Abstract

A new method based on quantum encoders is developed for implementing quantum computational gates, especially the CNOT gate. Photonic qubits are used where ∣0⟩ and ∣1⟩ represent horizontal H and vertical V polarized photons, respectively. By the use of polarizing beam splitters (PBSs) and entangled two-photon states in a certain experimental scheme, each input qubit state is copied, with a certain probability of success, into four equivalent input qubit states. The present method concentrates on the special case where the initial two-qubit state is separable. The CNOT gate (multiplied by 2) is decomposed into a summation of four unitary matrices each of dimension 4×4 and where each of these unitary matrices operates on a corresponding copy of the input two-qubit state. By adding the amplitudes of the four two-qubit states, which is obtained in a decoding interference experiment, the CNOT gate is implemented.

© 2008 Optical Society of America

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References

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  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2001).
  2. G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information, Volume 1: Basic Concepts (World Scientific, 2005).
  3. R. Barak and Y. Ben-Aryeh, “Quantum fast Fourier transform and quantum computation by linear optics,” J. Opt. Soc. Am. B 24, 231-240 (2007).
    [CrossRef]
  4. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
    [CrossRef]
  5. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
    [CrossRef] [PubMed]
  6. T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65, 012314 (2001).
    [CrossRef]
  7. J. D. Franson, M. M. Donegan, and B. C. Jacobs, “Generation of entangled ancilla states for use in linear optics quantum computing,” Phys. Rev. A 69, 052328 (2004).
    [CrossRef]
  8. A. P. Lund, T. B. Bell, and T. C. Ralph, “Comparison of linear optics quantum-computation control sign gates with ancilla inefficiency and an improvement to functionality under these conditions,” Phys. Rev. A 68, 022313 (2003).
    [CrossRef]
  9. F. M. Spedaliery, H. Lee, and J. W. Dowling, “High fidelity linear optical quantum computing with polarization encoding,” Phys. Rev. A 73, 012334 (2006).
    [CrossRef]
  10. A. J. F. Hayes, A. Gilchrist, C. R. Myers, and T. C. Ralph, “Utilizing encoding in scalable linear optics quantum computing,” J. Opt. B: Quantum Semiclassical Opt. 6, 533-541 (2004).
    [CrossRef]
  11. X. Zou, K. Pahlke, and W. Mathis, “Teleportation implementation of nondeterministic quantum logic operations by using linear optical elements,” Phys. Rev. A 65, 064305 (2002).
    [CrossRef]
  12. J. Fiura'sek, “Linear-optics quantum Toffoli and Fredkin gates,” Phys. Rev. A 73, 062313 (2006).
    [CrossRef]
  13. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarization beam splitters,” Phys. Rev. A 64, 062311 (2001).
    [CrossRef]
  14. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
    [CrossRef] [PubMed]
  15. T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum encoder for single-photon qubits,” Phys. Rev. A 69, 042306 (2004).
    [CrossRef]
  16. A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1998).
  17. M. Nakahara, “Quantum computing: an overview,” in Mathematical Aspects of Quantum Computing 2007, M.Nakahara, R.Rahimi, and A.SaiToh, eds. (World Scientific, 2008).
    [CrossRef]
  18. A. Yariv, Optical Electronics (Saunders College Publishing, 1991).
  19. Y. Ben-Aryeh, A. Mann, and B. C. Sanders, “Empirical state determination of entangled two-level systems and its relation to information theory,” Found. Phys. 29, 1963-1975 (1999).
    [CrossRef]

2007 (2)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

R. Barak and Y. Ben-Aryeh, “Quantum fast Fourier transform and quantum computation by linear optics,” J. Opt. Soc. Am. B 24, 231-240 (2007).
[CrossRef]

2006 (2)

F. M. Spedaliery, H. Lee, and J. W. Dowling, “High fidelity linear optical quantum computing with polarization encoding,” Phys. Rev. A 73, 012334 (2006).
[CrossRef]

J. Fiura'sek, “Linear-optics quantum Toffoli and Fredkin gates,” Phys. Rev. A 73, 062313 (2006).
[CrossRef]

2004 (3)

