Abstract

Simple examples that illustrate the principles of one-way quantum computation using Gaussian continuous-variable cluster states are given. In these examples, only single-mode evolutions, realizable via linear clusters, are considered. In particular, one can focus on Gaussian single-mode transformations performed through the cluster state. These examples highlight the differences between cluster-based schemes and protocols in which special quantum states are prepared off-line and then used as a resource for the on-line computation.

© 2007 Optical Society of America

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  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).
  2. R. Raussendorf and H. J. Briegel, "A one-way quantum computer," Phys. Rev. Lett. 86, 5188-5191 (2001).
    [Crossref] [PubMed]
  3. S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
    [Crossref]
  4. S. Lloyd and S. L. Braunstein, "Quantum computation over continuous variables," Phys. Rev. Lett. 82, 1784-1787 (1999).
    [Crossref]
  5. N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
    [Crossref] [PubMed]
  6. J. Zhang and S. L. Braunstein, "Continuous-variable Gaussian analog of cluster states," Phys. Rev. A 73, 032318 (2006).
    [Crossref]
  7. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).
  8. S. L. Braunstein, "Squeezing as an irreducible resource," Phys. Rev. A 71, 055801 (2005).
    [Crossref]
  9. P. Grangier, J. A. Levenson, and J.-P. Poizat, "Quantum non-demolition measurements in optics," Nature 396, 537-542 (1998).
    [Crossref]
  10. B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413, 400-403 (2001).
    [Crossref] [PubMed]
  11. J. M. Geremia, J. K. Stockton, and H. Mabuchi, "Real-time quantum feedback control of atomic spin-squeezing," Science 304, 270-273 (2004).
    [Crossref] [PubMed]
  12. M. A. Nielsen, "Optical quantum computation using cluster states," Phys. Rev. Lett. 93, 040503 (2004).
    [Crossref] [PubMed]
  13. D. E. Browne and T. Rudolph, "Resource-efficient linear optical quantum computation," Phys. Rev. Lett. 95, 010501 (2005).
    [Crossref] [PubMed]
  14. S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, "Efficient classical simulation of continuous variable quantum information processes," Phys. Rev. Lett. 88, 097904 (2002).
    [Crossref] [PubMed]
  15. M. A. Nielsen, "Cluster-state quantum computation," Rep. Math. Phys. 57, 147-161 (2006).
    [Crossref]
  16. D. Gottesman and I. L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations," Nature 402, 390-393 (1999).
    [Crossref]
  17. E. Knill, R. Laflamme, and G. J. Milburn, "A scheme for efficient quantum computation with linear optics," Nature 409, 46-52 (2001).
    [Crossref] [PubMed]
  18. S. D. Bartlett and W. J. Munro, "Quantum teleportation of optical quantum gates," Phys. Rev. Lett. 90, 117901 (2003).
    [Crossref] [PubMed]
  19. P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer, and A. Zeilinger, "Experimental one-way quantum computing," Nature 434, 169-176 (2005).
    [Crossref] [PubMed]
  20. X. Zhou, D. W. Leung, and I. L. Chuang, "Methodology for quantum logic gate construction," Phys. Rev. A 62, 052316 (2000).
    [Crossref]
  21. D. Gottesman, A. Kitaev, and J. Preskill, "Encoding a qubit in an oscillator," Phys. Rev. A 64, 012310 (2001).
    [Crossref]
  22. R. Filip, P. Marek, and U. L. Andersen, "Measurement-induced continuous-variable quantum interactions," Phys. Rev. A 71, 042308 (2005).
    [Crossref]
  23. A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
    [Crossref]
  24. S. L. Braunstein and H. J. Kimble, "Teleportation of continuous quantum variables," Phys. Rev. Lett. 80, 869-872 (1998).
    [Crossref]
  25. Usually, the quantum optical phase-space displacement operator D(α)=exp(αâdagger−α*â) is used to describe the displacements in Alice's measurement basis and Bob's correction operation for continuous-variable quantum teleportation. Here, we prefer to stick to the WH group elements, where Z(v)X(u) differs from D(α) only by a global phase with α=u+iv. Alice's measurement basis is then described by [Z(v)X(u) big dot times 1]integral dx∣x›∣x›√π. Bob's correction operation would be Z(v)X(u) in this case. In a communication rather than computation scenario where Alice and Bob are spatially separated, of course, Alice needs to send the results u and v to Bob via a classical channel.

