REALIZING AN N-TWO-QUBIT QUANTUM LOGIC GATES IN A

Quantum Information and Computation, Vol. 16, No. 5&6 (2016) 0465–0482 c Rinton Press

REALIZING AN N-TWO-QUBIT QUANTUM LOGIC GATES IN A CAVITY QED WITH NEAREST QUBIT-QUBIT INTERACTION TAOUFIK SAID, ABDELHAQ CHOUIKH, AND KARIMA ESSAMMOUNI Laboratoire de Physique de la Matiere Condensee, Equipe Physique Quantique et Applications Faculte des Sciences Ben M’sik, Universite Hassan II Casablanca, B.P. 7955, Morroco MOHAMED BENNAI∗ Laboratoire de Physique de la Matiere Condensee, Equipe Physique Quantique et Applications Faculte des Sciences Ben M’sik, Universite Hassan II Casablanca, B.P. 7955, Morroco and LPHE-Modelisation et Simulation, Faculte des Sciences, Universite Mohamed V Rabat, Morroco

Received March 9, 2015 Revised January 28, 2016

We propose an effective way for realizing a three quantum logic gates (NTCP gate, NTCP-NOT gate and NTQ-NOT gate) of one qubit simultaneously controlling N target qubits based on the qubit-qubit interaction. We use the superconducting qubits in a cavity QED driven by a strong microwave field. In our scheme, the operation time of these gates is independent of the number N of qubits involved in the gate operation. These gates are insensitive to the initial state of the cavity QED and can be used to produce an analogous CNOT gate simultaneously acting on N qubits. The quantum phase gate can be realized in a time (nanosecond-scale) much smaller than decoherence time and dephasing time (microsecond-scale) in cavity QED. Numerical simulation under the influence of the gate operations shows that the scheme could be achieved efficiently within current state-of-the-art technology. Keywords: Superconducting qubit, qubit-qubit interaction, cavity QED, NTCP gate, NTCP-NOT gate, NTQ-NOT gate Communicated by: S Braunstein & H Zbinden

1

Introduction

In the past decade, various physical systems have been considered for building up quantum information processors [1, 2]. The cavity QED with neutral atoms is a very promising approach for quantum information processing, because a cavity can manipulate as a quantum bus the couple qubits efficiently, and information can be stored in certain atomic energy levels with long coherence time. So far, a large number of theoretical proposals for realizing two-qubit gates in many physical systems have been proposed. Moreover, two-qubit controlled-not (CNOT) gate, single-qubit Z gate, controlled phase (CP) gates and SWAP gate have been experimentally demonstrated in, such as cavity QED [3, 4], ion traps [5, 6], nuclear magnetic ∗ [email protected],

[email protected]

465

466

Realizing an N-two-qubit quantum logic gate in a cavity QED with nearest qubit–qubit interaction

resonance NMR [7, 8], quantum dots [9, 10], superconducting qubits [11], and Transmon qubits [12, 13]. Recently, Yang et al. proposed a scheme for implementing multiqubit tunable phase gate (NTCP gate) of one superconducting qubit simultaneously controlling n qubits selected from N qubits (1 < n < N ) in a cavity [14]. In ref. [15], the authors proposed a method for realizing a multiqubit gate, the procedure will become complicated as the number of qubits increases. Furthermore, it is significant to realize multiqubit gates directly [16, 17, 18]. In recent work, we have studied the atom-cavity interaction and the dipole-dipole interaction in a cavity QED [19]. In this work, we present and demonstrate a method for realizing a NTCP gate, NTCP-NOT gate and NTQ-NOT gate by using qubit-qubit interaction with one control superconducting qubit simultaneously controlling N target qubits in a cavity QED by adding a strong microwave field. These gates can be realized in a time of nanosecondscale much shorter than the decoherence time and dephasing time. Thus, the operation time required for the gate implementation is independent of the number N of qubits, this type of controlled gate with N target qubits is useful in quantum information processing. For instance, it has applications in entanglement preparation [20, 21, 22], error correction [23], Grover search algorithm [24, 25], quantum discrete Fourier transform [26, 27], Deutsch-Jozsa algorithm [28, 29], quantum dense coding [30, 31], and quantum cloning [32, 33]. The unitary operator representing this type of multiqubit gate is given by [29] Up =

