- Email: [email protected]
One-step generation of cluster states in multiple flux qubits coupled with a nanomechanical resonator
ARTICLE IN PRESS Physica B 405 (2010) 3334–3336
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
One-step generation of cluster states in multiple flux qubits coupled with a nanomechanical resonator Feng-Yang Zhang, Pei Pei, Chong Li , He-Shan Song School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 27 April 2010 Accepted 30 April 2010
We propose a scheme to efficiently produce large cluster states by using many superconducting Josephson junction flux qubits coupling to the quantized nanomechanical resonator (NAMR). This coupling is independently controlled by an external coupling magnetic field. Through controlling the parameter of system, the cluster states are generated by this scheme. Also, our approach is convenient for implementing one way quantum computing. & 2010 Elsevier B.V. All rights reserved.
Keywords: Josephson junction Nanomechanical resonator Quantum computing
1. Introduction Entangled state is an important resources in the quantum information field. Therefore, the best way to prepare entangled states is still an open question. Recently, many schemes have been exploited to prepare entangled states in theory and experiment [1–4]. Particularly, the cluster states are proposed in Ref. [5], and they demonstrate the advantage compare with other entangled states. How to produce highly entangled cluster states and to implement quantum computing with cluster states have already become a favorite topic. Such as, Walther et al. [6] proposed an efficient scheme to generate a cluster state and implement smallscale one-way quantum computing by quantum optics experiment. Due to the difficulty of large-scale integration in the optical devices, the solid-devices are renewed attention because these may be integratability [7]. With the rapid development of fabrication technique, the solid-devices superconducting Josephson junction qubits provide an arena to produce cluster states and implement one-way quantum computing [8–11]. The basic types of superconducting qubits include charge qubit, flux qubit, and phase qubit [12,13]. In Ref. [8] presented an efficient protocol to produce the cluster states with the superconducting qubits in one-step. There is difficult to engineer the initial state. Then You et al. [9] proposed using superconducting quantum circuits to generate the cluster states. These circuits are based on superconducting charge qubits. Compare with former, this scheme is convenient to prepare the initial state and to perform local single qubit measurement. However, these proposals are still challenging in the experiment.
Recently, the NAMR attracted great interest, because its inherent properties that high frequency, low dissipation and small mass. The NAMR coupled with superconducting charge qubits have been extensively studied [14–17]. The quantum nondemolition measurement has been reported [18], two-qubit quantum gate has been implemented [19], and the squeezed states have been produced [20] in this coupling system. Nevertheless, few paper reported the flux qubit coupled with NAMR until now. In this paper, we employ the multiple flux qubits couple to NAMR [21] to produce the highly entangled cluster states. And oneway quantum computation can be implemented in this scheme. The coupling is adjusted by a magnetic field that perpendicular to the NAMR in the coplanar of the flux qubit and NAMR. Compared with other schemes, our work exist several advantages: (i) In contrast to charge qubits, the flux qubit has longer coherence time [21]. Therefore, we do not require all qubits work at the degeneracy point. (ii) The evolution operator is far less sensitive to the thermal state of the NAMR. (iii) Compare to the optical devices, this coupling fabrication is easily combined in experiment. The lithographic technique is rather mature for the NAMR. This paper is organized as follows. In Section 2, we describe the proposed model and write the Hamiltonian for the flux qubit interacting with NAMR. In Section 3, we simplify the Hamiltonian and obtain an efficient evolution operator. The discussions and conclusions of this paper are given in Section 4.
2. Model 2.1. The NAMR
Corresponding author. Tel.: + 86 41184706201. Corresponding author.
