## Physical optimization of quantum error correction circuits with spatially separated quantum dot spins |

Optics Express, Vol. 21, Issue 10, pp. 12484-12494 (2013)

http://dx.doi.org/10.1364/OE.21.012484

Acrobat PDF (782 KB)

### Abstract

We propose an efficient protocol for optimizing the physical implementation of three-qubit quantum error correction with spatially separated quantum dot spins via virtual-photon-induced process. In the protocol, each quantum dot is trapped in an individual cavity and each two cavities are connected by an optical fiber. We propose the optimal quantum circuits and describe the physical implementation for correcting both the bit flip and phase flip errors by applying a series of one-bit unitary rotation gates and two-bit quantum iSWAP gates that are produced by the long-range interaction between two distributed quantum dot spins mediated by the vacuum fields of the fiber and cavity. The protocol opens promising perspectives for long distance quantum communication and distributed quantum computation networks.

© 2013 OSA

## 1. Introduction

37. D. Stepanenko and G. Burkard, “Quantum gates between capacitively coupled double quantum dot two-spin qubits,” Phys. Rev. B **75**, 085324 (2007) [CrossRef] .

41. T. Meunier, V. E. Calado, and L. M. K. Vandersypen, “Efficient controlled-phase gate for single-spin qubits in quantum dots,” Phys. Rev. B **83**, 121403(R)(2011) [CrossRef] .

42. R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. **79**, 1217–1265 (2007) [CrossRef] .

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

*et al.*[44

44. A. Imamoḡlu, 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] .

45. C. Y. Hsieh and P. Hawrylak, “Quantum circuits based on coded qubits encoded in chirality of electron spin complexes in triple quantum dots,” Phys. Rev. B **82**, 205311 (2010) [CrossRef] .

46. C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,” Phys. Rev. B **83**, 115303 (2011) [CrossRef] .

*et al.*[47

47. A. Majumdar, E. D. Kim, Y. Gong, M. Bajcsy, and J. Vučković, “Phonon mediated off-resonant quantum dot-cavity coupling under resonant excitation of the quantum dot,” Phys. Rev. B **84**, 085309 (2011) [CrossRef] .

*η*and frequency detunings

## 2. Model and Hamiltonian

*A*and

*B*connected by a single-transverse-mode optical fiber, as shown in Fig. 1(a). Each of two QDs is embedded inside a microdisk, put into a single-mode microcavity tuned to frequency

*m*= −1/2〉 and |

_{x}*m*= 1/2〉, which correspond the logical zero and one states, respectively, |

_{x}*m*= −1/2〉 ≡ |0〉 and |

_{x}*m*= 1/2〉 ≡ |1〉. In the short fiber limit

_{x}*Lν̄*/(2

*πc*) ≤ 1, where

*L*is the length of the fiber and

*ν̄*is the decay rate of the cavity field into a continuum of the fiber modes, only one fiber mode essentially interacts with the cavity modes. In this case the total Hamiltonian of the system is written as (

*h̄*= 1) [44

44. A. Imamoḡlu, 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] .

*m*,

*n*∈ {0, 1,

*v*},

*j*th cavity mode,

*g*and Ω

_{j}*denote the coupling constants of the*

_{j}*j*th dot interacting with the cavity mode and the classical laser field, respectively.

*k*∈ {0, 1,

*v*}) stands for the energy of level |

*k*〉 of the

*j*th dot,

*b̂*is the annihilation operator of the fiber mode, and

*η*is the cavity-fiber coupling strength.

*g*=

_{j}*g*, Ω

*= Ω,*

_{j}*v*〉

*can be eliminated adiabatically, the Hamiltonian describing the QD-cavity and cavity-fiber interactions in the interaction picture is thus given by here*

_{j}*Ĥ*is denoted by Eq. (2) and where

_{CF}*ξ*= Ω

^{2}/Δ

*,*

_{L}*ζ*=

*g*

^{2}/Δ

*,*

_{c}*δ*= Δ

*− Δ*

_{c}*, and*

_{L}*λ*=

*g*Ω(1/Δ

*+ 1/Δ*

_{c}*)/2. The first and second terms in Hamiltonian*

_{L}*and |0〉*

_{j}*, which are induced by the cavity fields and classical fields, respectively. The last term describes the Raman coupling of the two states |0〉*

_{j}*and |1〉*

_{j}*. As done in Refs. [48*

_{j}48. A. Serafini, S. Mancini, and S. Bose, “Distributed Quantum Computation via Optical Fibers,” Phys. Rev. Lett. **96**, 010503 (2006) [CrossRef] [PubMed] .