N +1 Y j=2

(Ij − 2|−1 i|−j ih−1 |h−j |),

(1)

where the subscript 1 represents the control qubit 1, while j represents the target P qubit j, and Ij is the identity operator for the qubit pair (1, j), which is given by Ij = rs |r1 sj ihr1 sj |, with r, s ∈ {+, −}. From the operator of the “Eq. (1)”, it can be seen that the operator Up induces a phase flip (from the + sign to the − sign) to the logical state |−i of each target qubit when the control qubit 1 is initially in the state |−i, and nothing happens otherwise. In this paper, we will present an effective method to realize the NTCP gate, NTQ-NOT gate, and NTCP-NOT gate by using the system of N + 1 charge qubits coupled to a resonator in a cavity QED with nearest qubit–qubit interaction, we calculated the evolution operator a three-step, we used the overall evolution operator for obtaining these logic gates, we also calculate the implementation time, and discuss the result. Our numerical calculation shows that implementation of these gates is feasible in the cavity QED. 2

Basic theory of one superconducting qubit simultaneously controlling N qubits in cavity QED

The superconducting charge qubit consists of a small box, connected to a symmetric superconducting quantum interference device (SQUID) with capacitance CJ0 ,and Josephson coupling energy EJ0 , pierced by an external magnetic flux Φ = Φ0 /2 (Φ0 is the flux quantum), permit tuning of the effective Josephson energy [Figure2(C)]. Let us consider N + 1 identical superconducting charge qubits placed in a single-mode cavity QED are involved in the gate operation[34] [Figure 2(D)]. A control gate voltage Vg is connected to the system via a gate capacitor Cg . We suppose that Vg = Vgdc + Vgac + Vgqu , where Vgdc (Vgac ) is the dc (ac) part of the gate voltage and Vgqu is the quantum part of the gate voltage, which is caused by the electric field of the resonator mode when the qubit is coupled to a resonator. Correspondingly, ac qu dc dc ac ac qu qu we have ng = ndc g + ng + ng where ng = Cg Vg /2e, ng = Cg Vg /2e and ng = Cg Vg /2e. The goal of this section is to demonstrate the manner to obtain a NTCP gate in the case of

T. Said, A. Chouikh, K. Essammouni, and M. Bennai

467

apply a resonant pulse to each qubit in a cavity QED. We Consider N + 1 qubits interacting with the cavity mode. Therefore, the total Hamiltonian of the system (assuming ~ = 1) is [35, 36, 37, 38] Htotal

=

Ez Sz,j + ωc a+ a − EJ (Φ)Sx + (a+ + a)Sz + ε(a+ e−iωd t + a− eiωd t ) +

N +1 X

Γij σx,i σx,j ,

(2)

i,j=1 i6=j

PN +1 Sz and Sx are the collective operators for the (1, 2, ..., N + 1) qubits, where Sz = j=1 gj σz,j PN +1 and Sx = j=1 σx,j , with Pauli operators σz,j = 12 (|ej ihej | − |gj ihgj |), σx,j = 21 (|gj ihej | + |ej ihgj |), |ej i(|gj i) is the excited state (ground state) of the qubit, ωc is the cavity mode frequency, ωd is the frequency of the external drive, a+ , a are the creation and annihilation of the cavity mode, ε is the amplitude of the microwave, gj is the coupling constant between the charge qubit and the resonator mode, and Γij is the force qubit-qubit coupling. Note 2 that the Ez = −2Ec (1 − 2ndc g ) with the charge energy Ec = e /2CΣ (EJ0 ≪ Ec with CΣ = Cg + 2CJ0 ) and ng = Cg Vg /2e. The effective Josephson coupling energy is given by EJ (Φ) = 2EJ0 cos(πΦ/Φ0 ) . The qubits are capacitively coupled to each other via the cavity mode. The fourth term of “Eq. (2)” is given by Vgqu = V0qu (a + a+ ) where V0qu = (~ωc )1/2 (Lc0 )−1/2 . Finally, the coupling constant gj is given by gj = 2Ec Cg V0qu /(~e). To get useful logical gate rates, we work with large amplitude driving fields, in this case, the quantum fluctuations in the drive are very small with respect to the drive amplitude and the drive can be considered, for all practical purposes, as a classical field. Here, it is convenient to displace the field operators + ∗ using the time-dependent displacement operator [36, 39] D(α) = e(αa −α a) . Under this + + ∗ transformation, the field a and a goes to (α + a) and (a + α ), respectively, where α is a complex number representing the classical part of the field. The displaced Hamiltonian reads [36, 40] H

