- Email: [email protected]
Quantum gate operation with non-instantaneous unitary kicks
Optics Communications 284 (2011) 1099–1104
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Quantum gate operation with non-instantaneous unitary kicks Xiao-Qiang Shao a,b, Li Chen a,b, Shou Zhang a,b,⁎, Yong-Fang Zhao a, Kyu-Hwang Yeon c a b c
Center for the Condensed-Matter Science and Technology, Department of Physics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, PR China Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, PR China BK21 Program Physics & Department of Physics, College of Natural Science, Chungbuk National University, Cheonju, Chungbuk 361-763, Republic of Korea
a r t i c l e
i n f o
Article history: Received 4 June 2010 Received in revised form 10 September 2010 Accepted 13 October 2010 Keywords: Quantum Zeno effect Unitary kicks
a b s t r a c t A two-qubit controlled-z gate is presented based on the non-instantaneous unitary kicks. Instead of putting two atoms through the cavity simultaneously, we make the atoms cross the cavity sequentially. The interaction between the second atom and the cavity plays the role for kicking the evolution of the system consisting of the first atom and cavity. By repeating the whole process N times, we obtain the controlled-z gate with a high fidelity. The effects of decoherence such as spontaneous emission and the loss of cavity on the average gate fidelity are investigated in virtue of master equation. Furthermore the method for achieving the multi-qubit controlled-z gate is also proposed. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The existing algorithms exposit that the quantum computer would be more powerful than the classical computer [1,2]. This matter of fact has triggered in the past years many studies on the theoretical and the practical aspects of quantum computing. Generally speaking, the quantum computers are divided into two categories according to the principle for calculation, i.e., the measurement-based quantum computer and the circuit-based quantum computer. Quantum logic gate is the central and basic element for actualizing the circuit-based quantum computer. In 1995, DiVincenzo pointed out that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit [3]. Thus the investigation on the universal quantum logic gate becomes the research focus and different physical models have been established in recent years, such as ion-trap system [4], cavity quantum electrodynamics (QED) [5,6], nuclear magnetic resonance (NMR) system [7], linear optics [8], quantum dot [9], and superconducting quantum interference device (SQUID) system [10]. The quantum Zeno effect put forward by Misra and Sudarshan describes the peculiar behavior of unstable particles under frequent measurement [11], i.e., the dissipation of an unstable quantum system is suppressed or slowed down, and the transition between quantum states is frozen if measurements are performed frequently. It has been shown that the quantum Zeno effect is not only useful for the decoherence suppression, but also has many applications in other fields of quantum information including quantum computation [12], quantum switching [13,14], and entanglement preparation [15], etc. In 2004, Facchi et al. unified the quantum Zeno effect and the “bang-
⁎ Corresponding author. E-mail address: [email protected] (S. Zhang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.10.048
bang" decoupling method for suppressing decoherence in open quantum systems [16]. They found the evolution of a controlled system Q alternately undergoes N “kicks" Ukick (instantaneous unitary transformations) and free evolutions in a time interval t will be freezed in the large N limit, of which the quantum Zeno effect is obtained by unitary evolutions instead of nonunitary evolutions induced by von Neumann measurements, and they termed this phenomenon iconically as “unitary kicks" in their following works [17]. Recently, Rossi, JR. et al. presented a strategy to control the evolution of a quantum system by applying unitary interactions between the system of interest and a single auxiliary system [18]. We find the potentially physical principle can be brought into the theory of unitary kicks of Facchi's by making a little modification, i.e., replacing the instantaneous unitary transformation (Ukick) with noninstantaneous unitary transformation (time evolution unitary operator Ukick(t)). In practice, the unitary kicks will not be instantaneous, and decoherence will be present during the kicks. In the present work, we utilize the non-instantaneous unitary kicks to implement a two-qubit controlled-z gate. Compared with previous schemes for implementing two-qubit quantum gate with two atoms simultaneously interacting with the cavity field [19–21], the main advantage of the current scheme is that interactions between the cavity field and only a single atom at a time are needed, which may contribute to reducing the experimental complexity for control. The structure of the paper is organized as follows. The principle of non-instantaneous unitary kicks is briefly introduced in Sec. II. Then the two-qubit controlled-z gate is put forward with approximate and accurate calculation in Sec. III. The effects of decoherence involving atomic spontaneous emission and cavity decay on the average fidelity of the gate are investigated in Sec. IV. The generalization for achieving the multi-qubit controlled-z gate is provided in Sec. V. A summary will appear in Sec. VI.
1100
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
2. Preliminary: the evolution of quantum state under noninstantaneous unitary kicks In this section, we give a brief introduction to the principle of unitary kicks by replacing the instantaneous unitary transformation with non-instantaneous unitary transformation [16]. Consider the dynamics system whose evolution is governed by the following operator N t t UN ðt1 + t2 Þ = Ukick 2 U 1 ; N N
U ðt1 Þ =
ð1Þ
ˆ = ∑ P HP ; HZ = PH n n
1 N−1 †N k †k N ∑ U ðU U Þ Ukick HUkick ðUkick U Þ Ukick : N k = 0 kick kick
ð6Þ
is called Zeno Hamiltonian, Pn is the spectral projections of Ukick belonging to the eigenvalue λn and Ukick
t2 −iλ t = N = ∑ e n 2 Pn : N n
ð7Þ
Finally by combining Eq. (2) with Eq. (5), the whole system is governed by the limiting evolution operator
ð2Þ UN ðt1 + t2 Þ∼Ukick
N N t2 t U ðt1 Þ = Ukick 2 expð−iHZ t1 Þ N N = exp −i ∑ λn Pn t2 + Pn HPn t1 :
ð8Þ
n
ð3Þ
with HN ðt1 Þ =
ð5Þ
where
where V N(0) = I and d i V ðt Þ = HN ðt1 ÞV N ðt1 Þ; dt1 N 1
lim V N ðt1 Þ∼expð−iHZ t1 Þ; N→∞
n
where U(t1/N) = exp(− iHt1/N) is the free evolution operator for the system of interest, H is the corresponding Hamiltonian, and Ukick(t2/N) denotes the kick unitary transformation. For convenience, we rewrite the above operator in the following form N †N t t UN ðt1 + t2 Þ = Ukick 2 Ukick 2 UN ðt1 + t2 Þ N N N t = Ukick 2 V N ðt1 Þ; N
We suppose the effect of Ukick is more powerful than U [22], so in the large N limit, the limiting evolution operator
ð4Þ
The action of Ukick is just like an observer performing the repeatedly projective measurement on the system of interest. Once the number of measurement N approaches to infinity, a Zeno subspace will be generated. Eq. (8) is the important result governing our following works.
