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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 10 — May. 20, 2013
  • pp: 12484–12494
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Physical optimization of quantum error correction circuits with spatially separated quantum dot spins

Hong-Fu Wang, Ai-Dong Zhu, and Shou Zhang  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 12484-12494 (2013)
http://dx.doi.org/10.1364/OE.21.012484


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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

Recently there has been growing interest in pursuing semiconductor quantum dots (QD) [37

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

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] .

] as potential qubit candidates for solid-state-based quantum computation and quantum information processing due to their relative isolation from the environment [42

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] .

] and the fact that the electronic spin degrees of freedom in semiconductors typically have decoherence times. Semiconductor QDs have potentially excellent scalability in the implementation of quantum computer and have wide applicability in the field of quantum information rooting in their unique advantages, such as large electric-dipole moments of intersubband transitions, high nonlinear optical susceptibility, and great flexibility in designing devices. For example, a quantum computation protocol based on electron spins in QDs has been previously proposed by Loss and DiVincenzo [43

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

]. In this protocol, they described a universal set of one- and two-qubit gates using the spin states of coupled single-electron QDs based on local exchange interactions controlled by electrodes. Imamoḡlu 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] .

] proposed a new quantum information processing protocol based on QDs strongly coupled to a microcavity mode. They realized the parallel, long-range transverse spin-spin interactions between conduction-band electrons, mediated by the cavity mode, which is an effective way to carry out quantum computation. Hsieh and Hawrylak [45

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] .

] proposed a microscopic theory of quantum circuits based on coded qubits encoded in chirality of electron spin complexes in lateral gated semiconductor triple QD molecules with one-electron spin in each dot. Their work established both single- and double-qubit operations necessarily used for performing quantum computation with chirality-based coded qubits. Hu and Rarity [46

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] .

] proposed an efficient protocol for loss-resistant state teleportation and entanglement swapping using a single QD spin in an optical microcavity based on giant circular birefringence. Majumdar 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] .

] proposed a phonon-mediated off-resonant QD-cavity coupling model. Based on this model, they successfully explained recently observed resonant QD spectroscopic results and showed that phonon-mediated processes effectively extended the detuning range in which off-resonant QD-cavity coupling might occur beyond that given by pure dephasing processes.

In this paper, we propose an efficient protocol for optimal implementation of three-qubit quantum error correction with QD spins trapped in spatially separated cavities. We propose the optimal quantum circuits and describe the physical implementation for correcting both the bit flip and phase flip errors by using one-bit unitary rotation gates and two-bit quantum iSWAP gates that are produced by the long-range interaction between two distributed QD spins mediated by the vacuum fields of the fiber and cavity via virtual-photon-induced process. The present protocol has the following merits: (i) quantum error correction and entanglement among spatially separated quantum nodes is very useful for distributed quantum computation and quantum communication networks; (ii) the selective coupling between two QD spins is much easily manipulated by controlling the optical switch and it is unnecessary to address each QD selectively; (iii) the proposed protocol is robust with respect to systemic parameters, such as the fiber-cavity coupling strength η and frequency detunings Δαj(α=c,L).

2. Model and Hamiltonian

We consider a QD-cavity-fiber coupling system consisting of two distant cavities 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 ωcj and illuminated by a laser field tuned to frequency ωLj. The QDs are doped such that each dot has a full valence band and a single conduction band electron, which is modeled by a three-level atom, as shown in Fig. 1(b), where each QD interacts with the cavity and the laser field with respective detunings Δcj and ΔLj. The QD qubit is defined by the conduction band states |mx = −1/2〉 and |mx = 1/2〉, which correspond the logical zero and one states, respectively, |mx = −1/2〉 ≡ |0〉 and |mx = 1/2〉 ≡ |1〉. In the short fiber limit Lν̄/(2πc) ≤ 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 ( = 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] .

