Realizing flexible two-qubit controlled phase gate with a hybrid solid-state system Feng-yang Zhang∗ , Ying Shi, Chong Li,† and He-shan Song School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024 China (Dated: October 11, 2011)
arXiv:1110.1901v1 [quant-ph] 10 Oct 2011
We propose a theoretical scheme for realizing flexible two-qubit controlled phase gate. A transmission line resonator is used to induce the coupling between nitrogen-vacancy (N-V) in diamond and superconducting qubit. The N-V center acts as control qubit and the superconducting qubit as target qubit. Through adjusting external flux, we obtain desired coupling between random superconducting qubit and transmission line resonator. Moreover, our protocol might be implemented via the current experimental technology. PACS numbers: 03.67.Lx, 76.30.Mi, 85.25.Dq
Quantum computer is a magical machine, which is more efficient than classical computers[1–3] in solving hard computational problems . Therefore, how to successfully construct quantum computer is still an open issue. The core element of quantum computer is quantum logic gate. It is well known that a quantum-computing network can be decomposed into one qubit rotations gate and two-qubit gates[4]. So far, many theoretical and experimental proposals demonstration two-qubit gates in different physical system, for example, two-qubit controlled phase gates have been experimental realized in cavity QED[5], quantum dot[6], NMR[7], ion traps system[8], respectively. Quantum information processing (QIP) need repeated implement quantum gate operations. However, the decoherence of system is the main stumbling block for QIP. Seeking the nice candidate for quantum information carrier is still a popular study field. Due to the sufficient long electronic spin lifetime and the possibility of coherent manipulation at room temperature[9], the N-V center in diamond[10] consisting of a substitutional nitrogen atom and an adjacent vacancy, provides an arena to study various macroscopic quantum phenomena and acts as a perfect candidate toward QIP[11]. The two-qubit conditional quantum gate has been realized in experiment[12], then the theoretical scheme of achieving multiqubit conditional phase gate was proposed in Ref.[13]. Also, the coherent coupling between superconducting qubit and N-V center in diamond has already been achieved[14, 15]. In this paper, we propose an alternative scheme to realize flexible two-qubit controlled phase gate by a transmission line resonator which induces the interaction between N-V center in diamond and random superconducting qubit. The meaning of flexible two-qubit controlled phase gate is a controlling qubit to control random desired target qubit. Because of the long coherence time of N-V center, it can act as controlling qubit (first qubit), and the superconducting qubit acts as target qubit (sec-
∗ Email:
[email protected] † Email:
[email protected]
L
(a )
(b )
FIG. 1: (Color online) (a) Schematic diagram of the random superconducting qubit (black frame) coupled to N-V center (yellow lozenge) mediated by the transmission line resonator. (b) Detailed schematic of a charge qubit placed in antinode of the high-quality single-mode quantum transmission line resonator.
ond qubit). All the coupling (decoupling) between target qubits and the transmission line resonator can be controlled by adjusting its external flux. So the expectative two-qubit phase gate can be constructed. As shown below, this proposal has the following advantages: (i) the coupling technologies between the transmission line resonator and the superconducting qubit[16], the N-V center[17] are rather mature in current experiments; (ii) compared with the conventional scheme[18], there is no need to adjust the qubit’s frequency and the transmission line’s frequency throughout the entire operation; (iii) the control qubit has relative long decoherence time and the target qubit has flexible controllability. We consider the coupling of N-V center in diamond and