3. Two-qubit controlled-Z gate We consider two identical three-level atoms repeatedly interacting with the cavity via the resonant interaction sequentially. Each atomic states is denoted by |g i, |e i, and |i i, as shown in Fig. 1. The state |i i is decoupled due to the large detuning, and it is not affected during the atomcavity interaction. The Hamiltonians for the interaction between atom a (atom b) and cavity in the Schrödinger picture are given by † † Hac = ωa ð jea ihea j−jga ihga j Þ + ωc a a + ℏga jea ihga ja + jga ihea ja ;
ð9Þ
† † Hbc = ωb ð jeb iheb j−jgb ihgb j Þ + ωc a a + ℏgb jeb ihgb ja + jgb iheb ja ;
ð10Þ
and
where ωa, ωb, and ωc are the eigenvalues of the free Hamiltonian for atom a, atom b, and cavity, respectively. a† (a) is the creation (annihilation) operator of the cavity mode. ga(b) is the coupling constant between the atom a (b) and the cavity. For simplicity, we suppose ωa = ωb = ωc and move into the interaction picture in the unit of ℏ. The quantum information is encoded into a subspace spanned by the states {|ga, gb i, |ga, ib i, |ea, gb i, |ea, ib i} with the cavity in the vacuum state. The states |ga, gb, 0c i and |ga, ib, 0c i are unchanged because of HIac(bc)|ga, gb, 0c i (|ga, ib, 0c i) = 0. In what follows, we mainly discuss the time evolutions of the states |ea, gb, 0c i and |ea, ib, 0c i.
Fig. 1. The level configuration of the atom. The transition between |g i and |e i is coupled resonantly to the cavity mode with the coupling constant g, and the level |i i is decoupled to the atom-cavity interaction. The quantum information is encoded into the subspace {|ga, gb i, |ga, ib i, |ea, gb i, |ea, ib i}.
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
1101
For the initial state |ea, gb, 0c i, we obtain a closed subspace during the whole evolution process, i.e., {|ga, eb, 0c i, |ga, gb, 1c i, |ea, gb, 0c i}, hence the unitary operators for each cycle between atom and cavity can be written as 0 θ I = Uac N
1 0 B θ B0 cos B N B @ θ 0 −i sin N
0
1 θC C NC C; θ A
−i sin cos
ð11Þ
N
and 0
ϕ B cos N B ϕ I B ϕ =B Ubc B −i sin N @ N 0
1 0C C C C; 0C A 1
ϕ N ϕ cos N 0
−i sin
ð12Þ
where θ = gat1 and ϕ = gbt2. Then the total evolution operator for this subspace reads N ϕ I θ I I U UN ðt1 + t2 Þ = Ubc ; N ac N
ð13Þ
where UIbc(ϕ/N) (|2π − (ϕ mod 2π)| ≫ θ) acts as a kick and its spectral projections are
P1 =
0 1 0 1 −1 0 1 1 1@ 1 −1 1 0 A; P2 = @ 1 1 2 2 0 0 0 0 0
1 0 0 A; 0
ð14Þ
corresponding to the eigenvalues − gb and gb, respectively. In the large N limit, according to Eq. (8), we obtain the final effective evolution operator of the subspace as 0
I UN ðt1
0 + t2 Þ∼exp@ −igb t2 0
−igb t2 0 0
1 0 cosðgb t2 Þ 0 A 0 = @ −i sinðgb t2 Þ 0 0
−i sinðgb t2 Þ cosðgb t2 Þ 0
1 0 0 A; 1
ð15Þ
which implies that the state |ea, gb, 0c i does not evolve if the rotation angle ϕ is relatively larger than θ. For the initial state |ea, ib, 0c i, it only couples to state |ga, ib, 1c i with coupling constant ga, and we obtain the following evolution form jea ; ib ; 0c i→cosðga t Þjea ; ib ; 0c i−i sinðga t Þjga ; ib ; 1c i;
ð16Þ
thus the phase shift of -1 will be acquired after the interaction time t = π/ga. Then we are able to achieve a two-qubit controlled-z gate in the computation basis {|ga, gb i, |ga, ib i, |ea, gb i, |ea, ib i} as 0
1 B0 CZ = B @0 0
0 1 0 0
1 0 0 0 0 C C: 1 0 A 0 −1
ð17Þ
Note that in the above process, the two-qubit controlled-z gate is carried out in the condition of a large N limit. It is necessary for us to quantify the range of N for governing the future experimental operations. All we need to do is to find out the accurate result of Eq. (13), which can be done by mapping the action of UIbc(ϕ/N)UIac(θ/N) on the three dimensional Hilbert subspace {|ga, eb, 0c i, − i|ga, gb, 1c i, |ea, gb, 0c i} onto a rotation on the real Euclidian subspace [18]. Through the straightway algebraic calculation, we obtain 20 cos
ϕ N
6B 6B N 6B ϕ I θ I B Uac =6 Ubc 6B sin ϕ N N 6B N 4@ 0 0
−sin cos 0
ϕ N
ϕ N
10 0
1
0
CB CB CB 0 cos θ CB N B 0C CB A@ θ 0 sin 1 N
1−a2 cosðNφÞ + a2
0
ϕ B cos N C7 B B θ C7 7 B ϕ −sin C 7 B NC C7 = B sin N C7 B B θ A5 @ cos 0 N 0
13N
ab½1−cosðNφÞ−c sinðNφÞ B B B = B ab½1−cosðNφÞ + c sinðNφÞ 1−b2 cosðNφÞ + b2 B @ ac½1−cosðNφÞ−b sinðNφÞ bc½1−cosðNφÞ + a sinðNφÞ
1N ϕ θ sin N N C C C ϕ θ ϕ θC cos cos −cos sin C N N N NC C C A θ θ cos sin N N 1 ac½1−cosðNφÞ + b sinðNφÞ C C C bc½1−cosðNφÞ−a sinðNφÞ C; C A 1−c2 cosðNφÞ + c2
−sin
ϕ θ cos N N
sin
ð18Þ
1102
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
where "
ϕ θ cos φ = arcsin 2 cos 2N 2N
2 3 ϕ θ ϕ θ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# cos + cos + cos cos −1 ϕ θ ϕ θ 2 2 2 6 7 N N N N + sin −sin sin = arccos4 sin2 5; 2 2N 2N 2N 2N
ð19Þ
sin Nθ cos Nϕ + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a= ϕ ϕ ϕ θ θ θ 4 cos 2N sin2 2N cos 2N + sin2 2N −sin2 2N sin2 2N
ð20Þ
b=
sin Nϕ sin Nθ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; ϕ ϕ ϕ θ θ θ sin2 2N 4 cos 2N cos 2N + sin2 2N −sin2 2N sin2 2N
ð21Þ
c=
sin Nϕ cos Nθ + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: ϕ ϕ ϕ θ θ θ sin2 2N 4 cos 2N cos 2N + sin2 2N −sin2 2N sin2 2N
ð22Þ