]
H^=H^C+H^QD+H^int+H^CF,
(1)
with
H^C=j=A,Bωcja^cja^cj,H^QD=j=A,B(εj(0)σ^j00+εj(1)σ^j11+εj(v)σ^jvv),H^int=j=A,B[gj(a^cjσ^j1v+a^cjσ^jv1)+Ωj(eiωLjtσ^j0v+eiωLjtσ^jv0)],H^CF=[ηb^(a^cA+a^cB)+H.c.],
(2)
where σ^jmn=|mjjn| with m, n ∈ {0, 1, v}, a^cj(a^cj) is the annihilation (creation) operator for the jth cavity mode, gj and Ωj denote the coupling constants of the jth dot interacting with the cavity mode and the classical laser field, respectively. εj(k) (k ∈ {0, 1, v}) stands for the energy of level |k〉 of the jth dot, is the annihilation operator of the fiber mode, and η is the cavity-fiber coupling strength.

Fig. 1 (a) Schematic of QD-cavity-fiber-coupling system. (b) The relevant levels of QD.

For the sake of convenience, we set gj = g, Ωj = Ω, Δcj=Δc, and ΔLj=ΔL. Under the large-detuning conditions Δcjgj and ΔLjΩj, the level |vj can be eliminated adiabatically, the Hamiltonian describing the QD-cavity and cavity-fiber interactions in the interaction picture is thus given by
H^I=H^I0+H^CF,
(3)
here ĤCF is denoted by Eq. (2) and
H^I0=j=A,B[ξ|0jj0|+ζa^cja^cj|1jj1|+λ(eiδta^cjσ^j10+eiδta^cjσ^j01)],
(4)
where ξ = Ω2L, ζ = g2c, δ = Δc − ΔL, and λ = gΩ(1/Δc + 1/ΔL)/2. The first and second terms in Hamiltonian H^I0 represent the Stark shifts for the states |1〉j 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

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

51

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

], we define
c^0=12(a^cAa^cB),c^1=12(a^cA+a^cB+2b^),c^2=12(a^cA+a^cB2b^),
(5)
here ĉ0, ĉ1, and ĉ2 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
H^CF=2ηc^1c^12ηc^2c^2,H^I0=j=A,B[ξ|0jj0|+ζ4(c^1c^1+c^2c^2+c^2c^2)|1jj1|]+12{λeiδt[(c^1+c^2+2c^0)σ^A10+(c^1+c^22c^0)σ^B10]+ζ2[(c^1c^2+2c^1c^0+2c^2c^0)|1AA1|+(c^1c^22c^1c^02c^2c^0)|1BB1|]+H.c.}.
(6)
We now take ĤCF as the “free Hamiltonian” and perform the unitary transformation eCFt, obtaining
H^I=j=A,B[ξ|0jj0|+ζ4(c^1c^1+c^2c^2+c^2c^2)|1jj1|]+{λ2[ei(δ2η)tc^1+ei(δ+2η)tc^2+2eiδtc^0]σ^A10+λ2[ei(δ2η)tc^1+ei(δ+2η)tc^22eiδtc^0]σ^B10+ζ4[ei22ηtc^1c^2+2ei2ηtc^1c^0+2ei2ηtc^2c^0]|1AA1|+ζ4[ei22ηtc^1c^22ei2ηtc^1c^02ei2ηtc^2c^0]|1BB1|+H.c.}.
(7)
Under the conditions δλ, |δ±2η|λ/2, and 2ηλ/2, 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