51. S. B. Zheng, “Quantum communication and entanglement between two distant atoms via vacuum fields,” Chin. Phys. B **19**, 064204 (2010) [CrossRef] .

*ĉ*

_{0},

*ĉ*

_{1}, and

*ĉ*

_{2}are three bosonic modes, which are linearly relative to the field modes of the cavities and fiber. Then the Hamiltonian of the whole system in Eq. (3) can be rewritten as

*Ĥ*as the “free Hamiltonian” and perform the unitary transformation

_{CF}*e*

^{iĤCFt}, obtaining

*δ*≫

*λ*,

*ζ*/4, the bosonic modes not only do not exchange quantum numbers with the QD system, but also do not exchange quantum numbers with each other. The Stark shifts and Heisenberg

*XY*coupling between the QDs are induced by the off-resonant Raman coupling. We thus have the effective Hamiltonian [49

49. H. F. Wang, S. Zhang, A. D. Zhu, and K. H. Yeon, “Fast and effective implementation of discrete quantum Fourier transform via virtual-photon-induced process in separate cavities,” J. Opt. Soc. Am. B **29**, 1078–1084 (2012) [CrossRef] .

51. S. B. Zheng, “Quantum communication and entanglement between two distant atoms via vacuum fields,” Chin. Phys. B **19**, 064204 (2010) [CrossRef] .

*ĉ*

_{0},

*ĉ*

_{1}, and

*ĉ*

_{2}are conserved during the interaction. Assume that the two cavity modes

*A*and

*B*and the fiber mode are all initially in the vacuum state. Then the three bosonic modes

*ĉ*

_{0},

*ĉ*

_{1}, and

*ĉ*

_{2}remain in the vacuum state during the evolution. In this case the effective Hamiltonian reduces to where The quantum information in the present protocol is stored in the state |0〉 and |1〉. For QD-cavity-fiber interaction system shown in Fig. 1, two-qubit operation produced by Hamiltonian (10) acting on qubits

*A*and

*B*can thus be expressed as

*|0〉*

_{A}*, |0〉*

_{B}*|1〉*

_{A}*, |1〉*

_{B}*|0〉*

_{A}*, and |1〉*

_{B}*|1〉*

_{A}*. With the choice of*

_{B}*ςt*=

*π*/2 and performing the single-qubit phase shifts: |0〉

*→*

_{j}*e*|0〉

^{i}^{ξ}^{t}*and |1〉*

_{j}*→*

_{j}*e*

^{i}^{(}

^{π}^{+}

^{ϑt}^{)}|1〉

*(*

_{j}*j*=

*A*,

*B*), an iSWAP gate is obtained such that The conventional two-qubit controlled-NOT (CNOT) gate can be constructed by applying iSWAP gate twice (together with six single-qubit rotation gates), namely

*α*=

*x*,

*z; i*= 1, 2) are the Pauli operators acting on the

*i*th qubit. In the following we show how to implement three-qubit quantum error correction for both bit flip and phase flip errors based on the iSWAP gate produced by the QD-cavity-fiber interaction model under Hamiltonian (10).

## 3. Optimal quantum circuit and physical implementation of three-qubit quantum error correction

_{1}=

*α*|0〉

_{1}+

*β*|1〉

_{1}state, and the bottom two qubits are initialized to the |0〉

_{2}|0〉

_{3}state. To protect the state |Ψ〉

_{1}from external decoherence, the state |Ψ〉

_{1}is encoded by applying one- and two-qubit gate operations to qubits 1, 2, and 3, as shown in Fig. 2, where

*is subject to external noise (i.e., (simulated) external decoherence partly disrupts the state), leading to a (partial) bit flip of one of the spins, |Ψ〉*