In Fig. 2, we plot the evolution of the probability amplitude Ceg, 0 for the initial state |ea, gb, 0c i versus the operating steps N by setting θ = π. The three curves correspond to different values of ϕ, i.e., ϕ = Nπ/2 for the black, ϕ = Nπ/5 for the red and ϕ = Nπ/10 for the green, respectively. We are able to see the evolution of the probability amplitude Ceg, 0 depends on two factors, one is the number of operating steps N, and the other is the ratio between the two rotation angles ϕ/θ. For all these three cases, the probability amplitude Ceg, 0 can reach to unity with a good approximation for large N. However, a large ϕ to θ ratio may reduce the number of operating steps N. For the case ϕ = Nπ/2, a stable and high value of Ceg, 0 could be obtained for N ≥ 8. Similarly, we see that N ≥ 20 is large enough for achieving a high fidelity logic gate corresponding to ϕ = Nπ/5 and N ≥ 40 is good for the case ϕ = Nπ/10. Thus we can conclude that for implementing the two-qubit controlled-z gate with a high fidelity, a finite N is sufficiently good if the rotation angles satisfying the relation ϕ/θ ≥ 4 (ϕ ≠ 2L × Nπ, L being an arbitrary integer). 4. The effects of decoherence on the gate fidelity Since no quantum system can be completely isolated from its environment, the irreversible interaction between the quantum system and its environment would lead to decoherence when the two-qubit controlled-z gate is put into experiment. So it is essential to quantify the effect on the gate fidelity. The dissipation channels for the current model include the spontaneous emission of atom (at the rate γ) and the cavity decay (at the rate κ). For the case of realizing the gate with N steps, we need to solve N pairs master equations in the following form
h i κ α † α α † α † I ρ˙ Kth = −i Hac a aρKth −2aρKth a + ρKth a a ; ρα Kth − 2 γej a α a α a α a a σee ρKth −2σje ρKth σej + ρKth σee − ∑ 2 j = g;i ej γ b α b α b α b + b σee ρKth −2σje ρKth σej + ρKth σee ; 2 h i κ β I β † β β † β † ρ˙ Kth = −i Hbc ; ρKth − a aρKth −2aρKth a + ρKth a a 2 ej γa a β a β a β a σee ρKth −2σje ρKth σej + ρKth σee − ∑ 2 j = g;i γej b β b β b β b + b σee ρKth −2σje ρKth σej + ρKth σee ; 2
½
½
ð23Þ
master equations numerically with the quantum optics toolbox [23]. The quality of the controlled-z gate can be assessed by the average gate fidelity defined as [24,25]
F ðε; U Þ =
h i ∑j tr UUj† U † ε Uj + d2 d2 ðd + 1Þ
;
ð24Þ
where d = 4 for two qubits and Uj to be the tensor of Pauli matrices II, IX, IY, ⋯, ZZ, U is the perfect controlled-z gate and ε is the trace-preserving quantum operation obtained through our controlled-z gate. By substituting the result acquired in Eq. (23) into Eq. (24), we plot the evolution of the average gate fidelity versus the decoherence parameter κ/g with γ = κ and ga = gb = g in Fig. 3. It can be seen the present twoqubit controlled-z gate is immune to both the cavity decay and the atomic spontaneous emission, since for the condition κ = γ = 0.01g, the fidelity remains about 94%. The reason is that the current model takes advantage of resonant atom-cavity interaction, it spends a short interaction time which is able to avoid the effect of decoherence efficiently. Although we have proposed our scheme from a theoretical view point, we still look forward that the conditions for realizing our
where κ denotes the decay rate of cavity, γej n represents the branching ration of the atom decay from level |en i to | jn i (n = a, b) and we ei assume γeg n = γn = γ/2 for simplicity. In each pair, the first master equation with superscript α represents the interaction between atom a and the cavity, and the second one with superscript β denotes the interaction between atom b and the cavity. The final state obtained in the Kth step will be the initial state of the (K + 1)th step, i.e., ρ(K + 1)th (0) = ρKth(t). By designing a suitable algorithm, we can solve these
Fig. 2. The evolution of the probability amplitude Ceg, 0 for the initial state |ea, gb, 0c i versus the operating steps N by setting θ = π. The three curves corresponding to different values of ϕ, i.e., ϕ = Nπ/2 for the black, ϕ = Nπ/5 for the red and ϕ = Nπ/10 for the green.
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
1103
Fig. 3. The fidelity of the two-qubit controlled-z gate versus κ/g. Other parameters: γ = κ, ϕ = Nπ/2 and N = 8.
gate may be reached in the near future. The requirements for the intended experiment are the three-level atoms and a resonant cavity. We may employ a cesium atom for our proposal. The low states |g i corresponds to F = 3, m = 2 hyperfine state of 62S1/2 electronic ground state, |i i corresponds to F = 4, m = 4 hyperfine state of 62S1/2 electronic ground state, and the excited state |e i corresponds to F = 4, m = 3 hyperfine states of 62P1/2 electronic excited states, respectively. In recent experiments [26], the suitability of toroidal microcavities for strongcoupling cavity QED has been investigated and a strong atom-cavity coupling rate can be reached as λ/2π = 750MHz with the decoherence parameters γ/2π = 2.62MHz, κ/2π = 3.5MHz. By substituting these typical parameters into Eq. (24), we obtain the average fidelity of the controlled-z gate is 97.5%.