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

]
H^eff=j=A,B{ξ|0jj0|+ζ4(c^1c^1+c^2c^2+c^2c^2)|1jj1|+λ24[(2δc^0c^0+1δ2ηc^1c^1+1δ+2ηc^2c^2)|0jj0|+(2δc^0c^0+1δ2ηc^1c^1+1δ2ηc^2c^2)|1jj1|]}+ζ2322η[(c^1c^1c^2c^2)(|1AA1|+|1BB1|)2+4(c^1c^1c^2c^2)(|1AA1||1BB1|)2]ς(σ^A10σ^B01+σ^A01σ^B10),
(8)
where
ς=λ24(2δ1δ2η1δ+2η).
(9)
The quantum numbers of the bosonic modes ĉ0, ĉ1, and ĉ2 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 ĉ0, ĉ1, and ĉ2 remain in the vacuum state during the evolution. In this case the effective Hamiltonian reduces to
H^eff=j=A,B(ξ|0jj0|+ϑ|1jj1|)ς(σ^A10σ^B01+σ^A01σ^B10),
(10)
where
ϑ=14[3ζ+λ2(2δ+1δ2η+1δ+2η)].
(11)
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
U^eff(t)=eiHefft=(ei2ξt0000ei(ξ+ϑ)tcosςtiei(ξ+ϑ)tsinςt00iei(ξ+ϑ)tsinςtei(ξ+ϑ)tcosςt0000ei2ϑt),
(12)
where the basis is ordered as |0〉A|0〉B, |0〉A|1〉B, |1〉A|0〉B, and |1〉A|1〉B. With the choice of ςt = π/2 and performing the single-qubit phase shifts: |0〉jeiξt |0〉j and |1〉jei(π+ϑt)|1〉j (j = A, B), an iSWAP gate is obtained such that
|0A|0B=|0A|0B,|1A|1B=|1A|1B,|0A|1B=i|1A|0B,|1A|0B=i|0A|1B.
(13)
The conventional two-qubit controlled-NOT (CNOT) gate can be constructed by applying iSWAP gate twice (together with six single-qubit rotation gates), namely
U^CNOT=ei(π/4)ei(π/2)σ^z1ei(π/4)σ^z2U^iSWAPei(π/4)σ^x1U^iSWAPei(π/4)σ^z2ei(π/4)σ^x2ei(π/4)σ^z1,
(14)
where σαi (α = x, z; i = 1, 2) are the Pauli operators acting on the ith 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

Three-qubit quantum error correction can correct either one bit flip error or one phase flip error. The quantum circuit for the implementation of quantum error correction for correcting one bit flip error by applying iSWAP gates is shown in Fig. 2, including encoding, decoding, and error-correction steps. The top one qubit, which carries the information to be encoded, is initialized to the |Ψ〉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
u11=(eiπ/400eiπ/4),u12=(cos(π/4)sin(π/4)sin(π/4)cos(π/4)),u13=(ei3π/4cos(π/4)ei3π/4sin(π/4)ei3π/4sin(π/4)ei3π/4cos(π/4)).
(15)
After that, a highly entangled three-qubit state is obtained,
|ΨE=E^(|Ψ1|02|03)=α|02|03|01+β|12|13|11.
(16)
Then the encoded state |Ψ〉E 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.

Fig. 2 Three-bit quantum circuit representation for correcting one bit flip error. Here the gate • − • corresponds to a two-qubit iSWAP gate and uij are single-qubit rotation gates.

For correcting phase errors instead of bit errors, the initial state 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 |±=(|0±|1)/2, and
u21=u22=u23=12(1111),u11=v11u13,u22=u12v12u13,u12=v13u13,u24=u12v13,u13=v13u11,u26=u12v12u13,u14=u12v14,u32=u12v12u13,u21=u23=u25=u31=u11,
(17)
with
v11=(100i),v12=1422(112211),v13=(100eiπ/4),v14=1422(121i(21)i).
(18)
After the decoding procedure, the original state of the first qubit can then be restored by applying a three-qubit Toffoli gate, whose quantum circuit is shown in Fig. 3(b) [52

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

], while the other two qubits contain information about the error which occurred due to the effect of external noise. In this way the full error-correction process, including the final Toffoli gate, is implemented making that the expected state preservation of an arbitrary coherent input.

Fig. 3 (a) Three-bit quantum circuit representation for correcting one bit phase error. (b) Quantum circuit implementation of three-qubit Toffoli gate by applying two-qubit iSWAP gates and single-qubit rotation gates.

Fig. 4 Schematic setup to implement the quantum circuits shown in Fig. 2 and Fig. 3 for correcting both the bit flip error and phase flip error. Here s1 and s2 are optical switches, which control two adjacent cavities whether have interaction or not.