= =

˙ D(α)Htotal D(α) − iD+ (α)D(α)

Ez Sz,j + ωc a+ a − EJ (Φ)Sx + (a+ + a)Sz − g(α∗ + α)Sz +

N +1 X

Γij σx,i σx,j ,

(3)

i,j=1 i6=j

we have chosen α(t) to satisfy α˙ = −iωr α − iεe−i(ωd t+ϕ) , where ϕ is the initial phase of the pulse. This choice of α is made so as to eliminate the direct drive of microwave field on the cavity mode, which is described by ε(a+ e−iωd t + aeiωd t ). Then, we get α = − ωε e−i(ωd t+ϕ) , where ω = ωc − ωd , and for convenience, we assume that we can tuned the coupling strength Γij and the coupling constant gj at the same time to have Γij = Γ and gj = g. Then the Hamiltonian H becomes [36, 39]

H

=

Ez Sz,j + ωc a+ a − EJ (Φ)Sx + (a+ + a)Sz + Ω cos(ωd t + ϕ)Sz +Γ

N +1 X

i,j=1 i6=j

σx,i σx,j ,

(4)

468


PN +1 where Ω = 2gε/ω and Sz = g j=1 σz,j . In addition, by setting Ez = 0 (i.e., ng = 1/2) for each qubit and defining ω0 = EJ (Φ), the Hamiltonian (4) reduces to H

=

ωc a+ a − ω0 Sx + (a+ + a)Sz + Ω cos(ωd t + ϕ)Sz +Γ

N +1 X

(5)

σx,i σx,j ,

i,j=1 i6=j

Define the new basis [14, 41] |+j i = Hamiltonian H becomes H

= =

√1 (|gj i 2

+ |ej i), |−j i =

√1 (|gj i 2

− |ej i). Then, the

H0 + H 1 + H 2 + H 3 ωc a+ a − ω0 Sz + (a+ + a)Sx + Ω cos(ωd t + ϕ)Sx +Γ

N +1 X

σz,i σz,j ,

(6)

i,j=1 i6=j

with ωc a+ a − ω0 Sz

H0

=

H1 H2

= =

Ω cos(ωd t + ϕ)Sx (a+ + a)Sx

H3

=

Γ

N +1 X

σz,i σz,j ,

(7) (8) (9) (10)

i,j=1 i6=j

H0 is the free Hamiltonian of the qubits and the cavity mode, H1 is the interaction Hamiltonian between the qubits and the classical pulse, H2 is the interaction Hamiltonian between the qubits and the cavity mode, and H3 is the interaction Hamiltonian between qubits. In the interaction picture with respect to H0 , the Hamiltonians H1 , H2 and H3 are rewritten, respectively, as (under the assumption that ωd = ω0 )

H1

=

Ω

N +1 X

(eiϕ σj− + σj+ e−iϕ )

(11)

(aσj+ eiδt + a+ σj− e−iδt )

(12)

i,j=1

H2

=

g

N +1 X

i,j=1

H3

=

Γ

N +1 X

σz,i σz,j ,

(13)

i,j=1 i6=j

where δ = ωc −ω0 (the detuning between the atomic transition frequency ωc and the frequency of the cavity mode ω0 ), σx,j = 21 (|+j ih+j | − |−j ih−j |), σz,j = 21 (|−j ih+j | + |+j ih−j |), σj+ = |+j ih−j |, and σj− = |−j ih+j |.


469

Fig. 1. Representation of different detunings δ = ωc − ω0 . In (A), the detuning δ < 0. In (B), the detuning δ ′ > 0. We will use the first case (A) with ϕ = π and the second case (B) with ϕ = 0 to obtain the logic gate. We use the same symbols ωd , ωc , and ω0 for the frequency of the external drive, the atom transition frequency, and the cavity mode frequency. The two horizontal solid lines represent the qubit energy levels for the states |g > and |e > [14].