A. J. F. Hayes, A. Gilchrist, C. R. Myers, and T. C. Ralph, “Utilizing encoding in scalable linear optics quantum computing,” J. Opt. B: Quantum Semiclassical Opt. 6, 533-541 (2004).
[CrossRef]

J. D. Franson, M. M. Donegan, and B. C. Jacobs, “Generation of entangled ancilla states for use in linear optics quantum computing,” Phys. Rev. A 69, 052328 (2004).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum encoder for single-photon qubits,” Phys. Rev. A 69, 042306 (2004).
[CrossRef]

2003 (1)

A. P. Lund, T. B. Bell, and T. C. Ralph, “Comparison of linear optics quantum-computation control sign gates with ancilla inefficiency and an improvement to functionality under these conditions,” Phys. Rev. A 68, 022313 (2003).
[CrossRef]

2002 (2)

X. Zou, K. Pahlke, and W. Mathis, “Teleportation implementation of nondeterministic quantum logic operations by using linear optical elements,” Phys. Rev. A 65, 064305 (2002).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
[CrossRef] [PubMed]

2001 (3)

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarization beam splitters,” Phys. Rev. A 64, 062311 (2001).
[CrossRef]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65, 012314 (2001).
[CrossRef]

1999 (1)

Y. Ben-Aryeh, A. Mann, and B. C. Sanders, “Empirical state determination of entangled two-level systems and its relation to information theory,” Found. Phys. 29, 1963-1975 (1999).
[CrossRef]

Barak, R.

Bell, T. B.

A. P. Lund, T. B. Bell, and T. C. Ralph, “Comparison of linear optics quantum-computation control sign gates with ancilla inefficiency and an improvement to functionality under these conditions,” Phys. Rev. A 68, 022313 (2003).
[CrossRef]

Ben-Aryeh, Y.

R. Barak and Y. Ben-Aryeh, “Quantum fast Fourier transform and quantum computation by linear optics,” J. Opt. Soc. Am. B 24, 231-240 (2007).
[CrossRef]

Y. Ben-Aryeh, A. Mann, and B. C. Sanders, “Empirical state determination of entangled two-level systems and its relation to information theory,” Found. Phys. 29, 1963-1975 (1999).
[CrossRef]

Benenti, G.

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information, Volume 1: Basic Concepts (World Scientific, 2005).

Casati, G.

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information, Volume 1: Basic Concepts (World Scientific, 2005).

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2001).

Donegan, M. M.

J. D. Franson, M. M. Donegan, and B. C. Jacobs, “Generation of entangled ancilla states for use in linear optics quantum computing,” Phys. Rev. A 69, 052328 (2004).
[CrossRef]

Dowling, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Dowling, J. W.

F. M. Spedaliery, H. Lee, and J. W. Dowling, “High fidelity linear optical quantum computing with polarization encoding,” Phys. Rev. A 73, 012334 (2006).
[CrossRef]

Fiura'sek, J.

J. Fiura'sek, “Linear-optics quantum Toffoli and Fredkin gates,” Phys. Rev. A 73, 062313 (2006).
[CrossRef]

Franson, J. D.

J. D. Franson, M. M. Donegan, and B. C. Jacobs, “Generation of entangled ancilla states for use in linear optics quantum computing,” Phys. Rev. A 69, 052328 (2004).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum encoder for single-photon qubits,” Phys. Rev. A 69, 042306 (2004).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
[CrossRef] [PubMed]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarization beam splitters,” Phys. Rev. A 64, 062311 (2001).
[CrossRef]

Gilchrist, A.

A. J. F. Hayes, A. Gilchrist, C. R. Myers, and T. C. Ralph, “Utilizing encoding in scalable linear optics quantum computing,” J. Opt. B: Quantum Semiclassical Opt. 6, 533-541 (2004).
[CrossRef]

Hayes, A. J. F.