2 random superconducting qubit induced by transmission line resonator (see the Fig.1). The typical superconducting qubits include charge qubit, flux qubit, and phase qubit. Any two-level system of these qubits are commonly described by a general Hamiltonian with Pauli matrix σx and σz [19]. In this paper, we choose charge qubit as an example, the Hamiltonian describes the qubit is[19, 20] 1 1 − ng σz − EJ (Φ)σx , (1) Hq = −4Ec 2 2 where the Coulomb energy Ec = e2 /(Cg + 2CJ ), the dimensionless gate charge ng = Cg Vg /2e, EJ (Φ) = EJ0 cos(πΦ/Φ0 ). Here, Cg expresses the gate capacitance, Vg is the gate voltage, Φ is the external flux through the superconducting loop, the flux quantum Φ0 = h/2e, and EJ0 is Josephson coupling energy. If the qubit is assumed to be biased at the degeneracy point ng = 1/2, the Hamiltonian reduces to Hq = − 21 EJ (Φ)σx . Also, we get the evolution operator U1 (τ ) = exp[iζσx τ /2],
(2)
where ζ = EJ (Φ) and we take ~ = 1 (in entire paper). The N-V center contains a vacancy that results in broken bonds in the system. We utilize C3v group theory to analyse the level structure of N-V center in diamond[10, 21]. The level structure (see Fig.2 (a)) includes the spin triplet ground state 3 A, excited state 3 E, and the spin singlet metastable 1 A. Also, the ground states have a zero-field splitting Dgs = 2.87 GHz between degenerated sublevels ms = ±1 and ms = 0[22]. We can remove the degeneracy of spin sublevels ms = ±1 by external magnetic field B (see the Fig.2 (b)). The two sublevels ms = 0 and ms = −1 can be coupled by a microwave field, so we can encode the qubits. Under a frame rotating with microwave field frequency, the Hamiltonian of the N-V center could be written as[15, 23] HN −V = (ω0 −ωr )|−1ih−1|+
Ω (|0ih−1|+|−1ih0|), (3) 2
where ω0 = Dgs + γ|B| expresses the energy gap of the sublevels, γ is the gyromagnetic ration of an electron. ωr and Ω are the frequency and the Rabi frequency of the microwave pulse with the N-V center, respectively. If the microwave field resonates to the N-V center, i.e. ω0 = ωr , the Hamiltonian (3) reduces to HN −V =
Ω Sx , 2
(4)
where Sx = |0ih−1| + | − 1ih0|. Also, we obtain the evolution operator U2 (τ ) = exp[iξSx τ /2], where ξ = −Ω.
(5)
(a ) 3
E{ 1
A
(b ) ms =± 1 3
A
{
D gs
}
ms =+ 1
ms =− 1
Ω
ms = 0 ms = 0
FIG. 2: (Color online) (a) Energy level scheme of the N-V center in diamond. It includes the excited state 3 E, metastable 1 A, and ground state 3 A. Dgs is the zero-field splitting between the ground state sublevels ms = 0 and ms = ±1. (b) The energy-level diagram for the ground triplet state of the electron spin within a N-V center. We encode qubits in the subspace, the down state ms = 0 and the up state ms = −1, which were driven by a microwave field Ω.
In this section, we discuss how to realize flexible twoqubit phase gate with our scheme. The total Hamiltonian of the hybrid system is ζ ξ H = ωa† a − σx − Sx + g(a + a† )σx 2 2 +G(a + a† )(S+ eiωr t + S− e−iωr t ),
(6)
where the first term describes the quantized√transmission line resonator with the frequency ω = π/ LC, and L(C) is the total self-inductance (capacitance). a† and a are the creation and annihilation operators of the transmission line resonator, respectively. The last two terms describe the interactions, g depicts the effective coupling strength between the transmission line resonator and superconducting qubit, G denotes the effective coupling strength between the transmission line resonator and NV center. In the interaction picture, under the rotating wave approximation, we obtain the Hamiltonian HI = g(a† eiωt + ae−iωt )σx + G(a† S− ei∆t + aS+ e−i∆t ), (7) where ∆ = ω − ωr . The external driving of the transmission line resonator can be modeled by Hd = ε(t)(a† e−iωd t + aeiωd t ),
(8)
where ε(t) expresses the drive amplitude and ωd is the frequency of the external drive. Under the strong coupling limit (the coupling strength G is much lager than the resonator linewidth κ), Eq.(8) can be approximately described by Hd′ = Ω′ Sx , where Ω′ = Gε(t)/∆. Thus, the whole Hamiltonian is [24] HT = g(a† eiωt + ae−iωt )σx +G(a† S− ei∆t + aS+ e−i∆t ) + Ω′ Sx .