5. Generalization for implementing multi-qubit controlled-Z gate In this section, we will generalize the idea for implementing the twoqubit controlled-z gate to the case of the M-qubit controlled-z gate. Similar to the case of two atoms, the quantum information is stored in the |g1 i, |e1 i, |gk i and |ik i, and the cavity field is in the vacuum state. In each step, we put the atom 1 through the cavity followed by the rest atoms crossing the cavity simultaneously. In this case, the Hamiltonians of the system in the interaction picture are HI1c =g1(|e1 i h g1|a+|g1 i h e1|a†) and HIrc =∑M k = 2 gk (|ek i h gk|a+|gk i h ek|a†). If the first atom is in the state |g N and all other atoms are in the states |g i or |i i, the system does not experience any dynamics. For the initial state |e1 i | ⋯ g k1 ⋯ gk 2 ⋯ gkm ⋯ i |0c i, where the qubits k1, k2,..., km(1 ≤ m ≤ M − 1) are in the state |g i, and all other statesnare in the states |i i, we obtain a three dimensional subspace as p1ffiffiffi ∑m l = 1 jg1 ij⋯gk1 ⋯ekl ⋯gkm ⋯ij0c i, |g1 i | ⋯ gk1 ⋯ gk2 ⋯ gkm ⋯ i m |1 c i,|e1 i | ⋯ g k1 ⋯ gk 2 ⋯ g k m ⋯ i |0c i}. It is apparent that the kick is performed by the m atoms collectively, which leads to this initial state unchanged for a large number of operation steps N. The phase shift of -1 is acquired only by initializing the first atom in state |e i and others in state |i i. In conclusion, we have actualized the M-qubit controlled-z gate with the non-instantaneous unitary kicks. Note that the rotation angle ϕ of the kick unitary transformation cannot take the value of 2L × Nπ, where L is an arbitrary integer. Based on this reason there is a limitation for the generalization. To see this fact clearly, we plot the evolutions of probability amplitudes for the states |e1 i | ⋯ gk1 ⋯ gk2 ⋯ gkm ⋯ i |0c i versus M-1 in Fig. 4 corresponding to the parameters illustrated in Fig. 2. This figure implies that for the case ϕ=Nπ/2, N=8, only (M−1)≤10 is meaningful for realizing the quantum logic gate. The reason is that with the increase of number of atoms M, the rotation angle performed by the kick is approaching 2Nπ, which destroys the dominant role played by the kick and make our scheme failed. Therefore, a relatively smaller rotation angle of the kick will provide an opportunity for achieving a multi-qubit
Fig. 4. The evolutions of probability amplitudes for the states |e1 i | ⋯ gk1 ⋯ gk2 ⋯ gkm ⋯ i |0c i versus M-1 corresponding to the parameters illustrated in Fig. 2. M denotes the number of qubits.
logic gate with more qubit, as shown for the cases of ϕ=Nπ/5, N=20 and ϕ=Nπ/10, N=40. 6. Summary In summary, we have proposed a scheme for implementing a twoqubit controlled-z gate via repeatedly letting two atoms across the cavity sequentially. This quantum Zeno-like effect is obtained by noninstantaneous unitary kick rather than instantaneous unitary kick. The strictly numerical and analytical results show the fidelity of our scheme is relatively high. Moreover, it can be generalized to the case of multi-qubit controlled-z gate and the expansibility is discussed under different conditions. We hope our work may be useful for the experimental realization of quantum computation in the near future. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant No. 61068001 and 11064016. References [1] P.W. Shor, Proc. 35th Annual Symp. on the Foundation of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. 124–134. [2] L.K. Grover, Phys. Rev. Lett. 80 (1998) 4329. [3] D.P. DiVincenzo, Phys. Rev. A 51 (1995) 1015.
1104 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74 (1995) 4091. A. Barenco, D. Deutsch, A. Ekert, R. Jozsa, Phys. Rev. Lett. 74 (1995) 4083. T. Sleator, H. Weinfurter, Phys. Rev. Lett. 74 (1995) 4087. N.A. Gershenfeld, I.L. Chuang, Science 275 (1997) 350. E. Knill, R. Laflamme, G.J. Milburn, Nature 409 (2001) 46. D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. C.P. Yang, S. Han, Phys. Rev. A 73 (2006) 032317. B. Misra, E.C.G. Sudarshan, J. Math. Phys. 18 (1977) 756. J.D. Franson, B.C. Jacobs, T.B. Pittman, Phys. Rev. A 70 (2004) 062302. B.C. Jacobs, J.D. Franson, Phys. Rev. A 79 (2009) 063830. L. Zhou, S. Yang, Y.X. Liu, C.P. Sun, F. Nori, Phys. Rev. A 80 (2009) 062109. X.B. Wang, J.Q. You, F. Nori, Phys. Rev. A 77 (2008) 062339. P. Facchi, D.A. Lidar, S. Pascazio, Phys. Rev. A 69 (2004) 032314. P. Facchi, G. Marmo, S. Pascazio, e-print, arXiv:0711.4280v22009. R. Rossi Jr., A.R. Bosco de Magalhães, J.G. Peixoto de Faria, M.C. Nemes, Phys. Rev. A 79 (2009) 012103. [19] S.B. Zheng, Phys. Rev. A 71 (2005) 062335.