_{E}*→ |Ψ′〉*

_{E}*. After the decoding network, the state |Ψ〉*

_{E}*is decoded (recovered). Finally the qubits 2 and 3 are measured. If both qubits are in state |1〉, then the qubit 1 is flipped, otherwise it is left unchanged.*

_{E}*α*|0〉

_{1}|0〉

_{2}|0〉

_{3}+

*β*|1〉

_{1}|0〉

_{2}|0〉

_{3}should be encoded as

*α*|+〉

_{2}|+〉

_{3}|+〉

_{1}+

*β*|−〉

_{2}|−〉

_{3}|−〉

_{1}in the encoding procedure, as shown in Fig. 3(a), where

52. N. Schuch and J. Siewert, “Natural two-qubit gate for quantum computation using the XY interaction,” Phys. Rev. A **67**, 032301 (2003) [CrossRef] .

*τ*=

*π*/2

*ς*such that the two-qubit iSWAP gate operating on the nearest-neighbor two QD spins is achieved. Furthermore, the physical implementations of quantum measurements on spins in QDs have been proposed in previous Refs. [43

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

53. G. Burkard, D. Loss, and D. P. DiVincenzo, “Coupled quantum dots as quantum gates,” Phys. Rev. B **59**, 2070–2078 (1999) [CrossRef] .

54. D. P. DiVincenzo, “Quantum computing and single-qubit measurements using the spin-filter effect,” J. Appl. Phys. **85**, 4785–4787 (1999) [CrossRef] .

## 4. Analysis and discussion

*g*= 0.5 meV, Ω =

*g*, Δ

*= 10*

_{L}*g*, Δ

*= 11*

_{c}*g*,

*κ*=

*γ*= Γ = 0.001

*g*, with

*κ*,

*γ*, and Γ being the decay rates for the cavity modes, the fiber mode, and the excited state of QD, respectively. In this way we have

*δ*= Δ

*− Δ*

_{c}*=*

_{L}*g*≈ 10.5

*λ*, which satisfy the conditions Δ

*≫*

_{c}*g*, Δ

*≫ Ω,*

_{L}*δ*≫

*λ*,

*ζ*/4. Our calculations show that (i) the probabilities that QD undergo transitions from |0〉 and |1〉 states to |

*v*〉 state are

*ĉ*

_{0},

*ĉ*

_{1}, and

*ĉ*

_{2}are excited due to nonresonant coupling with the classical modes is Therefore, the effective Hamiltonian

*Ĥ*′

_{eff}in Eq. (10) is valid; (iii) the effective decoherence rates due to the decay of the bosonic modes and the spontaneous emissions of QD are

*κ*′ =

*P*

_{1}

*κ*≈ 0.709 × 10

^{−5}

*g*, Γ′

_{0}

*=*

_{v}*P*

_{0}

*Γ = 10*

_{v}^{−5}

*g*, and Γ′

_{1}

*=*

_{v}*P*

_{1}

*Γ ≈ 0.826 × 10*

_{v}^{−5}

*g*, respectively; (iv) the time required for realizing the iSWAP gate is

*t*=

*π*/2

*ς*≈ 2.59 × 10

^{2}/

*g*, the average infidelity induced by the decoherence is thus about

*F̄*

_{inf}= 1.62 × 10

^{−3}.

55. B. Schumacher, “Sending entanglement through noisy quantum channels,” Phys. Rev. A **54**, 2614–2628 (1996) [CrossRef] [PubMed] .

*ε*= 0.1, the entanglement fidelities

*F*

_{bf}> 99% corresponding to correct bit flip error and

*F*

_{pf}> 97% to phase flip error. On the other hand, when the thermal photons in the environment can be negligible, the protocol is insensitive to the cavity decay, fiber loss, and the spontaneous emission of QD since the long-range interaction between two distributed QDs is mediated by the vacuum fields of the fiber and cavity and the total system evolves in the decoherence-free subspace in which neither of the subsystems is excited. These features make the protocol more feasible for experimental realization and very promising for the implementations of scalable quantum communication networks and distributed quantum computation. Therefore, the proposed protocol is efficient and feasible.