We will work on two special cases: (ϕ = π and δ < 0 ) and (ϕ = 0 and δ > 0). The operators of developments that will get these two particular cases will be used in Sec.2.3 for obtained the logic gates (NTCP gate, NTQ-NOT gate, and NTCP-NOT gate). 2.1

Evolution operator for pulse phase ϕ = π and detuning δ < 0

In this case of pulse phase ϕ = π and the negative detuning δ = (ωc −ω0 ) < 0, the Hamiltonian H1 becomes H1 = −2ΩSx

(14)

where Sx =

N +1 N +1 X 1 X − σx,j . (σj + σj+ ) = 2 j=1 j=1

(15)

By solving the Schrodinger equation d|Ψ(t)i = (H2 + H3 )|Ψ(t)i, dt

(16)

|Ψ(t)i = e−iH1 t |Ψ′ (t)i,

(17)

d|Ψ′ (t)i = HI′ |Ψ′ (t)i, (where HI′ = HI2 + HI3 ) dt

(18)

i with

we obtain i

470


with HI2

= =

eiH1 t H2 e−iH1 t N +1 X 1 1 + 1 − 2iΩt 1 1 + 1 − −2iΩt + −iδt σx,j + a e g σz,j + σj − σj e σz,j − σj + σj e − 2 2 2 2 2 2 j=1 +H.C,

(19)

and HI3 = eiH1 t H3 e−iH1 t = Γ

N +1 X

σzi σzj ,

(20)

i,j=1 i6=j

In the strong driving region 2Ω ≫ δ, g, Γ, we can eliminate the terms oscillating fast. Then the Hamiltonian HI reduces to [42, 43] ′

HI

=

HI2 + HI3

(21)

g(a+ eiδt + ae−iδt )Sx + Γ

=

N +1 X

σzi σzj .

(22)

i,j=1 i6=j ′

The evolution operator of the Hamiltonian HI can be written as [38, 44] 2

′

U (t) = e−iA(t)Sx e−iB(t)aSx e−iB

∗

(t) a+ Sx −C(t)X

e

, (X =

N +1 X

σzi σzj ).

(23)

i,j=1 i6=j

By solving the Schrodinger equation ′

i

′ ′ dU (t) = HI U (t), dt

(24)

we obtain C(t) B(t) A(t)

=

1 2

Z

t

′

Γdt = Γt, 0 t

(25)

g iδt (e − 1), iδ 0 Z t ′ ′ ′ 1 g2 t + (e−iδt − 1) . B(t )e−iδt dt = = ig δ iδ 0 = g

Z

′

′

eiδt dt =

(26) (27)

Setting t = τ = 2π/ |δ| and δt = 2π, we have B(τ ) = 0 and A(τ ) = g 2 τ /δ. Then, the evolution operator U ′ (t) become ′

2

U (τ ) = eiλSx τ

N +1 Y

i,j=1 i6=j

e−iΓσzi σzj ,

(28)


where λ =

−g 2 δ

471

> 0. Then, we obtain the evolution operator of the system as U (τ )

=

e−iH0 τ U ′ (τ )

=

e2iΩτ Sx eiλτ Sx

N +1 Y

2

e−iΓτ σzi σzj .

(29)

i,j=1 i6=j

The evolution operator U (τ ) will be needed in the first step for realizing the logic gates (NTCP gate, NTQ-NOT gate, and NTCP-NOT gate). 2.2 Evolution operator for pulse phase ϕ = 0 and detuning δ > 0 We consider the positive detuning δ > 0 and the pulse phase ϕ = 0, where the atom-cavity coupling constant varies when the detuning δ changes. We suppose that the qubit 1 is decoupled from the cavity and the pulse. In this case, we will adjusted the level spacing of ′ ′ ′ the qubit 1. Replace now the notation Ω, δ, g and Γ by Ω , δ , g ′ and Γ respectively to distinguish the case ϕ = 0 and δ > 0 from the case ϕ = π and δ < 0. In the following and for simplicity of this work, we use the same symbols ωc and ω0 [Figure 1]. Then, we obtain from the Hamiltonians H1 , H2 and H3 (assuming ωc = ω0 ) ′

H1 ′

H2

= =

H1′ = 2Ω′ Sx′ N +1 X

g′

(30) ′

′

(aσj+ eiδ t + a+ σj− e−iδ t )

(31)

i,j=1 ′

H3

=

Γ′

N +1 X

σz,i σz,j ,

(32)

i,j=1 i6=j

PN +1 PN +1 ′ ′ where δ ′ = (ωc − ω0 ) > 0 and Sx′ = 21 j=2 (σj− + σj+ ) = j=2 σx,j . In the case 2Ω′ ≫ g , δ ′ when the evolution time t = τ = 2δπ′ , the Hamiltonian HI′ is ′