2006 (4)

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[Crossref] [PubMed]

J. Zhang and S. L. Braunstein, "Continuous-variable Gaussian analog of cluster states," Phys. Rev. A 73, 032318 (2006).
[Crossref]

M. A. Nielsen, "Cluster-state quantum computation," Rep. Math. Phys. 57, 147-161 (2006).
[Crossref]

A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
[Crossref]

2005 (5)

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

R. Filip, P. Marek, and U. L. Andersen, "Measurement-induced continuous-variable quantum interactions," Phys. Rev. A 71, 042308 (2005).
[Crossref]

D. E. Browne and T. Rudolph, "Resource-efficient linear optical quantum computation," Phys. Rev. Lett. 95, 010501 (2005).
[Crossref] [PubMed]

S. L. Braunstein, "Squeezing as an irreducible resource," Phys. Rev. A 71, 055801 (2005).
[Crossref]

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
[Crossref]

2004 (2)

J. M. Geremia, J. K. Stockton, and H. Mabuchi, "Real-time quantum feedback control of atomic spin-squeezing," Science 304, 270-273 (2004).
[Crossref] [PubMed]

M. A. Nielsen, "Optical quantum computation using cluster states," Phys. Rev. Lett. 93, 040503 (2004).
[Crossref] [PubMed]

2003 (1)

S. D. Bartlett and W. J. Munro, "Quantum teleportation of optical quantum gates," Phys. Rev. Lett. 90, 117901 (2003).
[Crossref] [PubMed]

2002 (1)

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, "Efficient classical simulation of continuous variable quantum information processes," Phys. Rev. Lett. 88, 097904 (2002).
[Crossref] [PubMed]

2001 (4)

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

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

B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413, 400-403 (2001).
[Crossref] [PubMed]

D. Gottesman, A. Kitaev, and J. Preskill, "Encoding a qubit in an oscillator," Phys. Rev. A 64, 012310 (2001).
[Crossref]

2000 (1)

X. Zhou, D. W. Leung, and I. L. Chuang, "Methodology for quantum logic gate construction," Phys. Rev. A 62, 052316 (2000).
[Crossref]

1999 (2)

S. Lloyd and S. L. Braunstein, "Quantum computation over continuous variables," Phys. Rev. Lett. 82, 1784-1787 (1999).
[Crossref]

D. Gottesman and I. L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations," Nature 402, 390-393 (1999).
[Crossref]

1998 (2)

P. Grangier, J. A. Levenson, and J.-P. Poizat, "Quantum non-demolition measurements in optics," Nature 396, 537-542 (1998).
[Crossref]

S. L. Braunstein and H. J. Kimble, "Teleportation of continuous quantum variables," Phys. Rev. Lett. 80, 869-872 (1998).
[Crossref]

Andersen, U. L.

R. Filip, P. Marek, and U. L. Andersen, "Measurement-induced continuous-variable quantum interactions," Phys. Rev. A 71, 042308 (2005).
[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 434, 169-176 (2005).
[Crossref] [PubMed]

Bartlett, S. D.

S. D. Bartlett and W. J. Munro, "Quantum teleportation of optical quantum gates," Phys. Rev. Lett. 90, 117901 (2003).
[Crossref] [PubMed]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, "Efficient classical simulation of continuous variable quantum information processes," Phys. Rev. Lett. 88, 097904 (2002).
[Crossref] [PubMed]

Braunstein, S. L.