[20] C.Y. Chen, M. Feng, K.L. Gao, Phys. Rev. A 73 (2006) 064304. [21] X.B. Zou, Y.L. Dong, G.C. Guo, Phys. Rev. A 74 (2006) 032325. [22] The evolution operator U and the kick unitary transformation operator Ukick can be written as U=exp(-iHt1) and Ukick=exp(-iHkickt2), respectively. The effect of Ukick is more powerful than U means Hkickt2≫Ht1. There are two ways for satisfying the condition: First, for t1= t2, we should make the strength of Hkick much stronger than the strength of H; Second, if Hkick and H have the equivalent strength, a choice of t1≫ t2 will complete the same task. In the following, we will adopt the later method since the interaction strength between each atom and cavity is similar. [23] S.M. Tan, J. Opt. B: Quantum Semiclassical Opt. 1 (1999) 424. [24] A.G. White, A. Gilchrist, G.J. Pryde, J.L. O'Brien, M.J. Bremner, N.K. Langford, J. Opt. Soc. Am. B 24 (2007) 172. [25] M.A. Nielsen, Phys. Lett. A 303 (2002) 249. [26] S.M. Spillane, T.J. Kippenberg, K.J. Vahala, K.W. Goh, E. Wilcut, H.J. Kimble, Phys. Rev. A 71 (2005) 013817.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Quantum gate operation with non-instantaneous unitary kicks Xiao-Qiang Shao a,b, Li Chen a,b, Shou Zhang a,b,⁎, Yong-Fang Zhao a, Kyu-Hwang Yeon c a b c
Center for the Condensed-Matter Science and Technology, Department of Physics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, PR China Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002, PR China BK21 Program Physics & Department of Physics, College of Natural Science, Chungbuk National University, Cheonju, Chungbuk 361-763, Republic of Korea
a r t i c l e
i n f o
Article history: Received 4 June 2010 Received in revised form 10 September 2010 Accepted 13 October 2010 Keywords: Quantum Zeno effect Unitary kicks
a b s t r a c t A two-qubit controlled-z gate is presented based on the non-instantaneous unitary kicks. Instead of putting two atoms through the cavity simultaneously, we make the atoms cross the cavity sequentially. The interaction between the second atom and the cavity plays the role for kicking the evolution of the system consisting of the first atom and cavity. By repeating the whole process N times, we obtain the controlled-z gate with a high fidelity. The effects of decoherence such as spontaneous emission and the loss of cavity on the average gate fidelity are investigated in virtue of master equation. Furthermore the method for achieving the multi-qubit controlled-z gate is also proposed. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The existing algorithms exposit that the quantum computer would be more powerful than the classical computer [1,2]. This matter of fact has triggered in the past years many studies on the theoretical and the practical aspects of quantum computing. Generally speaking, the quantum computers are divided into two categories according to the principle for calculation, i.e., the measurement-based quantum computer and the circuit-based quantum computer. Quantum logic gate is the central and basic element for actualizing the circuit-based quantum computer. In 1995, DiVincenzo pointed out that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit [3]. Thus the investigation on the universal quantum logic gate becomes the research focus and different physical models have been established in recent years, such as ion-trap system [4], cavity quantum electrodynamics (QED) [5,6], nuclear magnetic resonance (NMR) system [7], linear optics [8], quantum dot [9], and superconducting quantum interference device (SQUID) system [10]. The quantum Zeno effect put forward by Misra and Sudarshan describes the peculiar behavior of unstable particles under frequent measurement [11], i.e., the dissipation of an unstable quantum system is suppressed or slowed down, and the transition between quantum states is frozen if measurements are performed frequently. It has been shown that the quantum Zeno effect is not only useful for the decoherence suppression, but also has many applications in other fields of quantum information including quantum computation [12], quantum switching [13,14], and entanglement preparation [15], etc. In 2004, Facchi et al. unified the quantum Zeno effect and the “bang-
⁎ Corresponding author. E-mail address: [email protected] (S. Zhang). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.10.048
bang" decoupling method for suppressing decoherence in open quantum systems [16]. They found the evolution of a controlled system Q alternately undergoes N “kicks" Ukick (instantaneous unitary transformations) and free evolutions in a time interval t will be freezed in the large N limit, of which the quantum Zeno effect is obtained by unitary evolutions instead of nonunitary evolutions induced by von Neumann measurements, and they termed this phenomenon iconically as “unitary kicks" in their following works [17]. Recently, Rossi, JR. et al. presented a strategy to control the evolution of a quantum system by applying unitary interactions between the system of interest and a single auxiliary system [18]. We find the potentially physical principle can be brought into the theory of unitary kicks of Facchi's by making a little modification, i.e., replacing the instantaneous unitary transformation (Ukick) with noninstantaneous unitary transformation (time evolution unitary operator Ukick(t)). In practice, the unitary kicks will not be instantaneous, and decoherence will be present during the kicks. In the present work, we utilize the non-instantaneous unitary kicks to implement a two-qubit controlled-z gate. Compared with previous schemes for implementing two-qubit quantum gate with two atoms simultaneously interacting with the cavity field [19–21], the main advantage of the current scheme is that interactions between the cavity field and only a single atom at a time are needed, which may contribute to reducing the experimental complexity for control. The structure of the paper is organized as follows. The principle of non-instantaneous unitary kicks is briefly introduced in Sec. II. Then the two-qubit controlled-z gate is put forward with approximate and accurate calculation in Sec. III. The effects of decoherence involving atomic spontaneous emission and cavity decay on the average fidelity of the gate are investigated in Sec. IV. The generalization for achieving the multi-qubit controlled-z gate is provided in Sec. V. A summary will appear in Sec. VI.
1100
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
2. Preliminary: the evolution of quantum state under noninstantaneous unitary kicks In this section, we give a brief introduction to the principle of unitary kicks by replacing the instantaneous unitary transformation with non-instantaneous unitary transformation [16]. Consider the dynamics system whose evolution is governed by the following operator N t t UN ðt1 + t2 Þ = Ukick 2 U 1 ; N N
U ðt1 Þ =
ð1Þ
ˆ = ∑ P HP ; HZ = PH n n
1 N−1 †N k †k N ∑ U ðU U Þ Ukick HUkick ðUkick U Þ Ukick : N k = 0 kick kick
ð6Þ
is called Zeno Hamiltonian, Pn is the spectral projections of Ukick belonging to the eigenvalue λn and Ukick
t2 −iλ t = N = ∑ e n 2 Pn : N n
ð7Þ
Finally by combining Eq. (2) with Eq. (5), the whole system is governed by the limiting evolution operator
ð2Þ UN ðt1 + t2 Þ∼Ukick
N N t2 t U ðt1 Þ = Ukick 2 expð−iHZ t1 Þ N N = exp −i ∑ λn Pn t2 + Pn HPn t1 :
ð8Þ
n
ð3Þ
with HN ðt1 Þ =
ð5Þ
where
where V N(0) = I and d i V ðt Þ = HN ðt1 ÞV N ðt1 Þ; dt1 N 1
lim V N ðt1 Þ∼expð−iHZ t1 Þ; N→∞
n
where U(t1/N) = exp(− iHt1/N) is the free evolution operator for the system of interest, H is the corresponding Hamiltonian, and Ukick(t2/N) denotes the kick unitary transformation. For convenience, we rewrite the above operator in the following form N †N t t UN ðt1 + t2 Þ = Ukick 2 Ukick 2 UN ðt1 + t2 Þ N N N t = Ukick 2 V N ðt1 Þ; N
We suppose the effect of Ukick is more powerful than U [22], so in the large N limit, the limiting evolution operator
ð4Þ
The action of Ukick is just like an observer performing the repeatedly projective measurement on the system of interest. Once the number of measurement N approaches to infinity, a Zeno subspace will be generated. Eq. (8) is the important result governing our following works.