E-mail addresses: [email protected] (C. Li), [email protected] (H.-S. Song). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.04.075
Our scheme only consider the fundamental flexural mode of the NAMR, because the coupling much smaller between other
ARTICLE IN PRESS F.-Y. Zhang et al. / Physica B 405 (2010) 3334–3336
3335
modes and the flux qubit [22]. In this case, the NAMR can be modeled as a harmonic oscillator with a high-Q mode of frequency o and the mass m. Therefore, the Hamiltonian without dissipation can be expressed as p2x 1 þ mo2 x2 , 2m 2
HNAMR ¼
ð1Þ
where the canonical momentum p and coordinate x satisfy the commutation relation ½x,p ¼ i, here we choose the unit of ‘. The p and x can be represented by the creation operator ay ffi pffiffiffiffiffiffiffiffiffiffiffi y and annihilation operator a as x ¼ 1= 2m o þ aÞ and p ¼ ða pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i mo=2ðay aÞ, respectively. Thus, the Hamiltonian in Eq. (1) can be rewritten as HNAMR ¼ oðay a þ 12Þ:
ð2Þ
2.2. The flux qubit The structure of a flux qubit [23] is constituted by three Josephson junctions with a superconducting loop, and the charging energy in much smaller than the Josephson coupling energy for each junction. The two junctions have identical Josephson coupling energy EJ and the third one is aEJ . The total Josephson energy of the three junctions is U ¼ EJ cos j1 þ EJ cos j2 þ aEJ cosð2pFf =F0 j1 j2 Þ,
Hf ¼ oia szðiÞ þ Di sxðiÞ ,
ð4Þ
ðiÞ x
where s and s are the Pauli matrices. The two eigenstates of szðiÞ are macroscopically distinct states with the qubit persistent current circulating Ip in opposite direction, i.e., the clockwise state jmS and the counterclockwise state jkS. Di is the tunnel splitting between the two states, oia ¼ Ip ðFfi F0 =2Þ is the energy spacing of the two current states. For convenience, we denote the new basis state j0S and j1S
g
g
j0S ¼ cos jmSþ sin jkS, 2 2
g
ð5Þ
g
ð6Þ
where the tang ¼ Di =oia . By definition the Hamiltonian then becomes Hf ¼ Oi s
where the parameters Oi ¼
UðtÞ ¼ expðiOi Jz tÞexp i
gi2 t
o
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0i oi2 a þ Di and sz ¼ j0S/0jj1S/1j.
In this section, we illustrate our proposal for producing the cluster states and implementing one-way quantum computation in detail. For the single flux qubit coupling with the NAMR, the Hamiltonian can be given by Hi ¼ oay a þ Oi s0iz þ gi ða þay Þðcos gs0iz sin gs0ix Þ:
ð8Þ
To generate the highly entangled cluster states, we consider N identical flux qubits coupling with the NAMR (see Fig. 1). Simultaneously, we adjust the parameters to g ¼ kp, k ¼ 71, 72, 73 . . . : The Hamiltonian of coupling system becomes ð9Þ
gi2 sinot
o2
! # Jz2 expðioay atÞ
ng o exp i ½ay ðeiot 1Þaðeiot 1ÞJz :
ð10Þ
o
Interestingly, if we choose the interaction time t satisfies the condition t ¼ 2np=o, and n is an integer. The time evolution operator (10) reduces to ! 2npgi2 2 2npOi UðtÞ ¼ exp i ð11Þ Jz exp i Jz : 2
o
o
From Eq. (11) we get the NAMR and flux qubits are disentangled at the time t, and the evolution operator is insensitive to the thermal state of the NAMR. Up to an unimportant global phase factor, the evolution operator becomes 2 3 " # N N X X 0 0 0 s ðiÞ exp4izi s ðiÞs ðiþ 1Þ5, UðtÞ ¼ exp ili ð12Þ z
z
z
j4i ¼ 1
where the parameters li ¼ 2npOi =o and zi ¼ 2npgi2 =o2 . If li and zi satisfy the condition
li ¼ 2zi ¼
ð7Þ
3. Generation of the cluster states
H ¼ oay a þ Oi Jz þ gi ða þ ay ÞJz ,
"
i¼1
j1S ¼ sin jmS þ cos jkS, 2 2
0i z,
P 0ðiÞ where Jz ¼ N i ¼ 1 sz , gi represents the coupling coefficient. Therefore, the whole system time-dependent evolution operator is given by UðtÞ ¼ exp½iHt ¼ exp½ioay atiOi Jz tig i ða þ ay ÞJz t, which has been investigated in Refs. [10,19]. Through utilizing the Magnus’ formula [25] the above evolution operator can be written as