J. Zhang and S. L. Braunstein, "Continuous-variable Gaussian analog of cluster states," Phys. Rev. A 73, 032318 (2006).
[Crossref]

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
[Crossref]

S. L. Braunstein, "Squeezing as an irreducible resource," Phys. Rev. A 71, 055801 (2005).
[Crossref]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, "Efficient classical simulation of continuous variable quantum information processes," Phys. Rev. Lett. 88, 097904 (2002).
[Crossref] [PubMed]

S. Lloyd and S. L. Braunstein, "Quantum computation over continuous variables," Phys. Rev. Lett. 82, 1784-1787 (1999).
[Crossref]

S. L. Braunstein and H. J. Kimble, "Teleportation of continuous quantum variables," Phys. Rev. Lett. 80, 869-872 (1998).
[Crossref]

Briegel, H. J.

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

Browne, D. E.

D. E. Browne and T. Rudolph, "Resource-efficient linear optical quantum computation," Phys. Rev. Lett. 95, 010501 (2005).
[Crossref] [PubMed]

Chuang, I. L.

X. Zhou, D. W. Leung, and I. L. Chuang, "Methodology for quantum logic gate construction," Phys. Rev. A 62, 052316 (2000).
[Crossref]

D. Gottesman and I. L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations," Nature 402, 390-393 (1999).
[Crossref]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

Filip, R.

R. Filip, P. Marek, and U. L. Andersen, "Measurement-induced continuous-variable quantum interactions," Phys. Rev. A 71, 042308 (2005).
[Crossref]

Geremia, J. M.

J. M. Geremia, J. K. Stockton, and H. Mabuchi, "Real-time quantum feedback control of atomic spin-squeezing," Science 304, 270-273 (2004).
[Crossref] [PubMed]

Gottesman, D.

D. Gottesman, A. Kitaev, and J. Preskill, "Encoding a qubit in an oscillator," Phys. Rev. A 64, 012310 (2001).
[Crossref]

D. Gottesman and I. L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations," Nature 402, 390-393 (1999).
[Crossref]

Grangier, P.

P. Grangier, J. A. Levenson, and J.-P. Poizat, "Quantum non-demolition measurements in optics," Nature 396, 537-542 (1998).
[Crossref]

Grosse, N. B.

A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
[Crossref]

Gu, M.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[Crossref] [PubMed]

Jeong, H.

A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
[Crossref]

Julsgaard, B.

B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413, 400-403 (2001).
[Crossref] [PubMed]

Kimble, H. J.

S. L. Braunstein and H. J. Kimble, "Teleportation of continuous quantum variables," Phys. Rev. Lett. 80, 869-872 (1998).
[Crossref]

Kitaev, A.

D. Gottesman, A. Kitaev, and J. Preskill, "Encoding a qubit in an oscillator," Phys. Rev. A 64, 012310 (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]

Kozhekin, A.

B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413, 400-403 (2001).
[Crossref] [PubMed]

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]

Lam, P. K.

A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
[Crossref]

Lance, A. M.

A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
[Crossref]

Leung, D. W.

X. Zhou, D. W. Leung, and I. L. Chuang, "Methodology for quantum logic gate construction," Phys. Rev. A 62, 052316 (2000).
[Crossref]

Levenson, J. A.

P. Grangier, J. A. Levenson, and J.-P. Poizat, "Quantum non-demolition measurements in optics," Nature 396, 537-542 (1998).
[Crossref]

Lloyd, S.

S. Lloyd and S. L. Braunstein, "Quantum computation over continuous variables," Phys. Rev. Lett. 82, 1784-1787 (1999).
[Crossref]

Mabuchi, H.

J. M. Geremia, J. K. Stockton, and H. Mabuchi, "Real-time quantum feedback control of atomic spin-squeezing," Science 304, 270-273 (2004).
[Crossref] [PubMed]

Marek, P.

R. Filip, P. Marek, and U. L. Andersen, "Measurement-induced continuous-variable quantum interactions," Phys. Rev. A 71, 042308 (2005).
[Crossref]

Menicucci, N. C.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[Crossref] [PubMed]

Milburn, G. J.

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

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

Munro, W. J.

S. D. Bartlett and W. J. Munro, "Quantum teleportation of optical quantum gates," Phys. Rev. Lett. 90, 117901 (2003).
[Crossref] [PubMed]

Nemoto, K.

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, "Efficient classical simulation of continuous variable quantum information processes," Phys. Rev. Lett. 88, 097904 (2002).
[Crossref] [PubMed]

Nielsen, M. A.