## 5. Conclusions

## Acknowledgments

## References and links

1. | P. W. Shor, “Algorithms for quantum computer computation: discrete logarithms and factoring,” in Proceedings of the Symposium on the Foundations of Computer Science, Los Alamitos, California(IEEE Computer Society, 1994), pp. 124–134 |

2. | L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett. |

3. | M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight Bounds on Quantum Searching,” Fortschr. Phys. |

4. | A. Y. Kitaev, “Quantum measurements and the Abelian Stabilizer Problem,” quant-ph/9511026 . |

5. | D. Simon, “On the power of quantum computation,” inProceedings of the Symposium on the Foundations of Computer Science, Los Alamitos, California (IEEE Computer Society, 1994) , pp. 116–123 |

6. | R. Jozsa, “Quantum Algorithms and the Fourier Transform,” quant-ph/9707033 . |

7. | C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. |

8. | C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett. |

9. | C. H. Bennett, “Quantum cryptography using any two nonorthogonal states,” Phys. Rev. Lett. |

10. | F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A |

11. | F. G. Deng and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A |

12. | F. G. Deng and G. L. Long, “Secure direct communication with a quantum one-time pad,” Phys. Rev. A |

13. | P. W. Shor, “Fault-tolerant quantum computation,” in Proceedings of the 37th Symposium on Foundations of Computing(IEEE Computer Society, 1996), pp. 56–65; e-print quant-ph/9605011 . |

14. | D. P. DiVincenzo and P. W. Shor, “Fault-Tolerant Error Correction with Efficient Quantum Codes,” Phys. Rev. Lett. |

15. | M. A. Nielsen and I. L. Chuang, “ |

16. | H. F. Wang, S. Zhang, A. D. Zhu, X. X. Yi, and K. H. Yeon, “Local conversion of four Einstein-Podolsky-Rosen photon pairs into four-photon polarization-entangled decoherence-free states with non-photon-number-resolving detectors,” Opt. Express |

17. | P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A |

18. | R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, “Perfect Quantum Error Correcting Code,” Phys. Rev. Lett. |

19. | C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A |

20. | A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev. Lett. |

21. | E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Phys. Rev. A |

22. | D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, “Experimental quantum error correction,” Phys. Rev. Lett. |

23. | G. Burkard, D. Loss, D. P. DiVincenzo, and J. A. Smolin, “Physical optimization of quantum error correction circuits,” Phys. Rev. B |

24. | O. Moussa, J. Baugh, C. A. Ryan, and R. Laflamme, “Demonstration of Sufficient Control for Two Rounds of Quantum Error Correction in a Solid State Ensemble Quantum Information Processor,” Phys. Rev. Lett. |

25. | J. W. Pan, C Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature |

26. | Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A |

27. | Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A |

28. | F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A |

29. | H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled three-photon W states,” Eur. Phys. J. D |

30. | H. F. Wang, S. Zhang, and K. H. Yeon, “Linear-optics-based entanglement concentration of unknown partially entangled three-photon W states,” J. Opt. Soc. Am. B |

31. | H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Scheme for entanglement concentration of unknown atomic entangled states by interference of polarized photons,” J. Phys. B: At. Mol. Opt. Phys. |

32. | C. Wang, Y. Zhao, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum-dot spins in optical microcavities,” Phys. Rev. A |

33. | Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A |

34. | Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A |

35. | Y. B. Sheng, L. Zhou, L. Wang, and S. M. Zhao, “Efficient entanglement concentration for quantum dot and optical microcavities systems,” Quantum. Inf. Process. |

36. | Y. B. Sheng and L. Zhou, “Efficient W-state entanglement concentration using quantum-dot and optical micro-cavities,” J. Opt. Soc. Am. B |

37. | D. Stepanenko and G. Burkard, “Quantum gates between capacitively coupled double quantum dot two-spin qubits,” Phys. Rev. B |

38. | J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus, and M. D. Lukin, “Relaxation, dephasing, and quantum control of electron spins in double quantum dots,” Phys. Rev. B |

39. | K. D. Petersson, C. G. Smith, D. Anderson, P. Atkinson, G. A. C. Jones, and D. A. Ritchie, “Microwave-Driven Transitions in Two Coupled Semiconductor Charge Qubits,” Phys. Rev. Lett. |