A. J. F. Hayes, A. Gilchrist, C. R. Myers, and T. C. Ralph, “Utilizing encoding in scalable linear optics quantum computing,” J. Opt. B: Quantum Semiclassical Opt. 6, 533-541 (2004).
[CrossRef]

Jacobs, B. C.

J. D. Franson, M. M. Donegan, and B. C. Jacobs, “Generation of entangled ancilla states for use in linear optics quantum computing,” Phys. Rev. A 69, 052328 (2004).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum encoder for single-photon qubits,” Phys. Rev. A 69, 042306 (2004).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
[CrossRef] [PubMed]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarization beam splitters,” Phys. Rev. A 64, 062311 (2001).
[CrossRef]

Knill, E.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

Lee, H.

F. M. Spedaliery, H. Lee, and J. W. Dowling, “High fidelity linear optical quantum computing with polarization encoding,” Phys. Rev. A 73, 012334 (2006).
[CrossRef]

Lund, A. P.

A. P. Lund, T. B. Bell, and T. C. Ralph, “Comparison of linear optics quantum-computation control sign gates with ancilla inefficiency and an improvement to functionality under these conditions,” Phys. Rev. A 68, 022313 (2003).
[CrossRef]

Mann, A.

Y. Ben-Aryeh, A. Mann, and B. C. Sanders, “Empirical state determination of entangled two-level systems and its relation to information theory,” Found. Phys. 29, 1963-1975 (1999).
[CrossRef]

Mathis, W.

X. Zou, K. Pahlke, and W. Mathis, “Teleportation implementation of nondeterministic quantum logic operations by using linear optical elements,” Phys. Rev. A 65, 064305 (2002).
[CrossRef]

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65, 012314 (2001).
[CrossRef]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65, 012314 (2001).
[CrossRef]

Myers, C. R.

A. J. F. Hayes, A. Gilchrist, C. R. Myers, and T. C. Ralph, “Utilizing encoding in scalable linear optics quantum computing,” J. Opt. B: Quantum Semiclassical Opt. 6, 533-541 (2004).
[CrossRef]

Nakahara, M.

M. Nakahara, “Quantum computing: an overview,” in Mathematical Aspects of Quantum Computing 2007, M.Nakahara, R.Rahimi, and A.SaiToh, eds. (World Scientific, 2008).
[CrossRef]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2001).

Pahlke, K.

X. Zou, K. Pahlke, and W. Mathis, “Teleportation implementation of nondeterministic quantum logic operations by using linear optical elements,” Phys. Rev. A 65, 064305 (2002).
[CrossRef]

Peres, A.

A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1998).

Pittman, T. B.

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum encoder for single-photon qubits,” Phys. Rev. A 69, 042306 (2004).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
[CrossRef] [PubMed]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarization beam splitters,” Phys. Rev. A 64, 062311 (2001).
[CrossRef]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

A. J. F. Hayes, A. Gilchrist, C. R. Myers, and T. C. Ralph, “Utilizing encoding in scalable linear optics quantum computing,” J. Opt. B: Quantum Semiclassical Opt. 6, 533-541 (2004).
[CrossRef]

A. P. Lund, T. B. Bell, and T. C. Ralph, “Comparison of linear optics quantum-computation control sign gates with ancilla inefficiency and an improvement to functionality under these conditions,” Phys. Rev. A 68, 022313 (2003).
[CrossRef]

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65, 012314 (2001).
[CrossRef]

Sanders, B. C.

Y. Ben-Aryeh, A. Mann, and B. C. Sanders, “Empirical state determination of entangled two-level systems and its relation to information theory,” Found. Phys. 29, 1963-1975 (1999).
[CrossRef]

Spedaliery, F. M.

F. M. Spedaliery, H. Lee, and J. W. Dowling, “High fidelity linear optical quantum computing with polarization encoding,” Phys. Rev. A 73, 012334 (2006).
[CrossRef]

Strini, G.

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information, Volume 1: Basic Concepts (World Scientific, 2005).

White, A. G.

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65, 012314 (2001).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics (Saunders College Publishing, 1991).

Zou, X.