M. A. Nielsen, "Cluster-state quantum computation," Rep. Math. Phys. 57, 147-161 (2006).
[Crossref]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[Crossref] [PubMed]

M. A. Nielsen, "Optical quantum computation using cluster states," Phys. Rev. Lett. 93, 040503 (2004).
[Crossref] [PubMed]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

Poizat, J.-P.

P. Grangier, J. A. Levenson, and J.-P. Poizat, "Quantum non-demolition measurements in optics," Nature 396, 537-542 (1998).
[Crossref]

Polzik, E. S.

B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413, 400-403 (2001).
[Crossref] [PubMed]

Preskill, J.

D. Gottesman, A. Kitaev, and J. Preskill, "Encoding a qubit in an oscillator," Phys. Rev. A 64, 012310 (2001).
[Crossref]

Ralph, T. C.

A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
[Crossref]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[Crossref] [PubMed]

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 434, 169-176 (2005).
[Crossref] [PubMed]

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 434, 169-176 (2005).
[Crossref] [PubMed]

D. E. Browne and T. Rudolph, "Resource-efficient linear optical quantum computation," Phys. Rev. Lett. 95, 010501 (2005).
[Crossref] [PubMed]

Sanders, B. C.

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, "Efficient classical simulation of continuous variable quantum information processes," Phys. Rev. Lett. 88, 097904 (2002).
[Crossref] [PubMed]

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 434, 169-176 (2005).
[Crossref] [PubMed]

Stockton, J. K.

J. M. Geremia, J. K. Stockton, and H. Mabuchi, "Real-time quantum feedback control of atomic spin-squeezing," Science 304, 270-273 (2004).
[Crossref] [PubMed]

Symul, T.

A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
[Crossref]

van Loock, P.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[Crossref] [PubMed]

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
[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 434, 169-176 (2005).
[Crossref] [PubMed]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

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 434, 169-176 (2005).
[Crossref] [PubMed]

Weedbrook, C.

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[Crossref] [PubMed]

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 434, 169-176 (2005).
[Crossref] [PubMed]

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 434, 169-176 (2005).
[Crossref] [PubMed]

Zhang, J.

J. Zhang and S. L. Braunstein, "Continuous-variable Gaussian analog of cluster states," Phys. Rev. A 73, 032318 (2006).
[Crossref]

Zhou, X.

X. Zhou, D. W. Leung, and I. L. Chuang, "Methodology for quantum logic gate construction," Phys. Rev. A 62, 052316 (2000).
[Crossref]

Nature (5)

P. Grangier, J. A. Levenson, and J.-P. Poizat, "Quantum non-demolition measurements in optics," Nature 396, 537-542 (1998).
[Crossref]

B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413, 400-403 (2001).
[Crossref] [PubMed]

D. Gottesman and I. L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations," Nature 402, 390-393 (1999).
[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]

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

Phys. Rev. A (6)

X. Zhou, D. W. Leung, and I. L. Chuang, "Methodology for quantum logic gate construction," Phys. Rev. A 62, 052316 (2000).
[Crossref]

D. Gottesman, A. Kitaev, and J. Preskill, "Encoding a qubit in an oscillator," Phys. Rev. A 64, 012310 (2001).
[Crossref]

R. Filip, P. Marek, and U. L. Andersen, "Measurement-induced continuous-variable quantum interactions," Phys. Rev. A 71, 042308 (2005).
[Crossref]

A. M. Lance, H. Jeong, N. B. Grosse, T. Symul, T. C. Ralph, and P. K. Lam, "Quantum-state engineering with continuous-variable postselection," Phys. Rev. A 73, 041801(R) (2006).
[Crossref]

J. Zhang and S. L. Braunstein, "Continuous-variable Gaussian analog of cluster states," Phys. Rev. A 73, 032318 (2006).
[Crossref]

S. L. Braunstein, "Squeezing as an irreducible resource," Phys. Rev. A 71, 055801 (2005).
[Crossref]

Phys. Rev. Lett. (8)