3. Two-qubit controlled-Z gate We consider two identical three-level atoms repeatedly interacting with the cavity via the resonant interaction sequentially. Each atomic states is denoted by |g i, |e i, and |i i, as shown in Fig. 1. The state |i i is decoupled due to the large detuning, and it is not affected during the atomcavity interaction. The Hamiltonians for the interaction between atom a (atom b) and cavity in the Schrödinger picture are given by † † Hac = ωa ð jea ihea j−jga ihga j Þ + ωc a a + ℏga jea ihga ja + jga ihea ja ;
ð9Þ
† † Hbc = ωb ð jeb iheb j−jgb ihgb j Þ + ωc a a + ℏgb jeb ihgb ja + jgb iheb ja ;
ð10Þ
and
where ωa, ωb, and ωc are the eigenvalues of the free Hamiltonian for atom a, atom b, and cavity, respectively. a† (a) is the creation (annihilation) operator of the cavity mode. ga(b) is the coupling constant between the atom a (b) and the cavity. For simplicity, we suppose ωa = ωb = ωc and move into the interaction picture in the unit of ℏ. The quantum information is encoded into a subspace spanned by the states {|ga, gb i, |ga, ib i, |ea, gb i, |ea, ib i} with the cavity in the vacuum state. The states |ga, gb, 0c i and |ga, ib, 0c i are unchanged because of HIac(bc)|ga, gb, 0c i (|ga, ib, 0c i) = 0. In what follows, we mainly discuss the time evolutions of the states |ea, gb, 0c i and |ea, ib, 0c i.
Fig. 1. The level configuration of the atom. The transition between |g i and |e i is coupled resonantly to the cavity mode with the coupling constant g, and the level |i i is decoupled to the atom-cavity interaction. The quantum information is encoded into the subspace {|ga, gb i, |ga, ib i, |ea, gb i, |ea, ib i}.
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
1101
For the initial state |ea, gb, 0c i, we obtain a closed subspace during the whole evolution process, i.e., {|ga, eb, 0c i, |ga, gb, 1c i, |ea, gb, 0c i}, hence the unitary operators for each cycle between atom and cavity can be written as 0 θ I = Uac N
1 0 B θ B0 cos B N B @ θ 0 −i sin N
0
1 θC C NC C; θ A
−i sin cos
ð11Þ
N
and 0
ϕ B cos N B ϕ I B ϕ =B Ubc B −i sin N @ N 0
1 0C C C C; 0C A 1
ϕ N ϕ cos N 0
−i sin
ð12Þ
where θ = gat1 and ϕ = gbt2. Then the total evolution operator for this subspace reads N ϕ I θ I I U UN ðt1 + t2 Þ = Ubc ; N ac N
ð13Þ
where UIbc(ϕ/N) (|2π − (ϕ mod 2π)| ≫ θ) acts as a kick and its spectral projections are
P1 =
0 1 0 1 −1 0 1 1 1@ 1 −1 1 0 A; P2 = @ 1 1 2 2 0 0 0 0 0
1 0 0 A; 0
ð14Þ
corresponding to the eigenvalues − gb and gb, respectively. In the large N limit, according to Eq. (8), we obtain the final effective evolution operator of the subspace as 0
I UN ðt1
0 + t2 Þ∼exp@ −igb t2 0
−igb t2 0 0
1 0 cosðgb t2 Þ 0 A 0 = @ −i sinðgb t2 Þ 0 0
−i sinðgb t2 Þ cosðgb t2 Þ 0
1 0 0 A; 1
ð15Þ
which implies that the state |ea, gb, 0c i does not evolve if the rotation angle ϕ is relatively larger than θ. For the initial state |ea, ib, 0c i, it only couples to state |ga, ib, 1c i with coupling constant ga, and we obtain the following evolution form jea ; ib ; 0c i→cosðga t Þjea ; ib ; 0c i−i sinðga t Þjga ; ib ; 1c i;
ð16Þ
thus the phase shift of -1 will be acquired after the interaction time t = π/ga. Then we are able to achieve a two-qubit controlled-z gate in the computation basis {|ga, gb i, |ga, ib i, |ea, gb i, |ea, ib i} as 0
1 B0 CZ = B @0 0
0 1 0 0
1 0 0 0 0 C C: 1 0 A 0 −1
ð17Þ
Note that in the above process, the two-qubit controlled-z gate is carried out in the condition of a large N limit. It is necessary for us to quantify the range of N for governing the future experimental operations. All we need to do is to find out the accurate result of Eq. (13), which can be done by mapping the action of UIbc(ϕ/N)UIac(θ/N) on the three dimensional Hilbert subspace {|ga, eb, 0c i, − i|ga, gb, 1c i, |ea, gb, 0c i} onto a rotation on the real Euclidian subspace [18]. Through the straightway algebraic calculation, we obtain 20 cos
ϕ N
6B 6B N 6B ϕ I θ I B Uac =6 Ubc 6B sin ϕ N N 6B N 4@ 0 0
−sin cos 0
ϕ N
ϕ N
10 0
1
0
CB CB CB 0 cos θ CB N B 0C CB A@ θ 0 sin 1 N
1−a2 cosðNφÞ + a2
0
ϕ B cos N C7 B B θ C7 7 B ϕ −sin C 7 B NC C7 = B sin N C7 B B θ A5 @ cos 0 N 0
13N
ab½1−cosðNφÞ−c sinðNφÞ B B B = B ab½1−cosðNφÞ + c sinðNφÞ 1−b2 cosðNφÞ + b2 B @ ac½1−cosðNφÞ−b sinðNφÞ bc½1−cosðNφÞ + a sinðNφÞ
1N ϕ θ sin N N C C C ϕ θ ϕ θC cos cos −cos sin C N N N NC C C A θ θ cos sin N N 1 ac½1−cosðNφÞ + b sinðNφÞ C C C bc½1−cosðNφÞ−a sinðNφÞ C; C A 1−c2 cosðNφÞ + c2
−sin
ϕ θ cos N N
sin
ð18Þ
1102
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
where "
ϕ θ cos φ = arcsin 2 cos 2N 2N
2 3 ϕ θ ϕ θ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# cos + cos + cos cos −1 ϕ θ ϕ θ 2 2 2 6 7 N N N N + sin −sin sin = arccos4 sin2 5; 2 2N 2N 2N 2N
ð19Þ
sin Nθ cos Nϕ + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a= ϕ ϕ ϕ θ θ θ 4 cos 2N sin2 2N cos 2N + sin2 2N −sin2 2N sin2 2N
ð20Þ
b=
sin Nϕ sin Nθ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; ϕ ϕ ϕ θ θ θ sin2 2N 4 cos 2N cos 2N + sin2 2N −sin2 2N sin2 2N
ð21Þ
c=
sin Nϕ cos Nθ + 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: ϕ ϕ ϕ θ θ θ sin2 2N 4 cos 2N cos 2N + sin2 2N −sin2 2N sin2 2N
ð22Þ
