quantum computation with quantum dots," Phys. Rev. A 57, 120-126 (1998).
[CrossRef]

Duan, L.-M.

L.-M. Duan and G.-C. Guo, "Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment," Phys. Rev. A 57, 737-741 (1998).
[CrossRef]

Goldman, J. R.

T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
[CrossRef]

Grover, L. K.

L. K. Grover, "Quantum computers can search rapidly by using almost any transformation," Phys. Rev. Lett. 80, 4329-4332 (1998).
[CrossRef]

Guo, G.-C.

L.-M. Duan and G.-C. Guo, "Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment," Phys. Rev. A 57, 737-741 (1998).
[CrossRef]

Itoh, K. M.

T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
[CrossRef]

Jozsa, R.

D. Deutsch and R. Jozsa, "Rapid solutions of problems by quantum computation," Proc. R. Soc. London Ser. A 439, 553-558 (1992).
[CrossRef]

Kane, B. E.

B. E. Kane, "A silicon-based nuclear spin quantum computer," Nature 292, 133-137 (1998).
[CrossRef]

Kitaev, A. Yu.

A. Yu. Kitaev, "Quantum measurements and the Abelian stabilizer problem," arXiv.org e-Print archive, quant-ph/9511026, November 20, 1995, http://arxiv.org/abs/quant-ph/9511026.

Knill, E.

E. Knill, R. Laflamme, and L. Viola, "Theory of quantum error correction for general noise," Phys. Rev. Lett. 84, 2525-2528 (2000).
[CrossRef] [PubMed]

Ladd, T. D.

T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
[CrossRef]

Laflamme, R.

E. Knill, R. Laflamme, and L. Viola, "Theory of quantum error correction for general noise," Phys. Rev. Lett. 84, 2525-2528 (2000).
[CrossRef] [PubMed]

Lidar, D. A.

D. A. Lidar, I. L. Chuang, and K. B. Whaley, "Decoherence-free subspaces for quantum computation," Phys. Rev. Lett. 81, 2594-2597 (1998).
[CrossRef]

Lidar, Daniel A.

Daniel A. Lidar and L.-A. Wu, "Decoherence-free subspaces and subsystems," arXiv.org e-Print archive, quant-ph/0301032, January 9, 2003, http://arxiv.org/abs/quant-ph/0301032.

Loss, D.

D. Loss and D. P. Divincenzo, "Quantum computation with quantum dots," Phys. Rev. A 57, 120-126 (1998).
[CrossRef]

Master, C. P.

C. P. Master, S. Utsunomiya, and Y. Yamamoto, "Algorithm-based analysis of collective decoherence in quantum search," Prog. Inform. 3, 5-18 (2006).
[CrossRef]

C. P. Master, "Quantum computing under real-world constraints: efficiency of an ensemble quantum algorithm and fighting decoherence by gate design," Ph.D. dissertation (Stanford University, 2006).

Milburn, G. J.

T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
[CrossRef]

Munro, W. J.

T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
[CrossRef]

Nemoto, K.

T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
[CrossRef]

Rasetti, M.

P. Zanardi and M. Rasetti, "Noiseless quantum codes," Phys. Rev. Lett. 79, 3306-3309 (1997).
[CrossRef]

Sakurai, J. J.

For example, see J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1993).

Shor, P. W.

P. W. Shor, "Algorithms for quantum computation: discrete log and factoring," in Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE, 1994), pp. 124-134.
[CrossRef]

Spiller, T.

T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
[CrossRef]

Utsunomiya, S.

C. P. Master, S. Utsunomiya, and Y. Yamamoto, "Algorithm-based analysis of collective decoherence in quantum search," Prog. Inform. 3, 5-18 (2006).
[CrossRef]

van Loock, P.

T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
[CrossRef]

Viola, L.

E. Knill, R. Laflamme, and L. Viola, "Theory of quantum error correction for general noise," Phys. Rev. Lett. 84, 2525-2528 (2000).
[CrossRef] [PubMed]

Whaley, K. B.

D. A. Lidar, I. L. Chuang, and K. B. Whaley, "Decoherence-free subspaces for quantum computation," Phys. Rev. Lett. 81, 2594-2597 (1998).
[CrossRef]

Wu, L.-A.

Daniel A. Lidar and L.-A. Wu, "Decoherence-free subspaces and subsystems," arXiv.org e-Print archive, quant-ph/0301032, January 9, 2003, http://arxiv.org/abs/quant-ph/0301032.

Yamaguchi, F.