40. | G. Shinkai, T. Hayashi, T. Ota, and T. Fujisawa, “Correlated Coherent Oscillations in Coupled Semiconductor Charge Qubits,” Phys. Rev. Lett. |

41. | T. Meunier, V. E. Calado, and L. M. K. Vandersypen, “Efficient controlled-phase gate for single-spin qubits in quantum dots,” Phys. Rev. B |

42. | R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. |

43. | D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A |

44. | A. Imamoḡlu, 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. |

45. | C. Y. Hsieh and P. Hawrylak, “Quantum circuits based on coded qubits encoded in chirality of electron spin complexes in triple quantum dots,” Phys. Rev. B |

46. | C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,” Phys. Rev. B |

47. | A. Majumdar, E. D. Kim, Y. Gong, M. Bajcsy, and J. Vučković, “Phonon mediated off-resonant quantum dot-cavity coupling under resonant excitation of the quantum dot,” Phys. Rev. B |

48. | A. Serafini, S. Mancini, and S. Bose, “Distributed Quantum Computation via Optical Fibers,” Phys. Rev. Lett. |

49. | H. F. Wang, S. Zhang, A. D. Zhu, and K. H. Yeon, “Fast and effective implementation of discrete quantum Fourier transform via virtual-photon-induced process in separate cavities,” J. Opt. Soc. Am. B |

50. | S. B. Zheng, “Virtual-photon-induced quantum phase gates for two distant atoms trapped in separate cavities,” Appl. Phys. Lett. |

51. | S. B. Zheng, “Quantum communication and entanglement between two distant atoms via vacuum fields,” Chin. Phys. B |

52. | N. Schuch and J. Siewert, “Natural two-qubit gate for quantum computation using the XY interaction,” Phys. Rev. A |

53. | G. Burkard, D. Loss, and D. P. DiVincenzo, “Coupled quantum dots as quantum gates,” Phys. Rev. B |

54. | D. P. DiVincenzo, “Quantum computing and single-qubit measurements using the spin-filter effect,” J. Appl. Phys. |

55. | B. Schumacher, “Sending entanglement through noisy quantum channels,” Phys. Rev. A |

**OCIS Codes**

(270.5580) Quantum optics : Quantum electrodynamics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: March 4, 2013

Revised Manuscript: April 23, 2013

Manuscript Accepted: April 23, 2013

Published: May 14, 2013

**Citation**

Hong-Fu Wang, Ai-Dong Zhu, and Shou Zhang, "Physical optimization of quantum error correction circuits with spatially separated quantum dot spins," Opt. Express **21**, 12484-12494 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12484