In Fig. 2, we plot the evolution of the probability amplitude Ceg, 0 for the initial state |ea, gb, 0c i versus the operating steps N by setting θ = π. The three curves correspond to different values of ϕ, i.e., ϕ = Nπ/2 for the black, ϕ = Nπ/5 for the red and ϕ = Nπ/10 for the green, respectively. We are able to see the evolution of the probability amplitude Ceg, 0 depends on two factors, one is the number of operating steps N, and the other is the ratio between the two rotation angles ϕ/θ. For all these three cases, the probability amplitude Ceg, 0 can reach to unity with a good approximation for large N. However, a large ϕ to θ ratio may reduce the number of operating steps N. For the case ϕ = Nπ/2, a stable and high value of Ceg, 0 could be obtained for N ≥ 8. Similarly, we see that N ≥ 20 is large enough for achieving a high fidelity logic gate corresponding to ϕ = Nπ/5 and N ≥ 40 is good for the case ϕ = Nπ/10. Thus we can conclude that for implementing the two-qubit controlled-z gate with a high fidelity, a finite N is sufficiently good if the rotation angles satisfying the relation ϕ/θ ≥ 4 (ϕ ≠ 2L × Nπ, L being an arbitrary integer). 4. The effects of decoherence on the gate fidelity Since no quantum system can be completely isolated from its environment, the irreversible interaction between the quantum system and its environment would lead to decoherence when the two-qubit controlled-z gate is put into experiment. So it is essential to quantify the effect on the gate fidelity. The dissipation channels for the current model include the spontaneous emission of atom (at the rate γ) and the cavity decay (at the rate κ). For the case of realizing the gate with N steps, we need to solve N pairs master equations in the following form
h i κ α † α α † α † I ρ˙ Kth = −i Hac a aρKth −2aρKth a + ρKth a a ; ρα Kth − 2 γej a α a α a α a a σee ρKth −2σje ρKth σej + ρKth σee − ∑ 2 j = g;i ej γ b α b α b α b + b σee ρKth −2σje ρKth σej + ρKth σee ; 2 h i κ β I β † β β † β † ρ˙ Kth = −i Hbc ; ρKth − a aρKth −2aρKth a + ρKth a a 2 ej γa a β a β a β a σee ρKth −2σje ρKth σej + ρKth σee − ∑ 2 j = g;i γej b β b β b β b + b σee ρKth −2σje ρKth σej + ρKth σee ; 2
½
½
ð23Þ
master equations numerically with the quantum optics toolbox [23]. The quality of the controlled-z gate can be assessed by the average gate fidelity defined as [24,25]
F ðε; U Þ =
h i ∑j tr UUj† U † ε Uj + d2 d2 ðd + 1Þ
;
ð24Þ
where d = 4 for two qubits and Uj to be the tensor of Pauli matrices II, IX, IY, ⋯, ZZ, U is the perfect controlled-z gate and ε is the trace-preserving quantum operation obtained through our controlled-z gate. By substituting the result acquired in Eq. (23) into Eq. (24), we plot the evolution of the average gate fidelity versus the decoherence parameter κ/g with γ = κ and ga = gb = g in Fig. 3. It can be seen the present twoqubit controlled-z gate is immune to both the cavity decay and the atomic spontaneous emission, since for the condition κ = γ = 0.01g, the fidelity remains about 94%. The reason is that the current model takes advantage of resonant atom-cavity interaction, it spends a short interaction time which is able to avoid the effect of decoherence efficiently. Although we have proposed our scheme from a theoretical view point, we still look forward that the conditions for realizing our
where κ denotes the decay rate of cavity, γej n represents the branching ration of the atom decay from level |en i to | jn i (n = a, b) and we ei assume γeg n = γn = γ/2 for simplicity. In each pair, the first master equation with superscript α represents the interaction between atom a and the cavity, and the second one with superscript β denotes the interaction between atom b and the cavity. The final state obtained in the Kth step will be the initial state of the (K + 1)th step, i.e., ρ(K + 1)th (0) = ρKth(t). By designing a suitable algorithm, we can solve these
Fig. 2. The evolution of the probability amplitude Ceg, 0 for the initial state |ea, gb, 0c i versus the operating steps N by setting θ = π. The three curves corresponding to different values of ϕ, i.e., ϕ = Nπ/2 for the black, ϕ = Nπ/5 for the red and ϕ = Nπ/10 for the green.
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
1103
Fig. 3. The fidelity of the two-qubit controlled-z gate versus κ/g. Other parameters: γ = κ, ϕ = Nπ/2 and N = 8.
gate may be reached in the near future. The requirements for the intended experiment are the three-level atoms and a resonant cavity. We may employ a cesium atom for our proposal. The low states |g i corresponds to F = 3, m = 2 hyperfine state of 62S1/2 electronic ground state, |i i corresponds to F = 4, m = 4 hyperfine state of 62S1/2 electronic ground state, and the excited state |e i corresponds to F = 4, m = 3 hyperfine states of 62P1/2 electronic excited states, respectively. In recent experiments [26], the suitability of toroidal microcavities for strongcoupling cavity QED has been investigated and a strong atom-cavity coupling rate can be reached as λ/2π = 750MHz with the decoherence parameters γ/2π = 2.62MHz, κ/2π = 3.5MHz. By substituting these typical parameters into Eq. (24), we obtain the average fidelity of the controlled-z gate is 97.5%.