T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
[CrossRef]

F. Yamaguchi and Y. Yamamoto, "Crystal lattice quantum computer," Appl. Phys. A Solids Surf. 68, 1-8 (1999).
[CrossRef]

Yamamoto, Y.

C. P. Master, S. Utsunomiya, and Y. Yamamoto, "Algorithm-based analysis of collective decoherence in quantum search," Prog. Inform. 3, 5-18 (2006).
[CrossRef]

T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
[CrossRef]

F. Yamaguchi and Y. Yamamoto, "Crystal lattice quantum computer," Appl. Phys. A Solids Surf. 68, 1-8 (1999).
[CrossRef]

Zanardi, P.

P. Zanardi, "Dissipation and decoherence in a quantum register," Phys. Rev. A 57, 3276-3284 (1998).
[CrossRef]

P. Zanardi and M. Rasetti, "Noiseless quantum codes," Phys. Rev. Lett. 79, 3306-3309 (1997).
[CrossRef]

Appl. Phys. A Solids Surf.

F. Yamaguchi and Y. Yamamoto, "Crystal lattice quantum computer," Appl. Phys. A Solids Surf. 68, 1-8 (1999).
[CrossRef]

Nature

B. E. Kane, "A silicon-based nuclear spin quantum computer," Nature 292, 133-137 (1998).
[CrossRef]

New J. Phys.

T. Spiller, K. Nemoto, S. L. Braunstein, W. J. Munro, P. van Loock, and G. J. Milburn, "Quantum computation by communication," New J. Phys. 8, 1-25 (2006).
[CrossRef]

Phys. Rev.

R. H. Dicke, "Coherence in spontaneous radiation processes," Phys. Rev. 93, 99-110 (1954).
[CrossRef]

Phys. Rev. A

L.-M. Duan and G.-C. Guo, "Reducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment," Phys. Rev. A 57, 737-741 (1998).
[CrossRef]

D. Loss and D. P. Divincenzo, "Quantum computation with quantum dots," Phys. Rev. A 57, 120-126 (1998).
[CrossRef]

P. Zanardi, "Dissipation and decoherence in a quantum register," Phys. Rev. A 57, 3276-3284 (1998).
[CrossRef]

Phys. Rev. Lett.

E. Knill, R. Laflamme, and L. Viola, "Theory of quantum error correction for general noise," Phys. Rev. Lett. 84, 2525-2528 (2000).
[CrossRef] [PubMed]

D. A. Lidar, I. L. Chuang, and K. B. Whaley, "Decoherence-free subspaces for quantum computation," Phys. Rev. Lett. 81, 2594-2597 (1998).
[CrossRef]

T. D. Ladd, J. R. Goldman, F. Yamaguchi, Y. Yamamoto, E. Abe, and K. M. Itoh, "All-silicon quantum computer," Phys. Rev. Lett. 89, 1-4 (2002).
[CrossRef]

P. Zanardi and M. Rasetti, "Noiseless quantum codes," Phys. Rev. Lett. 79, 3306-3309 (1997).
[CrossRef]

L. K. Grover, "Quantum computers can search rapidly by using almost any transformation," Phys. Rev. Lett. 80, 4329-4332 (1998).
[CrossRef]

Proc. R. Soc. London Ser. A

D. Deutsch and R. Jozsa, "Rapid solutions of problems by quantum computation," Proc. R. Soc. London Ser. A 439, 553-558 (1992).
[CrossRef]

Prog. Inform.

C. P. Master, S. Utsunomiya, and Y. Yamamoto, "Algorithm-based analysis of collective decoherence in quantum search," Prog. Inform. 3, 5-18 (2006).
[CrossRef]

Other

C. P. Master, "Quantum computing under real-world constraints: efficiency of an ensemble quantum algorithm and fighting decoherence by gate design," Ph.D. dissertation (Stanford University, 2006).

C. Cohen-Tannoudji, Quantum Mechanics: Vol 2 (Wiley-Interscience, 1977).

For example, see J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1993).

P. W. Shor, "Algorithms for quantum computation: discrete log and factoring," in Proceedings of the 35th Annual Symposium on the Foundations of Computer Science (IEEE, 1994), pp. 124-134.
[CrossRef]

A. Yu. Kitaev, "Quantum measurements and the Abelian stabilizer problem," arXiv.org e-Print archive, quant-ph/9511026, November 20, 1995, http://arxiv.org/abs/quant-ph/9511026.

Daniel A. Lidar and L.-A. Wu, "Decoherence-free subspaces and subsystems," arXiv.org e-Print archive, quant-ph/0301032, January 9, 2003, http://arxiv.org/abs/quant-ph/0301032.