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### References

- P. W. Shor, “Algorithms for quantum computer computation: discrete logarithms and factoring,” in Proceedings of the Symposium on the Foundations of Computer Science, Los Alamitos, California(IEEE Computer Society, 1994), pp. 124–134
- L. K. Grover, “Quantum mechanics helps in searching for a needle in a haystack,” Phys. Rev. Lett.79, 325–328 (1997). [CrossRef]
- M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, “Tight Bounds on Quantum Searching,” Fortschr. Phys.46, 493–505 (1998). [CrossRef]
- A. Y. Kitaev, “Quantum measurements and the Abelian Stabilizer Problem,” quant-ph/9511026 .
- D. Simon, “On the power of quantum computation,” inProceedings of the Symposium on the Foundations of Computer Science, Los Alamitos, California (IEEE Computer Society, 1994) , pp. 116–123
- R. Jozsa, “Quantum Algorithms and the Fourier Transform,” quant-ph/9707033 .
- C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett.70, 1895–1899 (1993). [CrossRef] [PubMed]
- C. H. Bennett and S. J. Wiesner, “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states,” Phys. Rev. Lett.69, 2881–2884 (1992). [CrossRef] [PubMed]
- C. H. Bennett, “Quantum cryptography using any two nonorthogonal states,” Phys. Rev. Lett.68, 3121–3124 (1992). [CrossRef] [PubMed]
- F. G. Deng, G. L. Long, and X. S. Liu, “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block,” Phys. Rev. A68, 042317 (2003). [CrossRef]
- F. G. Deng and G. L. Long, “Controlled order rearrangement encryption for quantum key distribution,” Phys. Rev. A68, 042315 (2003). [CrossRef]
- F. G. Deng and G. L. Long, “Secure direct communication with a quantum one-time pad,” Phys. Rev. A69, 052319 (2004) [CrossRef]
- P. W. Shor, “Fault-tolerant quantum computation,” in Proceedings of the 37th Symposium on Foundations of Computing(IEEE Computer Society, 1996), pp. 56–65; e-print quant-ph/9605011 .
- D. P. DiVincenzo and P. W. Shor, “Fault-Tolerant Error Correction with Efficient Quantum Codes,” Phys. Rev. Lett.77, 3260–3263 (1996). [CrossRef] [PubMed]
- M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information” (Cambridge University, 2000).
- H. F. Wang, S. Zhang, A. D. Zhu, X. X. Yi, and K. H. Yeon, “Local conversion of four Einstein-Podolsky-Rosen photon pairs into four-photon polarization-entangled decoherence-free states with non-photon-number-resolving detectors,” Opt. Express19, 25433–25440 (2011) [CrossRef]
- P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A52, 2493(R)–2496(R) (1995). [CrossRef]
- R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, “Perfect Quantum Error Correcting Code,” Phys. Rev. Lett.77, 198–201 (1996). [CrossRef] [PubMed]
- C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A54, 3824–3851 (1996). [CrossRef] [PubMed]
- A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev. Lett.77, 793–797 (1996). [CrossRef] [PubMed]
- E. Knill and R. Laflamme, “Theory of quantum error-correcting codes,” Phys. Rev. A55, 900–911 (1997). [CrossRef]
- D. G. Cory, M. D. Price, W. Maas, E. Knill, R. Laflamme, W. H. Zurek, T. F. Havel, and S. S. Somaroo, “Experimental quantum error correction,” Phys. Rev. Lett.81, 2152–2155 (1998). [CrossRef]
- G. Burkard, D. Loss, D. P. DiVincenzo, and J. A. Smolin, “Physical optimization of quantum error correction circuits,” Phys. Rev. B60, 11404–11416 (1999). [CrossRef]
- O. Moussa, J. Baugh, C. A. Ryan, and R. Laflamme, “Demonstration of Sufficient Control for Two Rounds of Quantum Error Correction in a Solid State Ensemble Quantum Information Processor,” Phys. Rev. Lett.107, 160501 (2011). [CrossRef] [PubMed]
- J. W. Pan, C Simon, Č. Brukner, and A. Zeilinger, “Entanglement purification for quantum communication,” Nature410, 1067–1070 (2001). [CrossRef] [PubMed]
- Y. B. Sheng, F. G. Deng, and H. Y. Zhou, “Efficient polarization-entanglement purification based on parametric down-conversion sources with cross-Kerr nonlinearity,” Phys. Rev. A77, 042308 (2008). [CrossRef]
- Y. B. Sheng and F. G. Deng, “Deterministic entanglement purification and complete nonlocal Bell-state analysis with hyperentanglement,” Phys. Rev. A81, 032307 (2010). [CrossRef]
- F. G. Deng, “One-step error correction for multipartite polarization entanglement,” Phys. Rev. A83, 062316 (2011). [CrossRef]