ð3Þ
where j1 and j2 are phase differences across the two junctions, respectively. the Ff is the external flux applied in the loop and F0 is the flux quantum. In the vicinity of Ff ¼ F0 =2, the flux qubit can be described as a two-level system [24]. The Hamiltonian for the i-th flux qubit is
ðiÞ z
Fig. 1. The coupling system of the NAMR (orange narrow box) and the multiple flux qubits. The coupling is controlled by a magnetic field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
li ¼ zi ¼
Z 4
Z 4
,
,
i ¼ 2 . . . N1, i ¼ 1,N:
ð13Þ
The long-range Ising-like unitary operator can be obtained by Eq. (12) " # N 1 X 1s0z ðiÞ 1s0z ði þ 1Þ : ð14Þ UðtÞ ¼ exp iZ 2 2 i¼1 If the initial statepffiffiffiof each flux qubit can be prepared to jgi S ¼ ðj0Si þ j1Si Þ= 2, the j0Si and j1Si are the eigenstates of operator s0z ðiÞ with the eigenvalues 71, the cluster states can be produced at the critical time value Z ¼ ð2m þ 1Þp as jCN S ¼
1 N # ½j0Si þj1Si s0z ðiþ 1Þ: 2N=2 i ¼ 1
ð15Þ
What need points out is, the highly entangled states Eq. (15) are different the typical cluster states as defined in Ref. [5]. They are formed from the dress states.
ARTICLE IN PRESS 3336
F.-Y. Zhang et al. / Physica B 405 (2010) 3334–3336
4. Discussion and conclusion
References
We now give a brief discussion on the experimental realization of the proposed scheme. The relation between the integers m and n easily can be obtained i.e. n ¼ ð2m þ 1ÞOi =8o. If m ¼0, the n get minimum value, and correspond to the minimum preparation time of the cluster states is Tmin ¼ 2pnmin =o ¼ 2p=o (where nmin ¼ 1Þ. We consider the frequency o ¼ 0:61 GHz and the decay time of the NAMR is TN ¼ 164 ms [10]. It is clearly seen that the preparation time much shorter than the decay time. In conclusion, we have proposed a different scheme to generate cluster states using NAMR coupling with multiple flux qubits. The cluster states can be efficiently generated in just one step by controlling time. From Eq. (11), it is easy to see, the flux qubit and NAMR are disentangled at the time t. Therefore, the unitary operator (14) is insensitive to the thermal state of NAMR. Also, we easily can see that the value of parameter Z decide the fidelity of cluster state. For the especial values Z ¼ mp with m being an integer, the entanglement disappears among flux qubits. If and only if the Z ¼ ð2m þ1Þp, the high-fidelity cluster states are generated in our scheme. Also, one-way quantum computing can be realized in our scheme.
[1] M. Steffen, M. Ansmann, R.C. Bialczak, N. Katz, E. Lucero, R. Mcdermott, M. Neeley, E.M. Weig, A.N. Cleland, J.M. Martinis, Science 313 (2006) 1423. [2] L.F. Wei, Y.X. Liu, F. Nori, Phys. Rev. Lett. 96 (2006) 246803. ¨ [3] B. Rothlisbeerger, J. Lehmann, D.S. Saraga, P. Traber, D. Loss, Phys. Rev. Lett. 100 (2008) 100502. [4] X.L. Zhang, K.L. Gao, M. Feng, Phys. Rev. A 74 (2006) 024303. [5] R. Raussendorf, H.J. Briegel, Phys. Rev. Lett. 86 (2001) 5188. [6] P. Walther, K.J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeter, A. Zeilinger, Nature (London) 434 (2005) 169. [7] A. Blais, R.S. Huang, A. Wallraff, S.M. Girvin, R.J. Schoelkopf, Phys. Rev. A 69 (2004) 062320. [8] T. Tanamoto, Y.X. Liu, S. Fujita, X. Hu, F. Nori, Phys. Rev. Lett. 97 (2006) 230501. [9] J.Q. You, X.B. Wang, T. Tanamoto, F. Nori, Phys. Rev. A 75 (2007) 052319. [10] G. Chen, Z.D. Chen, L.X. Yu, J.Q. Liang, Phys. Rev. A 76 (2007) 024301. [11] C.W. Wu, Y. Han, X.J. Zhong, P.X. Chen, C.Z. Li, Phys. Rev. A 81 (2010) 034301. ¨ A. Shnirman, Rev. Mod. Phys. 73 (2001) 