X. Zou, K. Pahlke, and W. Mathis, “Teleportation implementation of nondeterministic quantum logic operations by using linear optical elements,” Phys. Rev. A 65, 064305 (2002).
[CrossRef]

Found. Phys. (1)

Y. Ben-Aryeh, A. Mann, and B. C. Sanders, “Empirical state determination of entangled two-level systems and its relation to information theory,” Found. Phys. 29, 1963-1975 (1999).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

A. J. F. Hayes, A. Gilchrist, C. R. Myers, and T. C. Ralph, “Utilizing encoding in scalable linear optics quantum computing,” J. Opt. B: Quantum Semiclassical Opt. 6, 533-541 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46-52 (2001).
[CrossRef] [PubMed]

Phys. Rev. A (8)

T. C. Ralph, A. G. White, W. J. Munro, and G. J. Milburn, “Simple scheme for efficient linear optics quantum gates,” Phys. Rev. A 65, 012314 (2001).
[CrossRef]

J. D. Franson, M. M. Donegan, and B. C. Jacobs, “Generation of entangled ancilla states for use in linear optics quantum computing,” Phys. Rev. A 69, 052328 (2004).
[CrossRef]

A. P. Lund, T. B. Bell, and T. C. Ralph, “Comparison of linear optics quantum-computation control sign gates with ancilla inefficiency and an improvement to functionality under these conditions,” Phys. Rev. A 68, 022313 (2003).
[CrossRef]

F. M. Spedaliery, H. Lee, and J. W. Dowling, “High fidelity linear optical quantum computing with polarization encoding,” Phys. Rev. A 73, 012334 (2006).
[CrossRef]

X. Zou, K. Pahlke, and W. Mathis, “Teleportation implementation of nondeterministic quantum logic operations by using linear optical elements,” Phys. Rev. A 65, 064305 (2002).
[CrossRef]

J. Fiura'sek, “Linear-optics quantum Toffoli and Fredkin gates,” Phys. Rev. A 73, 062313 (2006).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum logic operations using polarization beam splitters,” Phys. Rev. A 64, 062311 (2001).
[CrossRef]

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Probabilistic quantum encoder for single-photon qubits,” Phys. Rev. A 69, 042306 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

T. B. Pittman, B. C. Jacobs, and J. D. Franson, “Demonstration of nondeterministic quantum logic operations using linear optical elements,” Phys. Rev. Lett. 88, 257902 (2002).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photomic qubits,” Rev. Mod. Phys. 79, 135-174 (2007).
[CrossRef]

Other (5)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2001).

G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information, Volume 1: Basic Concepts (World Scientific, 2005).

A. Peres, Quantum Theory: Concepts and Methods (Kluwer, 1998).

M. Nakahara, “Quantum computing: an overview,” in Mathematical Aspects of Quantum Computing 2007, M.Nakahara, R.Rahimi, and A.SaiToh, eds. (World Scientific, 2008).
[CrossRef]

A. Yariv, Optical Electronics (Saunders College Publishing, 1991).

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Figures (1)

Fig. 1
Fig. 1

Implementation of a probabilistic quantum encoder for copying input qubit four times. Using for each PBS a corresponding two-photon entangled state and 1AO1 detection process, an input qubit is copied into two output qubit states with probability of success equal to 1 2 . Repeating this process for three PBS the input qubit is copied into the four output qubits: out 2 , out l , out e and out b with probability 1 8 .

Equations (41)