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

S. Lloyd and S. L. Braunstein, "Quantum computation over continuous variables," Phys. Rev. Lett. 82, 1784-1787 (1999).
[Crossref]

N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, "Universal quantum computation with continuous-variable cluster states," Phys. Rev. Lett. 97, 110501 (2006).
[Crossref] [PubMed]

S. D. Bartlett and W. J. Munro, "Quantum teleportation of optical quantum gates," Phys. Rev. Lett. 90, 117901 (2003).
[Crossref] [PubMed]

M. A. Nielsen, "Optical quantum computation using cluster states," Phys. Rev. Lett. 93, 040503 (2004).
[Crossref] [PubMed]

D. E. Browne and T. Rudolph, "Resource-efficient linear optical quantum computation," Phys. Rev. Lett. 95, 010501 (2005).
[Crossref] [PubMed]

S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, "Efficient classical simulation of continuous variable quantum information processes," Phys. Rev. Lett. 88, 097904 (2002).
[Crossref] [PubMed]

S. L. Braunstein and H. J. Kimble, "Teleportation of continuous quantum variables," Phys. Rev. Lett. 80, 869-872 (1998).
[Crossref]

Rep. Math. Phys. (1)

M. A. Nielsen, "Cluster-state quantum computation," Rep. Math. Phys. 57, 147-161 (2006).
[Crossref]

Rev. Mod. Phys. (1)

S. L. Braunstein and P. van Loock, "Quantum information with continuous variables," Rev. Mod. Phys. 77, 513-577 (2005).
[Crossref]

Science (1)

J. M. Geremia, J. K. Stockton, and H. Mabuchi, "Real-time quantum feedback control of atomic spin-squeezing," Science 304, 270-273 (2004).
[Crossref] [PubMed]

Other (3)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge U. Press, 2000).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).

Usually, the quantum optical phase-space displacement operator D(α)=exp(αâdagger−α*â) is used to describe the displacements in Alice's measurement basis and Bob's correction operation for continuous-variable quantum teleportation. Here, we prefer to stick to the WH group elements, where Z(v)X(u) differs from D(α) only by a global phase with α=u+iv. Alice's measurement basis is then described by [Z(v)X(u) big dot times 1]integral dx∣x›∣x›√π. Bob's correction operation would be Z(v)X(u) in this case. In a communication rather than computation scenario where Alice and Bob are spatially separated, of course, Alice needs to send the results u and v to Bob via a classical channel.

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Figures (3)

Fig. 1
Fig. 1

Elementary teleportation circuit for continuous-variable cluster computation.

Fig. 2
Fig. 2

Using linear Gaussian cluster states for single-mode evolutions: a chain of coupled squeezed modes is generated via controlled Z gates; an input state ψ can be teleported through the cluster and potentially manipulated at each step; the form of the output state depends on the choice of measurement bases for detecting the observables p ̂ j = D j p ̂ j D j .

Fig. 3
Fig. 3

Off-line transformation of a quantum state ψ via continuous-variable quantum teleportation using a suitably modified two-mode squeezed state (here in the limit of infinite squeezing).

Equations (8)

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C Z ψ p = 0 = 1 π d x d y ψ ( x ) e 2 i x y x y ,
1 π d x d y ψ ( x ) e 2 i x ( y s ) y = X ( s ) F ψ ,
D j U WH = U WH D j ,
D j U WH = U WH D j ,
ψ out = X ( s 2 ) F D 2 X ( s 1 ) F D 1 ψ ,
ψ out = X ( s 2 ) Z ( s 1 ) F D 2 F D 1 ψ ,
F D 4 F D 3 F D 2 F D 1 = F 2 e i κ p ̂ 2 e i κ x ̂ 2 F 2 e i κ p ̂ 2 e i κ x ̂ 2 = e i κ 2 ( x ̂ p ̂ + p ̂ x ̂ ) + O ( κ 3 ) ,
ψ = X ( s 2 ) F Z ( κ s 1 ) X ( s 1 ) D 2 F D 1 ψ = X ( s 2 κ s 1 ) Z ( s 1 ) F 2 e i κ p ̂ 2 e i κ x ̂ 2 ψ ,

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