357. [12] Y. Makhlin, G. Schon, [13] J.Q. You, F. Nori, Phys. Today 58 (11) (2005) 42. [14] A.D. Armour, M.P. Blencowe, K.C. Schwab, Phys. Rev. Lett. 88 (2002) 148301. [15] E.K. Irish, K. Schwab, Phys. Rev. B 68 (2003) 155311. [16] P. Zhang, Y.D. Wang, C.P. Sun, Phys. Rev. Lett. 95 (2005) 097204. [17] M.D. LaHaye, J. Suh, P.M. Echternach, K.C. Schwab, M.L. Roukes, Nature (London) 459 (2009) 08093. [18] E. Buks, E. Arbel-Segev, S. Zaitsev, Eur. Phys. Lett. 81 (2008) 10001. [19] X.B. Zou, W. Mathis, Phys. Lett. A 324 (2004) 484. [20] X.X. Zhou, A. Mizel, Phys. Rev. Lett. 97 (2006) 267201. [21] F. Xue, Y.D. Wang, C.P. Sun, H. Okamoto, H. Yamaguchi, K. Semba, New J. Phys. 9 (2007) 35. [22] A.D. Armour, M.P. Blencowe, Phys. Rev. B 64 (2001) 035311. [23] J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van der Wal, S. Lloyd, Science 285 (1999) 1036. [24] T.P. Orlando, J.E. Mooij, L. Tian, C.H. van der Wal, L.S. Levitov, S. Lloyd, J.J. Mazo, Phys. Rev. B 60 (1999) 15398. [25] W. Magnus, Commun. Pure Appl. Math. 7 (1954) 649.
Acknowledgments This work is supported by the National Science Foundation of China under Grant nos. 60703100 and 10775023.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
One-step generation of cluster states in multiple flux qubits coupled with a nanomechanical resonator Feng-Yang Zhang, Pei Pei, Chong Li , He-Shan Song School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 27 April 2010 Accepted 30 April 2010
We propose a scheme to efficiently produce large cluster states by using many superconducting Josephson junction flux qubits coupling to the quantized nanomechanical resonator (NAMR). This coupling is independently controlled by an external coupling magnetic field. Through controlling the parameter of system, the cluster states are generated by this scheme. Also, our approach is convenient for implementing one way quantum computing. & 2010 Elsevier B.V. All rights reserved.
Keywords: Josephson junction Nanomechanical resonator Quantum computing
1. Introduction Entangled state is an important resources in the quantum information field. Therefore, the best way to prepare entangled states is still an open question. Recently, many schemes have been exploited to prepare entangled states in theory and experiment [1–4]. Particularly, the cluster states are proposed in Ref. [5], and they demonstrate the advantage compare with other entangled states. How to produce highly entangled cluster states and to implement quantum computing with cluster states have already become a favorite topic. Such as, Walther et al. [6] proposed an efficient scheme to generate a cluster state and implement smallscale one-way quantum computing by quantum optics experiment. Due to the difficulty of large-scale integration in the optical devices, the solid-devices are renewed attention because these may be integratability [7]. With the rapid development of fabrication technique, the solid-devices superconducting Josephson junction qubits provide an arena to produce cluster states and implement one-way quantum computing [8–11]. The basic types of superconducting qubits include charge qubit, flux qubit, and phase qubit [12,13]. In Ref. [8] presented an efficient protocol to produce the cluster states with the superconducting qubits in one-step. There is difficult to engineer the initial state. Then You et al. [9] proposed using superconducting quantum circuits to generate the cluster states. These circuits are based on superconducting charge qubits. Compare with former, this scheme is convenient to prepare the initial state and to perform local single qubit measurement. However, these proposals are still challenging in the experiment.