4. Analysis and discussion

We now briefly analyze and discuss some practical issues in relation to the experimental feasibility of the present protocol. For the presently available technology in QD experiments, we set g = 0.5 meV, Ω = g, ΔL = 10g, Δc = 11g, η=2g, and κ = γ = Γ = 0.001g, 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 = Δc − ΔL = g ≈ 10.5λ, which satisfy the conditions Δcg, ΔL ≫ Ω, δλ, |δ±2η|λ/2, and 2ηλ/2, ζ/4. Our calculations show that (i) the probabilities that QD undergo transitions from |0〉 and |1〉 states to |v〉 state are P0v=Ω2/ΔL2=102 and P1v=g2/Δc20.826×102; (ii) the probability that the three modes ĉ0, ĉ1, and ĉ2 are excited due to nonresonant coupling with the classical modes is
P1=λ24[1(δ2η)2+1(δ+2η)2+2δ2]0.709×102.
(19)
Therefore, the effective Hamiltonian Ĥeff 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′ = P1κ ≈ 0.709 × 10−5g, Γ′0v = P0vΓ = 10−5g, and Γ′1v = P1vΓ ≈ 0.826 × 10−5g, respectively; (iv) the time required for realizing the iSWAP gate is t = π/2ς ≈ 2.59 × 102/g, the average infidelity induced by the decoherence is thus about inf = 1.62 × 10−3.

For the quantum error correction process, ideally the output would be identical to the input. To examine the performance of the proposed protocol, we theoretically calculate the entanglement fidelity [55

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

], which is a useful measurement of how well the quantum information in the input is preserved, as shown in Fig. 5. We can see from Fig. 5 that even when the pulse error is chosen as ε = 0.1, the entanglement fidelities Fbf > 99% corresponding to correct bit flip error and Fpf > 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.

Fig. 5 The entanglement fidelities for one-qubit bit flip and phase flip errors correction, as the functions of pulse error ε.

5. Conclusions

In conclusion, we have proposed the optimal quantum circuits and described the physical implementation of quantum error correction by applying iSWAP gate produced by the long-range interaction between distributed QD spins mediated by the vacuum fields of the fiber and cavity. During the process of implementing quantum error correction, neither subsystem is excited and the protocol is insensitive to the cavity decay, the spontaneous emission of QD, and the fiber loss. The protocol is simple and feasible and the experimental realization of the protocol would be a useful step towards the implementation of more complex quantum computation in solid-state qubits.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11264042, 61068001, and 11165015; China Postdoctoral Science Foundation under Grant No. 2012M520612; the Program for Chun Miao Excellent Talents of Jilin Provincial Department of Education under Grant No. 201316; and the Talent Program of Yanbian University of China under Grant No. 950010001.

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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] .

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 84, 032307 (2011) [CrossRef] .

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 85, 012307 (2012) [CrossRef] .

34.

Y. B. Sheng, L. Zhou, and S. M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 042302 (2012) [CrossRef] .

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. 12, 1885–1895 (2013) [CrossRef] .

36.

Y. B. Sheng and L. Zhou, “Efficient W-state entanglement concentration using quantum-dot and optical micro-cavities,” J. Opt. Soc. Am. B 30, 678–686 (2013) [CrossRef] .

37.

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

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 76, 035315 (2007) [CrossRef] .

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. 103, 016805 (2009) [CrossRef] [PubMed] .

40.

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] .

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]

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] .

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] .

48.

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

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] .

50.

S. B. Zheng, “Virtual-photon-induced quantum phase gates for two distant atoms trapped in separate cavities,” Appl. Phys. Lett. 94, 154101 (2009) [CrossRef] .

51.

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

52.

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

55.

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

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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  36. 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]
  37. D. Stepanenko and G. Burkard, “Quantum gates between capacitively coupled double quantum dot two-spin qubits,” Phys. Rev. B75, 085324 (2007). [CrossRef]
  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. B76, 035315 (2007). [CrossRef]
  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.103, 016805 (2009). [CrossRef] [PubMed]
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  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]
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  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]
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  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. B83, 115303 (2011). [CrossRef]
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  50. S. B. Zheng, “Virtual-photon-induced quantum phase gates for two distant atoms trapped in separate cavities,” Appl. Phys. Lett.94, 154101 (2009). [CrossRef]
  51. S. B. Zheng, “Quantum communication and entanglement between two distant atoms via vacuum fields,” Chin. Phys. B19, 064204 (2010). [CrossRef]
  52. N. Schuch and J. Siewert, “Natural two-qubit gate for quantum computation using the XY interaction,” Phys. Rev. A67, 032301 (2003). [CrossRef]
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  54. D. P. DiVincenzo, “Quantum computing and single-qubit measurements using the spin-filter effect,” J. Appl. Phys.85, 4785–4787 (1999). [CrossRef]
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