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ψ = α 0 + β 1 ,
ψ in = { α 0 A 0 B + β 0 A 1 B + γ 1 A 0 B + δ 1 A 1 B } ,
ψ out = { α 0 A 0 B + β 0 A 1 B + γ 1 A 1 B + δ 1 A 0 B } .
ψ in = ( α 0 A + β 1 A ) ( γ 0 B + δ 1 B ) ,
ψ out = { α γ 0 A 0 B + α δ 0 A 1 B + β γ 1 A 1 B + β δ 1 A 0 B } .
1 2 ( α 000 + β 111 ) + 1 2 ψ ,
α 0 + β 1 α 0 1 0 2 + β 1 1 1 2 .
α 0 + β 1 ( α 0 + β 1 ) 1 ( α 0 + β 1 ) 2 ,
α 0 + β 1 α 0 1 0 2 0 3 0 4 + β 1 1 1 2 1 3 1 4 .
φ ab + = 1 2 ( 0 a 0 b + 1 a 1 b ) ; φ de + = 1 2 ( 0 d 0 e + 1 d 1 e ) ;
φ kl + = 1 2 ( 0 k 0 l + 1 k 1 l )
α 0 2 + β 1 2 α 0 2 0 b + β 1 2 1 b .
α 0 2 0 b + β 1 2 1 b α 0 2 0 b 0 e + β 1 2 1 b 1 e .
α 0 2 0 e 0 b + β 1 2 1 e 1 b α 0 2 0 e 0 b 0 l + β 1 2 1 e 1 b 1 l .
α 0 2 + β 1 2 α 0 2 0 e 0 b 0 l + β 1 2 1 e 1 b 1 l .
γ 0 2 + δ 1 2 γ 0 P 0 Q 0 R 0 S + δ 1 P 1 Q 1 R 1 S ,
( α 0 2 + β 1 2 ) ( γ 0 2 + δ 1 2 ) ( α 0 2 0 e 0 b 0 l + β 1 2 1 e 1 b 1 l ) ( γ 0 P 0 Q 0 R 0 S + δ 1 P 1 Q 1 R 1 S ) .
0 ( 1 0 ) ; 1 ( 0 1 ) .
I = ( 1 0 0 1 ) , σ 1 = ( 0 1 1 0 ) , σ 2 = ( 0 i i 0 ) , σ 3 = ( 1 0 0 1 ) ,
00 ( 1 0 ) ( 1 0 ) ( 1 0 0 0 ) ; 01 ( 1 0 ) ( 0 1 ) ( 0 1 0 0 )
10 ( 0 1 ) ( 1 0 ) ( 0 0 1 0 ) ; 11 ( 0 1 ) ( 0 1 ) ( 0 0 0 1 ) ,
U 2 = l , m = 0 3 t l . m σ l σ m .
t j , k = 1 4 Tr [ U 2 ( σ j σ k ) ] .
Tr [ ( σ j σ k ) ( σ l σ m ) ] = 4 δ j , l δ k , m .
CNOT = ( 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ) ,
CNOT = 1 2 { σ 3 I σ 3 σ 1 + I σ 1 + I I } .
ψ inc = α γ { 0 2 0 P } { 0 e 0 Q } { 0 b 0 R } { 0 l 0 S } + α δ { 0 2 1 P } { 0 e 1 Q } { 0 b 1 R } { 0 l 1 S } + β γ { 1 2 0 P } { 1 e 0 Q } { 1 b 0 R } { 1 l 0 S } + β δ { 1 2 1 P } { 1 e 1 Q } { 1 b 1 R } { 1 l 1 S } ,
σ 3 0 = 0 ; σ 3 1 1 ; σ 1 0 = 1 ; σ 1 1 = 0 ;
I 0 = 0 ; I 1 = 1 .
σ 3 I 0 0 = 0 0 , σ 3 σ 1 0 0 = 0 1 ,
I σ 1 0 0 = 0 1 , I I 0 0 = 0 0 .
σ 3 I 0 1 = 0 1 , σ 3 σ 1 0 1 = 0 0 ,
I σ 1 0 1 = 0 0 , I I 0 1 = 0 1 .
σ 3 I 1 0 = 1 0 , σ 3 σ 1 1 0 = 1 1 ,
I σ 1 1 0 = 1 1 , I I 1 0 = 1 0 .
σ 3 I 1 1 = 1 1 , σ 3 σ 1 1 1 = 1 0 .
I σ 1 1 1 = 1 0 , I I 1 1 = 1 1
4 00 in 4 00 out ; 4 01 in 4 01 out ;
4 10 in 4 10 out ; 4 11 in 4 11 out
00 4 00 2 00 ; 01 4 01 2 01 ;
10 4 10 2 11 ; 11 4 11 2 10 .

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