Recently, the NAMR attracted great interest, because its inherent properties that high frequency, low dissipation and small mass. The NAMR coupled with superconducting charge qubits have been extensively studied [14–17]. The quantum nondemolition measurement has been reported [18], two-qubit quantum gate has been implemented [19], and the squeezed states have been produced [20] in this coupling system. Nevertheless, few paper reported the flux qubit coupled with NAMR until now. In this paper, we employ the multiple flux qubits couple to NAMR [21] to produce the highly entangled cluster states. And oneway quantum computation can be implemented in this scheme. The coupling is adjusted by a magnetic field that perpendicular to the NAMR in the coplanar of the flux qubit and NAMR. Compared with other schemes, our work exist several advantages: (i) In contrast to charge qubits, the flux qubit has longer coherence time [21]. Therefore, we do not require all qubits work at the degeneracy point. (ii) The evolution operator is far less sensitive to the thermal state of the NAMR. (iii) Compare to the optical devices, this coupling fabrication is easily combined in experiment. The lithographic technique is rather mature for the NAMR. This paper is organized as follows. In Section 2, we describe the proposed model and write the Hamiltonian for the flux qubit interacting with NAMR. In Section 3, we simplify the Hamiltonian and obtain an efficient evolution operator. The discussions and conclusions of this paper are given in Section 4.
2. Model 2.1. The NAMR
Corresponding author. Tel.: + 86 41184706201. Corresponding author.
E-mail addresses: [email protected] (C. Li), [email protected] (H.-S. Song). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.04.075
Our scheme only consider the fundamental flexural mode of the NAMR, because the coupling much smaller between other
ARTICLE IN PRESS F.-Y. Zhang et al. / Physica B 405 (2010) 3334–3336
3335
modes and the flux qubit [22]. In this case, the NAMR can be modeled as a harmonic oscillator with a high-Q mode of frequency o and the mass m. Therefore, the Hamiltonian without dissipation can be expressed as p2x 1 þ mo2 x2 , 2m 2
HNAMR ¼
ð1Þ
where the canonical momentum p and coordinate x satisfy the commutation relation ½x,p ¼ i, here we choose the unit of ‘. The p and x can be represented by the creation operator ay ffi pffiffiffiffiffiffiffiffiffiffiffi y and annihilation operator a as x ¼ 1= 2m o þ aÞ and p ¼ ða pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i mo=2ðay aÞ, respectively. Thus, the Hamiltonian in Eq. (1) can be rewritten as HNAMR ¼ oðay a þ 12Þ:
ð2Þ
2.2. The flux qubit The structure of a flux qubit [23] is constituted by three Josephson junctions with a superconducting loop, and the charging energy in much smaller than the Josephson coupling energy for each junction. The two junctions have identical Josephson coupling energy EJ and the third one is aEJ . The total Josephson energy of the three junctions is U ¼ EJ cos j1 þ EJ cos j2 þ aEJ cosð2pFf =F0 j1 j2 Þ,
Hf ¼ oia szðiÞ þ Di sxðiÞ ,
ð4Þ
ðiÞ x
where s and s are the Pauli matrices. The two eigenstates of szðiÞ are macroscopically distinct states with the qubit persistent current circulating Ip in opposite direction, i.e., the clockwise state jmS and the counterclockwise state jkS. Di is the tunnel splitting between the two states, oia ¼ Ip ðFfi F0 =2Þ is the energy spacing of the two current states. For convenience, we denote the new basis state j0S and j1S
g
g
j0S ¼ cos jmSþ sin jkS, 2 2
g
ð5Þ
g
ð6Þ
where the tang ¼ Di =oia . By definition the Hamiltonian then becomes Hf ¼ Oi s
where the parameters Oi ¼
UðtÞ ¼ expðiOi Jz tÞexp i
gi2 t
o
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0i oi2 a þ Di and sz ¼ j0S/0jj1S/1j.