′

HI′ = g ′ (a+ eiδ t + ae−iδ t )Sx′ + Γ′

N +1 X

σzi σzj ,

(33)

σz,i σz,j

(34)

i,j=2 i6=j

then, the evolution operator U ′ (τ ′ ) is ′

U ′ (τ ′ ) = e−2iΩ τ

′

′ ′2 Sx −iλ′ τ ′ Sx

e

N +1 Y

′

eiΓ τ

′

,

i,j=2 i6=j

PN +1 2 where λ′ = gδ′ > 0, and Sx′ = j=2 σx,j , with σx,j = 12 (σj+ + σj− ). So, the evolution operator ′ ′ U (τ ) will be needed for the second step for realizing the logic gates. For realizing these logic gates, we will have the qubits decoupled from the cavity and applying a resonant pulse to each qubit. Therefore, we assumed that the Rabi frequency of the pulse applied to qubit 1 is Ω1 and the Rabi frequency of the pulse applied to qubits (2, ..., N + 1) is Ωr [Figure 3], where the initial phase for each pulse is ϕ = 0. So, in the interaction picture, we have the following interaction Hamiltonian for the qubit system and the pulses as H4 = 2Ω1 σx,1 + 2Ωr Sx′ .

(35)

472


The evolution operator for the Hamiltonian H4 in the evolution time τ is ′

U ′′ (τ ) = e−2iΩ1 τ σx,1 e−i2Ωr τ Sx .

(36)

The evolution operator U ′′ (τ ) given here, will be needed for realizing the logic gates (NTCP gate, NTQ-NOT gate, and NTCP-NOT gate). 3

Realizing a quantum logic gates

3.1 Preparation of the NTCP gate and discussion In this subsection, we will demonstrate how the NTCP gate can be realized based on the ′ ′ evolution operators U (τ ), U (τ ), and U ′′ (τ ). We consider N + 1 qubits moved to a cavity QED. The operations for the NTCP gate Realization and the evolutions operators after each step of operation are as follows First Step: With a detuning δ < 0 [Figure 3(A1 ) and Figure 3(A2 )], we Apply a resonant pulse (with ϕ = π) to each qubit. The pulse Rabi frequency is Ω. Thus, U (τ ) is the evolution π . operator for the N + 1 qubits system, where the interaction time τ = − 2δ Second Step: Apply a resonant pulse (with ϕ = 0) to each qubit with a detuning δ ′ > 0 [Figure 3(B1 ) and Figure 3(B2 )], where the pulse Rabi frequency is Ω′ . Thus, the U ′ (τ ′ ) is the evolution operator for the qubit system, where the interaction time τ ′ = 2δπ ′ . The combined time after these two steps is τ′ + τ =

π π − . 2δ ′ 2δ

(37)

We assumed Ω′ τ ′ = −Ωτ , λ′ τ ′ = λτ and Γ′ τ ′ = Γτ , which can be achieved by adjusting the δ and δ ′ (changing the ωc and ω0 ), the Ω and Ω′ (changing the intensity of the pulses) and the Γ and Γ′ . Then U (τ + τ ′ ) is U (τ + τ ′ )

=

2

′

′2

e2iΩτ (Sx −Sx ) eiλτ (Sx −Sx )

N +1 Y

eiΓτ σz,1 σz,j

j=2

=

′

e2iΩτ σx,1 e2iλτ σx,1 Sx

N +1 Y

eiΓτ σz,1 σz,j ,

(38)

j=2

with Sx − Sx′ = σx,1 , Sx2 − Sx′2 = I + 2σx,1 Sx′ (I is the identity operator for qubit1), where PN +1 Sx′ = j=2 σx,j , and σx,j = 21 (Sj+ + Sj− ). The third step: In the case of ϕ = 0, we applied the Rabi frequency for the pulse Ω1 to qubit 1 [Figure 3(C1 )]. Also, we applied the Rabi frequency of the pulse Ωr to qubits (2, ..., N + 1) [Figure 3(C2 )]. Then, we will obtained the time evolution operator U ′′ (τ ) (“Eq. (36)”) with τ is evolution time. After this three step operation, the combined time evolution operator of the N + 1 qubits system is Ugate