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Figures (6)

Fig. 1
Fig. 1

Projections P ( j , m ) onto the symmetrized states for the Deutsch–Jozsa algorithm; n = 8 and f ( x ) is one if the parity of x is odd. The plots correspond to the projections of ψ 1 (top left) through ψ 4 (bottom right). Note that the initial and final states for this example are ψ 1 = 0 and ψ 4 = 2 n 1 .

Fig. 2
Fig. 2

Projections P ( j , m ) for states ψ 1 through ψ 4 in the Deutsch–Jozsa algorithm, where n = 8 and f ( x ) equals one if the parity of the last four qubits is odd. Note that the initial and final states are ψ 1 = 0 and ψ 4 = 15 , respectively.

Fig. 3
Fig. 3

Projections P ( j , m ) for states ψ 1 through ψ 4 in the Deutsch–Jozsa algorithm, where n = 8 and f ( x ) equals one if the parity of the last four qubits is odd. Note that the initial and final states are ψ 1 = 0 and ψ 4 = 15 , respectively.

Fig. 4
Fig. 4

The normalized T 1 relaxation rates at each of the 50 computational steps for the n = 8 database search example; (a) applies to the target state τ = 255 , and (b) is for τ = 15 . The first Hadamard gate is followed by 12 Grover iterations consisting of 4 steps of H ̂ I ̂ 0 H ̂ I ̂ τ . The solid and dashed curves represent results for the standard implementation using an initial state ψ 1 = 0 and the improved implementation using an initial state ψ 1 = 15 = + + + + , respectively. (c) and (d) show the normalized T 2 relaxation rates for the n = 8 search algorithm, which are labelled as above in (a) and (b).

Fig. 5
Fig. 5

The normalized T 1 decay rates from ψ 1 to ψ 4 in the implementation of Shor’s algorithm for (a) M = 15 and x = 2 with four different second register states 1 2 (solid curve), 2 2 (dashed curve), 4 2 (dotted–dashed curve), and 8 2 (dotted curve) and (b) x = 4 with two different second register states 1 2 (solid curve) and 2 2 (dashed curve). (c) and (d) show the normalized T 2 relaxation rates for the n = 8 Shor’s algorithm, which are labelled as above in (a) and (b)

Fig. 6
Fig. 6

Normalized T 1 (left) and T 2 (right) decay rates from ψ 1 to ψ 4 with three different desired results, φ = 1 8 (solid curve), φ = 1 4 (dotted–dashed curve), and φ = 1 2 (dotted curve), in the implementation of Kitaev’s algorithm.

Equations (48)