In this section, we illustrate our proposal for producing the cluster states and implementing one-way quantum computation in detail. For the single flux qubit coupling with the NAMR, the Hamiltonian can be given by Hi ¼ oay a þ Oi s0iz þ gi ða þay Þðcos gs0iz sin gs0ix Þ:
ð8Þ
To generate the highly entangled cluster states, we consider N identical flux qubits coupling with the NAMR (see Fig. 1). Simultaneously, we adjust the parameters to g ¼ kp, k ¼ 71, 72, 73 . . . : The Hamiltonian of coupling system becomes ð9Þ
gi2 sinot
o2
! # Jz2 expðioay atÞ
ng o exp i ½ay ðeiot 1Þaðeiot 1ÞJz :
ð10Þ
o
Interestingly, if we choose the interaction time t satisfies the condition t ¼ 2np=o, and n is an integer. The time evolution operator (10) reduces to ! 2npgi2 2 2npOi UðtÞ ¼ exp i ð11Þ Jz exp i Jz : 2
o
o
From Eq. (11) we get the NAMR and flux qubits are disentangled at the time t, and the evolution operator is insensitive to the thermal state of the NAMR. Up to an unimportant global phase factor, the evolution operator becomes 2 3 " # N N X X 0 0 0 s ðiÞ exp4izi s ðiÞs ðiþ 1Þ5, UðtÞ ¼ exp ili ð12Þ z
z
z
j4i ¼ 1
where the parameters li ¼ 2npOi =o and zi ¼ 2npgi2 =o2 . If li and zi satisfy the condition
li ¼ 2zi ¼
ð7Þ
3. Generation of the cluster states
H ¼ oay a þ Oi Jz þ gi ða þ ay ÞJz ,
"
i¼1
j1S ¼ sin jmS þ cos jkS, 2 2
0i z,
P 0ðiÞ where Jz ¼ N i ¼ 1 sz , gi represents the coupling coefficient. Therefore, the whole system time-dependent evolution operator is given by UðtÞ ¼ exp½iHt ¼ exp½ioay atiOi Jz tig i ða þ ay ÞJz t, which has been investigated in Refs. [10,19]. Through utilizing the Magnus’ formula [25] the above evolution operator can be written as
ð3Þ
where j1 and j2 are phase differences across the two junctions, respectively. the Ff is the external flux applied in the loop and F0 is the flux quantum. In the vicinity of Ff ¼ F0 =2, the flux qubit can be described as a two-level system [24]. The Hamiltonian for the i-th flux qubit is
ðiÞ z
Fig. 1. The coupling system of the NAMR (orange narrow box) and the multiple flux qubits. The coupling is controlled by a magnetic field. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
li ¼ zi ¼
Z 4
Z 4
,
,
i ¼ 2 . . . N1, i ¼ 1,N:
ð13Þ
The long-range Ising-like unitary operator can be obtained by Eq. (12) " # N 1 X 1s0z ðiÞ 1s0z ði þ 1Þ : ð14Þ UðtÞ ¼ exp iZ 2 2 i¼1 If the initial statepffiffiffiof each flux qubit can be prepared to jgi S ¼ ðj0Si þ j1Si Þ= 2, the j0Si and j1Si are the eigenstates of operator s0z ðiÞ with the eigenvalues 71, the cluster states can be produced at the critical time value Z ¼ ð2m þ 1Þp as jCN S ¼
1 N # ½j0Si þj1Si s0z ðiþ 1Þ: 2N=2 i ¼ 1
ð15Þ
What need points out is, the highly entangled states Eq. (15) are different the typical cluster states as defined in Ref. [5]. They are formed from the dress states.
ARTICLE IN PRESS 3336
F.-Y. Zhang et al. / Physica B 405 (2010) 3334–3336
4. Discussion and conclusion
References
We now give a brief discussion on the experimental realization of the proposed scheme. The relation between the integers m and n easily can be obtained i.e. n ¼ ð2m þ 1ÞOi =8o. If m ¼0, the n get minimum value, and correspond to the minimum preparation time of the cluster states is Tmin ¼ 2pnmin =o ¼ 2p=o (where nmin ¼ 1Þ. We consider the frequency o ¼ 0:61 GHz and the decay time of the NAMR is TN ¼ 164 ms [10]. It is clearly seen that the preparation time much shorter than the decay time. In conclusion, we have proposed a different scheme to generate cluster states using NAMR coupling with multiple flux qubits. The cluster states can be efficiently generated in just one step by controlling time. From Eq. (11), it is easy to see, the flux qubit and NAMR are disentangled at the time t. Therefore, the unitary operator (14) is insensitive to the thermal state of NAMR. Also, we easily can see that the value of parameter Z decide the fidelity of cluster state. For the especial values Z ¼ mp with m being an integer, the entanglement disappears among flux qubits. If and only if the Z ¼ ð2m þ1Þp, the high-fidelity cluster states are generated in our scheme. Also, one-way quantum computing can be realized in our scheme.