5. Generalization for implementing multi-qubit controlled-Z gate In this section, we will generalize the idea for implementing the twoqubit controlled-z gate to the case of the M-qubit controlled-z gate. Similar to the case of two atoms, the quantum information is stored in the |g1 i, |e1 i, |gk i and |ik i, and the cavity field is in the vacuum state. In each step, we put the atom 1 through the cavity followed by the rest atoms crossing the cavity simultaneously. In this case, the Hamiltonians of the system in the interaction picture are HI1c =g1(|e1 i h g1|a+|g1 i h e1|a†) and HIrc =∑M k = 2 gk (|ek i h gk|a+|gk i h ek|a†). If the first atom is in the state |g N and all other atoms are in the states |g i or |i i, the system does not experience any dynamics. For the initial state |e1 i | ⋯ g k1 ⋯ gk 2 ⋯ gkm ⋯ i |0c i, where the qubits k1, k2,..., km(1 ≤ m ≤ M − 1) are in the state |g i, and all other statesnare in the states |i i, we obtain a three dimensional subspace as p1ffiffiffi ∑m l = 1 jg1 ij⋯gk1 ⋯ekl ⋯gkm ⋯ij0c i, |g1 i | ⋯ gk1 ⋯ gk2 ⋯ gkm ⋯ i m |1 c i,|e1 i | ⋯ g k1 ⋯ gk 2 ⋯ g k m ⋯ i |0c i}. It is apparent that the kick is performed by the m atoms collectively, which leads to this initial state unchanged for a large number of operation steps N. The phase shift of -1 is acquired only by initializing the first atom in state |e i and others in state |i i. In conclusion, we have actualized the M-qubit controlled-z gate with the non-instantaneous unitary kicks. Note that the rotation angle ϕ of the kick unitary transformation cannot take the value of 2L × Nπ, where L is an arbitrary integer. Based on this reason there is a limitation for the generalization. To see this fact clearly, we plot the evolutions of probability amplitudes for the states |e1 i | ⋯ gk1 ⋯ gk2 ⋯ gkm ⋯ i |0c i versus M-1 in Fig. 4 corresponding to the parameters illustrated in Fig. 2. This figure implies that for the case ϕ=Nπ/2, N=8, only (M−1)≤10 is meaningful for realizing the quantum logic gate. The reason is that with the increase of number of atoms M, the rotation angle performed by the kick is approaching 2Nπ, which destroys the dominant role played by the kick and make our scheme failed. Therefore, a relatively smaller rotation angle of the kick will provide an opportunity for achieving a multi-qubit
Fig. 4. The evolutions of probability amplitudes for the states |e1 i | ⋯ gk1 ⋯ gk2 ⋯ gkm ⋯ i |0c i versus M-1 corresponding to the parameters illustrated in Fig. 2. M denotes the number of qubits.
logic gate with more qubit, as shown for the cases of ϕ=Nπ/5, N=20 and ϕ=Nπ/10, N=40. 6. Summary In summary, we have proposed a scheme for implementing a twoqubit controlled-z gate via repeatedly letting two atoms across the cavity sequentially. This quantum Zeno-like effect is obtained by noninstantaneous unitary kick rather than instantaneous unitary kick. The strictly numerical and analytical results show the fidelity of our scheme is relatively high. Moreover, it can be generalized to the case of multi-qubit controlled-z gate and the expansibility is discussed under different conditions. We hope our work may be useful for the experimental realization of quantum computation in the near future. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant No. 61068001 and 11064016. References [1] P.W. Shor, Proc. 35th Annual Symp. on the Foundation of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. 124–134. [2] L.K. Grover, Phys. Rev. Lett. 80 (1998) 4329. [3] D.P. DiVincenzo, Phys. Rev. A 51 (1995) 1015.
1104 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
X.-Q. Shao et al. / Optics Communications 284 (2011) 1099–1104
J.I. Cirac, P. Zoller, Phys. Rev. Lett. 74 (1995) 4091. A. Barenco, D. Deutsch, A. Ekert, R. Jozsa, Phys. Rev. Lett. 74 (1995) 4083. T. Sleator, H. Weinfurter, Phys. Rev. Lett. 74 (1995) 4087. N.A. Gershenfeld, I.L. Chuang, Science 275 (1997) 350. E. Knill, R. Laflamme, G.J. Milburn, Nature 409 (2001) 46. D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. C.P. Yang, S. Han, Phys. Rev. A 73 (2006) 032317. B. Misra, E.C.G. Sudarshan, J. Math. Phys. 18 (1977) 756. J.D. Franson, B.C. Jacobs, T.B. Pittman, Phys. Rev. A 70 (2004) 062302. B.C. Jacobs, J.D. Franson, Phys. Rev. A 79 (2009) 063830. L. Zhou, S. Yang, Y.X. Liu, C.P. Sun, F. Nori, Phys. Rev. A 80 (2009) 062109. X.B. Wang, J.Q. You, F. Nori, Phys. Rev. A 77 (2008) 062339. P. Facchi, D.A. Lidar, S. Pascazio, Phys. Rev. A 69 (2004) 032314. P. Facchi, G. Marmo, S. Pascazio, e-print, arXiv:0711.4280v22009. R. Rossi Jr., A.R. Bosco de Magalhães, J.G. Peixoto de Faria, M.C. Nemes, Phys. Rev. A 79 (2009) 012103. [19] S.B. Zheng, Phys. Rev. A 71 (2005) 062335.
[20] C.Y. Chen, M. Feng, K.L. Gao, Phys. Rev. A 73 (2006) 064304. [21] X.B. Zou, Y.L. Dong, G.C. Guo, Phys. Rev. A 74 (2006) 032325. [22] The evolution operator U and the kick unitary transformation operator Ukick can be written as U=exp(-iHt1) and Ukick=exp(-iHkickt2), respectively. The effect of Ukick is more powerful than U means Hkickt2≫Ht1. There are two ways for satisfying the condition: First, for t1= t2, we should make the strength of Hkick much stronger than the strength of H; Second, if Hkick and H have the equivalent strength, a choice of t1≫ t2 will complete the same task. In the following, we will adopt the later method since the interaction strength between each atom and cavity is similar. [23] S.M. Tan, J. Opt. B: Quantum Semiclassical Opt. 1 (1999) 424. [24] A.G. White, A. Gilchrist, G.J. Pryde, J.L. O'Brien, M.J. Bremner, N.K. Langford, J. Opt. Soc. Am. B 24 (2007) 172. [25] M.A. Nielsen, Phys. Lett. A 303 (2002) 249. [26] S.M. Spillane, T.J. Kippenberg, K.J. Vahala, K.W. Goh, E. Wilcut, H.J. Kimble, Phys. Rev. A 71 (2005) 013817.