(9)
3 √ We introduce the new basis |±i = (|−1i±|0i)/ 2, which are the eigenstates of operator Sx with eigenvalues ±1. Under the frame rotating transformation of the unitary operator U = exp(iΩ′ Sx ), in the strong driving regime Ω′ ≫ {G, ∆}, we can eliminate the fast-oscillating terms of the Eq.(9) and obtain Hef f = g(a† eiωt + ae−iωt )σx +
G † i∆t (a e + ae−i∆t )Sx(10) . 2
superconducting qubit. If we omit the phase factor eiη , and choose η = π/8 + mπ/2 (for integer m), two-qubit controlled phase gate can be constructed[18] 1 0 0 0 0 1 0 0 . (16) U = 0 0 1 0 0 0 0 −1
By adjusting the external flux Φ threading the superconducting qubit loop, those external flux Φ which pass through the idle superconducting qubits are set to be nΦ0 /2. Therefore, the desired coupling between superconducting qubit and transmission line resonator is achieved and the flexible two-qubit controlled phase gate ∗ † UI (t) = e−iA(t)σx Sx e−iB(t)aσx e−iB (t)a σx is obtained as well. We briefly address the experimental feasibility of the −iC(t)aSx −iC ∗ (t)a† Sx −iD(t) e , (11) e ×e proposed scheme with the parameters already available in current experimental setups. In 2004, Blais et al. prowhere the time-dependent parameters posed the coupling between superconducting qubit and gG transmission line[26]. The size of the superconducting [cos ∆t − cos(ω − ∆)t] , A(t) = qubit 1 × 1µm2 has been reported[27]. Therefore, multiω ple qubits can be trapped in the space between the center conductor and the ground planes of transmission line resig onator. Later on, the transmon was proposed in Ref.[28], B(t) = (e−iωt − 1), ω and the relaxation and dephasing time T1 = 1.5µs and T2 = 2.05µs, respectively, were reported[29]. iG −i∆t One the other hand, in current experiments, the couC(t) = (e − 1), pling of transmission line resonator to an ensemble of 2∆ N-V center has been achieved[17]. Also, the dephasing and relaxation time of N-V center are reported to be 2 2 1 iωt 1 i∆t G g T2′ = 350µs[30] and T1 from 6ms at the room tempera(e − 1) − t + (e − 1) − t (12) . D(t) = ω iω 4∆ i∆ ture up to 28s at the low temperature[31], respectively. For the isotopically pure diamond sample, the dephasing If the evolution time t satisfies t = 2πn/ω = 2πp/∆ time T2′ = 2ms[32]. (n, p are integers), the parameters B(t) = C(t) = 0, i.e. We choose the following parameters, the effective the interaction between transmission line resonator and Josephson energy of superconducting qubit EJ = 2.2 × N-V center, and interaction between transmission line 2πGHz, the frequency of transmission line resonator resonator and superconducting qubit can be eliminated. ω = 1GHz, the coupling strength g = 19.71MHz[33], Therefore, up to a phase factor, the evolution operator the coupling constant of N-V center and transmission line Eq.(8) reduces as G = 11 × 2πMHz. Also, we take the integer n = 1, m = 0 and Rabi frequency Ω = 20×2πGHz. So, generating twoU3 (n) = exp[−iAσx Sx ]. (13) qubit phase gate’s time t ∼ 6ns is much shorter than the decoherence time of system. The preceding rough estiAfter aforementioned three-step operation (2), (5) and mates indicate that multiple operations can be performed (10), the joint evolution operator of the system is before decoherence happens. In summary, we have proposed a scheme to achieve U = U1 U2 U3 flexible two-qubit controlled phase gate by using solid = exp[iζσx τ /2] × exp[iξSx τ /2] × exp[−iAσx Sx(14) ]. hybrid system. Due to the N-V center has long decoherence time at the room temperature (especially, at the low Under the condition ζτ /2 = ξτ /2 = A = η, Eq.(11) can temperature, the coherence time is much longer), we utibe written as lize it as control qubit. The superconducting qubit has U = exp[iη(σx + Sx − σx Sx )]. (15) the characteristic advantage, which owns flexible controllability via adjusting the external parameters, therefore, In order to realize flexible two-qubit phase gate, we it is used as target qubit. Hence, we predict that this define the basis of √ qubit {|ggj i, |gej i, |egj i, |ee i}. Here j scheme might be realized with current technology. √ |gi = (| − 1i√− |0i)/ 2, |ei = (| − 1i +√|0i)/ 2, |gij = (| ↑ One of the authors (Feng-yang Zhang) thank Dr. Pei Pei and Jia-sen Jin for helpful discussions. This work is ij − | ↓ij )/ 2, |eij = (| ↑ij + | ↓ij )/ 2, and | ↑ij (| ↓ij ) supported by the Fundamental Research Funds for the is the eigenstate of the operator σzj for discretionary the The {σx Sx , aσx , a† σx , aSx , a† Sx , I} form a closed Lie algebra. Utilizing the Wei-Norman algebraic method[25], we obtain the evolution operator which can be written in a factorized way as
4 Central Universities No.DUT10LK10, the National Science Foundation of China under Grants No. 60703100,
No. 11175033, and No. 10875020.
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