[1] M. Steffen, M. Ansmann, R.C. Bialczak, N. Katz, E. Lucero, R. Mcdermott, M. Neeley, E.M. Weig, A.N. Cleland, J.M. Martinis, Science 313 (2006) 1423. [2] L.F. Wei, Y.X. Liu, F. Nori, Phys. Rev. Lett. 96 (2006) 246803. ¨ [3] B. Rothlisbeerger, J. Lehmann, D.S. Saraga, P. Traber, D. Loss, Phys. Rev. Lett. 100 (2008) 100502. [4] X.L. Zhang, K.L. Gao, M. Feng, Phys. Rev. A 74 (2006) 024303. [5] R. Raussendorf, H.J. Briegel, Phys. Rev. Lett. 86 (2001) 5188. [6] P. Walther, K.J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeter, A. Zeilinger, Nature (London) 434 (2005) 169. [7] A. Blais, R.S. Huang, A. Wallraff, S.M. Girvin, R.J. Schoelkopf, Phys. Rev. A 69 (2004) 062320. [8] T. Tanamoto, Y.X. Liu, S. Fujita, X. Hu, F. Nori, Phys. Rev. Lett. 97 (2006) 230501. [9] J.Q. You, X.B. Wang, T. Tanamoto, F. Nori, Phys. Rev. A 75 (2007) 052319. [10] G. Chen, Z.D. Chen, L.X. Yu, J.Q. Liang, Phys. Rev. A 76 (2007) 024301. [11] C.W. Wu, Y. Han, X.J. Zhong, P.X. Chen, C.Z. Li, Phys. Rev. A 81 (2010) 034301. ¨ A. Shnirman, Rev. Mod. Phys. 73 (2001) 357. [12] Y. Makhlin, G. Schon, [13] J.Q. You, F. Nori, Phys. Today 58 (11) (2005) 42. [14] A.D. Armour, M.P. Blencowe, K.C. Schwab, Phys. Rev. Lett. 88 (2002) 148301. [15] E.K. Irish, K. Schwab, Phys. Rev. B 68 (2003) 155311. [16] P. Zhang, Y.D. Wang, C.P. Sun, Phys. Rev. Lett. 95 (2005) 097204. [17] M.D. LaHaye, J. Suh, P.M. Echternach, K.C. Schwab, M.L. Roukes, Nature (London) 459 (2009) 08093. [18] E. Buks, E. Arbel-Segev, S. Zaitsev, Eur. Phys. Lett. 81 (2008) 10001. [19] X.B. Zou, W. Mathis, Phys. Lett. A 324 (2004) 484. [20] X.X. Zhou, A. Mizel, Phys. Rev. Lett. 97 (2006) 267201. [21] F. Xue, Y.D. Wang, C.P. Sun, H. Okamoto, H. Yamaguchi, K. Semba, New J. Phys. 9 (2007) 35. [22] A.D. Armour, M.P. Blencowe, Phys. Rev. B 64 (2001) 035311. [23] J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van der Wal, S. Lloyd, Science 285 (1999) 1036. [24] T.P. Orlando, J.E. Mooij, L. Tian, C.H. van der Wal, L.S. Levitov, S. Lloyd, J.J. Mazo, Phys. Rev. B 60 (1999) 15398. [25] W. Magnus, Commun. Pure Appl. Math. 7 (1954) 649.
Acknowledgments This work is supported by the National Science Foundation of China under Grant nos. 60703100 and 10775023.