=

U (2τ + τ ′ )

=

e−2iσx,1 τ (Ω1 −Ω) .e−2iΩr Sx τ e2iλτ σx,1 Sx

′

′

N +1 Y j=2

eiσz,1 σz,j τ Γ ,

(39)


Fig. 2. (A) Proposed circuit of a controlled-phase gate simultaneously controlling on N target qubits (2, 3, ..., N + 1). Here, Z represents a controlled-phase flip on each target qubit. If the control qubit (i.e., qubit 1) is in the state |ei, then the state |ei at each Z is phase-flipped as |ei −→ −|ei while the state |gi remains unchanged. However, if the control qubit is in the state |gi, nothing happens to either of the states |gi and |ei at each Z. (B) N -target-qubit controlledNOT gate, obtained from N -qubit CP gate. In the circuit (B), the symbol ⊕ represents a CNOT gate on each target qubit. If the control qubit is in the state |ei, then the state at ⊕ is bit flipped as |ei −→ |gi and |gi −→ |ei. However, when the control qubit is in the state |gi, the state at ⊕ remains unchanged. (C) Diagram of a superconducting charge qubit, in the charge regime ∆ ≫ Ec ≫ EJ0 ≫ kB T (here, ∆, Ec , EJ0 , kB and T are the Gap, charging energy, Josephson coupling energy, boltzmann constant, and temperature, respectively). (D) (1, 2, ..., N + 1) superconducting qubits are placed in a microwave cavity and are coupled to each other via the cavity mode for realized the NTCP gate.

473

474


Fig. 3. Representation of the three steps in the case the change of the qubit transition frequency ωc and the ac gate-voltage frequency ωd with superconducting charge qubits coupled to a resonator. The first step (A1 and A2 ), the second step (B1 and B2 ), and the third step (C1 and C2 ), where the figures (A1 ),(B1 ) and (C1 ) correspond to qubit 1, and the other figures (A2 ),(B2 ) and (C2 ) correspond to qubits (2, 3, ..., N + 1). In these figures, δ and δ ′ are small detuning between the cavity mode frequency ω0 , and the qubit transition frequency ωc , ∆ = ωc − ω0 is large detuning of the cavity mode, and ϕ is the initial stage of the pulse. The Ω(V0 ), Ω′ (V0′ ), Ω1 (V01 ), and Ωr (V0r ) are the Rabi frequencies of the amplitudes V0 , V0′ , V01 and V0r of the ac gate voltage, respectively, and each circle with a symbo ∽ represents an ac gate voltage. In each figure, the two horizontal solid lines represent the qubit levels |gi and |ei.


475

with the conditions λ′′

=

Ω1

=

λ 2 λ′′ N + Ω,

Ωr Γ

= =

λ′′ , 4β,

(40)

which can be achieved by adjusting the Rabi frequencies Ω, Ω1 and Ωr , the time evolution operator Ugate becomes N +1 Y UP (1, j), (41) Ugate = j=2

′′

with UP (1, j) = e−2iλ τ (σx,1 +σx,j −2σx,1 σx,j )+4iβτ σz,1 σz,j . According to the evolution operator UP (1, j) above, on the basis |+1 i = √12 (|g1 i + |e1 i) and |−1 i = √12 (|g1 i − |e1 i) of the Pauli operators σx,1 and σz,1 for qubit 1, so the basis |+j i = √12 (|gj i + |ej i) and |−j i = √12 (|gj i − |ej i) of the Pauli operators σx,j and σz,j for qubits (2, 3, ..., N + 1), we can obtain following evolutions Up (1, j)|+1 i|+j i

Up (1, j)|+1 i|−j i

Up (1, j)|−1 i|+j i

Up (1, j)|−1 i|−j i

= = = =

e−iλ

′′

τ

′′

e

−iλ τ

e

−iλ′′ τ

e

3iλ′′ τ

cos β|+1 i|+j i + ie3iλ cos β|+1 i|−j i + ie

cos β|−1 i|+j i + ie

cos β|−1 i|−j i + ie

′′

τ

′′

−iλ τ −iλ′′ τ

−iλ′′ τ

sin β|−1 i|−j i,

sin β|−1 i|+j i,

sin β|+1 i|−j i,

(42)

sin β|+1 i|+j i.