- H. F. Wang, S. Zhang, and K. H. Yeon, “Linear optical scheme for entanglement concentration of two partially entangled three-photon W states,” Eur. Phys. J. D56, 271–275 (2010). [CrossRef]
- H. F. Wang, S. Zhang, and K. H. Yeon, “Linear-optics-based entanglement concentration of unknown partially entangled three-photon W states,” J. Opt. Soc. Am. B27, 2159–2164 (2010). [CrossRef]
- H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Scheme for entanglement concentration of unknown atomic entangled states by interference of polarized photons,” J. Phys. B: At. Mol. Opt. Phys.43, 235501 (2010). [CrossRef]
- C. Wang, Y. Zhao, and G. S. Jin, “Entanglement purification and concentration of electron-spin entangled states using quantum-dot spins in optical microcavities,” Phys. Rev. A84, 032307 (2011). [CrossRef]
- Y. B. Sheng, L. Zhou, S. M. Zhao, and B. Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A85, 012307 (2012). [CrossRef]
- Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A85, 042302 (2012). [CrossRef]
- Y. B. Sheng, L. Zhou, L. Wang, and S. M. Zhao, “Efficient entanglement concentration for quantum dot and optical microcavities systems,” Quantum. Inf. Process.12, 1885–1895 (2013). [CrossRef]
- Y. B. Sheng and L. Zhou, “Efficient W-state entanglement concentration using quantum-dot and optical micro-cavities,” J. Opt. Soc. Am. B30, 678–686 (2013). [CrossRef]
- D. Stepanenko and G. Burkard, “Quantum gates between capacitively coupled double quantum dot two-spin qubits,” Phys. Rev. B75, 085324 (2007). [CrossRef]
- J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. Marcus, and M. D. Lukin, “Relaxation, dephasing, and quantum control of electron spins in double quantum dots,” Phys. Rev. B76, 035315 (2007). [CrossRef]
- K. D. Petersson, C. G. Smith, D. Anderson, P. Atkinson, G. A. C. Jones, and D. A. Ritchie, “Microwave-Driven Transitions in Two Coupled Semiconductor Charge Qubits,” Phys. Rev. Lett.103, 016805 (2009). [CrossRef] [PubMed]
- G. Shinkai, T. Hayashi, T. Ota, and T. Fujisawa, “Correlated Coherent Oscillations in Coupled Semiconductor Charge Qubits,” Phys. Rev. Lett.103, 056802 (2009). [CrossRef] [PubMed]
- T. Meunier, V. E. Calado, and L. M. K. Vandersypen, “Efficient controlled-phase gate for single-spin qubits in quantum dots,” Phys. Rev. B83, 121403(R)(2011). [CrossRef]
- R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys.79, 1217–1265 (2007). [CrossRef]
- D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A57, 120–126 (1998) [CrossRef]
- A. Imamoḡlu, 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]
- C. Y. Hsieh and P. Hawrylak, “Quantum circuits based on coded qubits encoded in chirality of electron spin complexes in triple quantum dots,” Phys. Rev. B82, 205311 (2010). [CrossRef]
- C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,” Phys. Rev. B83, 115303 (2011). [CrossRef]
- A. Majumdar, E. D. Kim, Y. Gong, M. Bajcsy, and J. Vučković, “Phonon mediated off-resonant quantum dot-cavity coupling under resonant excitation of the quantum dot,” Phys. Rev. B84, 085309 (2011). [CrossRef]
- A. Serafini, S. Mancini, and S. Bose, “Distributed Quantum Computation via Optical Fibers,” Phys. Rev. Lett.96, 010503 (2006). [CrossRef] [PubMed]
- H. F. Wang, S. Zhang, A. D. Zhu, and K. H. Yeon, “Fast and effective implementation of discrete quantum Fourier transform via virtual-photon-induced process in separate cavities,” J. Opt. Soc. Am. B29, 1078–1084 (2012). [CrossRef]
- S. B. Zheng, “Virtual-photon-induced quantum phase gates for two distant atoms trapped in separate cavities,” Appl. Phys. Lett.94, 154101 (2009). [CrossRef]
- S. B. Zheng, “Quantum communication and entanglement between two distant atoms via vacuum fields,” Chin. Phys. B19, 064204 (2010). [CrossRef]
- N. Schuch and J. Siewert, “Natural two-qubit gate for quantum computation using the XY interaction,” Phys. Rev. A67, 032301 (2003). [CrossRef]
- G. Burkard, D. Loss, and D. P. DiVincenzo, “Coupled quantum dots as quantum gates,” Phys. Rev. B59, 2070–2078 (1999). [CrossRef]
- D. P. DiVincenzo, “Quantum computing and single-qubit measurements using the spin-filter effect,” J. Appl. Phys.85, 4785–4787 (1999). [CrossRef]
- B. Schumacher, “Sending entanglement through noisy quantum channels,” Phys. Rev. A54, 2614–2628 (1996). [CrossRef] [PubMed]

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