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

H ̂ I = I k I ( a ̂ I J ̂ + a ̂ I J ̂ + ) .
J ̂ ± = l = 1 n J ̂ l ± = 1 2 l = 1 n σ l ± ,
H I = I g I a ̂ I + a ̂ I J ̂ z .
J ̂ z = j = 1 N J ̂ j z = 1 2 j = 1 n σ ̂ j z .
J ̂ 2 j 1 , j 2 , , j n ; j , m = j ( j + 1 ) j 1 , j 2 , , j n ; j , m ,
J ̂ z j 1 , j 2 , , j n ; j , m = j 1 , j 2 , , j n ; j , m .
j , m ± 1 , α J ̂ ± j , m , α 2 = ( j m ) ( j ± m + 1 ) .
γ = ( j + m ) ( j m + 1 ) γ 0 = n 2 ( n + 1 2 ) γ 0 .
ψ = j , m , α c j , m , α j , m , α .
P ( j , m ) = α c j , m , α 2 = α j , m , α ψ 2 = ψ α j , m , α j , m , α ψ .
γ γ 0 = j m P ( j , m ) [ j ( j + 1 ) m ( m 1 ) ] .
ψ = m 1 m 2 m n c m 1 m 2 m n m 1 m 2 m n .
ρ ̂ = m 1 , , m n m 1 , , m n c m 1 , , m n c m 1 , , m n e Γ 0 m m t m 1 , , m n m 1 , , m n ,
F = ψ ρ ̂ ψ .
F = ( I = 0 2 n 1 j = 0 2 n 1 c I 2 c j 2 e Γ I , j t ) 1 2 ,
F = 1 1 2 I = 0 2 n 1 j = 0 2 n 1 c I 2 c j 2 Γ 0 m I m j t 1 1 2 Γ t .
Γ Γ 0 = I = 0 2 n 1 j = 0 2 n 1 c I 2 c j 2 m I m j .
ψ 1 = 0 c 1 w .
H ̂ ψ 2 = 1 2 n x = 0 2 n 1 x c 1 2 ( 0 w 1 w ) .
U ̂ f c N ψ 3 = 1 2 n x = 0 2 n 1 ( 1 ) f ( x ) x c 1 2 ( 0 w 1 w ) .
H ̂ ψ 4 = 1 2 n y = 0 2 n 1 x = 0 2 n 1 ( 1 ) f ( x ) + x y y c 1 w .
f ( x ) = ( x 1 x 2 x 3 x 4 x ¯ 5 x ¯ 6 x ¯ 7 x ¯ 8 ) ( x 1 x 2 x 3 x ¯ 4 x 5 x ¯ 6 x ¯ 7 x ¯ 8 ) .
ψ 1 = 0 = .
H ̂ ψ 2 = H ̂ 0 = 1 2 n x = 0 2 n 1 x .
I ̂ τ ψ 3 = ( I ̂ 2 τ τ ) H ̂ 0 .
H ̂ ψ 4 = H ̂ ( I ̂ 2 τ τ ) H ̂ 0 .
I ̂ 0 ψ 5 = ( I ̂ 2 0 0 ) H ̂ ( I ̂ 2 τ τ ) H ̂ 0 .
ψ 5 = Q ̂ ψ 1 = ( I ̂ 2 0 0 ) H ̂ 2 0 + 2 ( I ̂ 2 0 0 ) H ̂ τ τ H ̂ 0 = ( 1 4 U τ 0 2 ) 0 + 2 U τ 0 H ̂ τ ,
Q ̂ H ̂ τ = I ̂ 0 H ̂ I ̂ τ H ̂ H ̂ τ = ( I ̂ 2 0 0 ) H ̂ τ = H ̂ τ 2 U τ 0 * 0 .
l tot ( π 2 ) θ = π 4 2 n ,
Q ̂ = ( I ̂ 2 γ γ ) H ̂ ( I ̂ 2 τ τ ) H ̂ ,
i d d t ψ ( t ) = ( H ̂ 0 + H ̂ I ) ψ ( t ) ,
ψ ( 0 ) = j , m , α Π I 0 .
ψ ( t ) = c m , 0 ( t ) e i ω m t j , m , α 0 + i c m 1 , 1 i ( t ) e i ω ( m 1 ) t i ω i t j , m 1 , α 1 i ,
i [ c ̇ m , 0 ( t ) e i ω m t j , m , α 0 ] + i c ̇ m 1 , 1 i ( t ) e i ω ( m 1 ) t i ω i t j , m 1 , α 1 i = i k i [ c m , 0 ( t ) e i ω m t j ( j + 1 ) m ( m 1 ) j , m 1 , α 1 i + c m 1 , 1 i ( t ) e i ω ( m 1 ) t i ω i t × j ( j + 1 ) m ( m 1 ) j , m , α 0 ] .
c ̇ m , 0 ( t ) = i i k i j ( j + 1 ) m ( m 1 ) e i ( ω ω i ) t c m 1 , 1 i ( t ) ,
c ̇ m 1 , 1 i ( t ) = i k i j ( j + 1 ) m ( m 1 ) e i ( ω ω i ) t c m , 0 ( t ) .
c m , 0 ( 0 ) = 1 ,
c m 1 , 1 i ( 0 ) = 0 .
c ̇ m , 0 ( t ) = i k i 2 [ j ( j + 1 ) m ( m 1 ) ] 0 t d t e i ( ω ω i ) ( t t ) c m , 0 ( t ) d t .
c ̇ m , 0 ( t ) = γ 2 c m , 0 ( t ) ,
γ = 2 π k 2 ( ω ) [ j ( j + 1 ) m ( m 1 ) ] ρ ( ω ) ,
i d d t ψ ( t ) = H ̂ I ψ ( t ) ,
U ̂ ( t ) = exp [ i g t J ̂ z a ̂ a ̂ ] .
ψ ( 0 ) = [ c m j , m , α + c m j , m , α ] s n c n n p .
ψ ( t ) = U ̂ ( t ) ψ ( 0 ) = n [ c m e i g m n t j , m , α s + c m e i g m n t j , m , α s ] c m n p ,
ρ ̂ s ( red ) Tr ρ ( ψ ψ ) = [ c m 2 j , m , α j , m , α + c m 2 j , m , α j , m , α + c m c m * n c n 2 e i g ( m m ) n t j , m , α j , m , α + c m c m * n c n 2 e i g ( m m ) n t j , m , α j , m , α ] .
Γ = c g m m Δ n ̂ 2 1 2 ,

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