By setting β = 2kπ (with k being an integer), we obtain Up (1, j)|+1 i|+j i

Up (1, j)|+1 i|−j i Up (1, j)|−1 i|+j i

Up (1, j)|−1 i|−j i

Where the term e−iλ integer), we have

′′

τ

= = = =

|+1 i|+j i,

|+1 i|−j i, |−1 i|+j i, e

4iλ′′ τ

(43)

|−1 i|−j i.

is omitted. By selecting 4λ′′ τ = (2k + 1)π (with k being an Up |+1 i|+j i

Up |+1 i|−j i Up |−1 i|+j i

Up |−1 i|−j i

= = = =

|+1 i|+j i,

|+1 i|−j i, |−1 i|+j i,

(44)

−|−1 i|−j i.

By this way, one can see that N two-qubit CP gates are simultaneously performed on the qubit pairs (1, 2), (1, 3), . . . , (1, N + 1), respectively. Hence, it is clear that the NTCP gate can be realised after the three-step process. Now, we give a brief discussion about our proposal. The decoherence time T1 = 1.87µs [45] and the coupling strength is g = 2π × 100M Hz [39] , which is experimentally available.

476

Realizing an N-two-qubit quantum logic gate in a cavity QED with nearest qubit–qubit interaction 2

With δ = g (We notice that δ satisfies the equation (g/δ) = 2k + 1, where k is an integer. So, when k = 0, δ takes maximum δ = g). Then, the operation time is top = 3τ = 2π × g3 . So, the direct calculation shows that the operation time required to implement the NTCP gate with superconducting qubits will be top = 30 ns, which is much shorter than T1 . The dephasing time T2 = 2.22µs which has been realized in recent experiments [45], is much longer than the operation time top . 3.2

NTCP-NOT gate and discussion

In this subsection, we focus on how to realize NTCP-NOT gate with one superconducting qubit simultaneously controlling N target qubits by introducting the qubit-qubit interaction. Our system consists of N + 1 identical charge qubit coupled to a single-mode resonator, then the overall evolution operator written as (see “Eq. (41)”) Ugate =

N +1 Y

UN P (1, j).

(45)

j=2

′′

with UN P (1, j) = e−2iλ τ (σx,1 +σx,j −2σx,1 σx,j )+4iβτ σz,1 σz,j . According to the evolution operator UP (1, j) above, on the basis |+1 i = √12 (|g1 i + |e1 i) and |−1 i = √12 (|g1 i − |e1 i) of the Pauli operators σx,1 and σz,1 for qubit 1, so the basis |+j i = √12 (|gj i+|ej i) and |−j i = √12 (|gj i−|ej i) of the Pauli operators σx,j and σz,j for qubits (2, 3, ..., N + 1), we can obtain UN p (1, j)|+1 i|+j i

UN p (1, j)|+1 i|−j i

UN p (1, j)|−1 i|+j i

UN p (1, j)|−1 i|−j i

= = = =

e−iλ e

τ

−iλ′′ τ

e−iλ e

′′

′′

′′

τ

3iλ τ

cos β|+1 i|+j i + ie3iλ

cos β|+1 i|−j i + ie

′′

−iλ′′ τ

cos β|−1 i|+j i + ie−iλ


τ

′′

τ

′′

−iλ τ


sin β|−1 i|+j i,

sin β|+1 i|−j i,

(46)

sin β|+1 i|+j i,

By setting β = (2k + 12 )π (with k being an integer), we obtain UN p (1, j)|+1 i|+j i

UN p (1, j)|+1 i|−j i

UN p (1, j)|−1 i|+j i

UN p (1, j)|−1 i|−j i Where the term eiλ we have

′′

τ

= = = =

′′

ie2iλ τ |−1 i|−j i, ′′

ie−2iλ τ |−1 i|+j i, ′′

ie−2iλ τ |+1 i|−j i,

(47)

′′

ie−2iλ τ |+1 i|+j i.

is omitted. By selecting 2λ′′ τ = (2k + 12 )π (with k being an integer), UN p (1, j)|+1 i|+j i UN p (1, j)|+1 i|−j i

UN p (1, j)|−1 i|+j i UN p (1, j)|−1 i|−j i

= = = =

−|−1 i|−j i, |−1 i|+j i,

|+1 i|−j i, |+1 i|+j i.

(48)


477

By this way, one can see that N two-qubit-CP-NOT gates are simultaneously performed on the qubit pairs (1, 2), (1, 3), . . . , (1, N +1), respectively. Hence, it is clear that the NTCP-NOT gate can be realised in the cavity QED. The decoherence time T1 = 1.87µs [45] and the coupling strength is g = 2π × 100M Hz ′ 3 [39], with δ = 2g. The implementation time is top = 2π × 2g . So, the direct calculation shows that the operation time required to implement the NTCP-NOT gate will be top = 15ns, which is much shorter than T1 . The dephasing time T2 = 2.22µs which has been realized in recent ′ experiments [45], is much longer than the implementation time top . 3.3 N two-qubit-NOT gate and discussion In this subsection, we will demonstrate how to realize a NTQ-NOT gate (N Two-Qubit-NOT gate) based on the following overall evolution operator of the N + 1 identical charge qubits (see “Eq. (41)”) N +1 Y UN (1, j). (49) Ugate = j=2

−2iλ′′ τ (σx,1 +σx,j −2σx,1 σx,j )+4iβτ σz,1 σz,j

. According to the evolution operator with UN (1, j) = e UN (1, j) above, on the basis |+1 i = √12 (|g1 i + |e1 i) and |−1 i = √12 (|g1 i − |e1 i) of the Pauli operators σx,1 and σz,1 for qubit 1, so the basis |+j i = √12 (|gj i+|ej i) and |−j i = √12 (|gj i−|ej i) of the Pauli operators σx,j and σz,j for qubits (2, 3, ..., N + 1), we can obtain UN (1, j)|+1 i|+j i UN (1, j)|+1 i|−j i

UN (1, j)|−1 i|+j i

UN (1, j)|−1 i|−j i

= = = =

e−iλ

′′

τ

′′

e

−iλ τ

e

−iλ′′ τ

e

3iλ′′ τ

cos β|+1 i|+j i + ie3iλ

cos β|+1 i|−j i + ie

cos β|−1 i|+j i + ie


′′

τ

′′

−iλ τ


sin β|−1 i|+j i,

−iλ′′ τ

−iλ′′ τ

sin β|+1 i|−j i,

(50)

sin β|+1 i|+j i.

By setting β = (2k + 12 )π (with k being an integer), we obtain UN (1, j)|+1 i|+j i

UN (1, j)|+1 i|−j i

UN (1, j)|−1 i|+j i

UN (1, j)|−1 i|−j i

= = = =

′′

ie3iλ τ |−1 i|−j i, ′′

ie−iλ τ |−1 i|+j i, ′′

ie−iλ τ |+1 i|−j i,

ie

−iλ′′ τ

(51)

|+1 i|+j i.

By selecting λ′′ τ = (2k + 12 )π (with k being an integer), we have UN (1, j)|+1 i|+j i

UN (1, j)|+1 i|−j i UN (1, j)|−1 i|+j i

UN (1, j)|−1 i|−j i

= = = =

|−1 i|−j i,

|−1 i|+j i, |+1 i|−j i,

(52)

|+1 i|+j i.

By this way, one can see that N two-superconducting qubit-NOT gates are simultaneously performed on the qubit pairs (1, 2), (1, 3), . . . , (1, N + 1), respectively. Each twosuperconducting qubit-NOT operation includes the same control qubit and a different target

478


qubit. Thus, the N two-superconducting qubit-NOT gate are implemented simultaneously between one superconducting charge qubit and N qubits. Hence, it is clear that the NTQ-NOT gate can be realised in cavity QED. Finally, we give a brief discussion about our proposal. The decoherence time T1 = √ 1.87µs[45] and the coupling strength is g = 2π × 100M Hz[39], with δ = 2g. Then, the ′′ operation time is top = 2π × √32g . So, the direct calculation shows that the operation time ′′

required to implement the NTQ-NOT gate will be top = 21.21 ns, which is much shorter than ′′ T1 . The dephasing time T2 = 2.22µs[45] is much longer than the operation time top .

3.4 Fidelity Let us now study the fidelity of the gate operations. In order to check the validity of our proposal, we define the following fidelity to characterize the deviation of how much the output states |Ψ(t)i deviate in amplitude and phase from the ideal logical gate transformation for the different input states[35, 46]: F

= =

|hΨ(t)|U (t)|Ψ(0)i| 4 cos( π ( δ )) 5 2Ω

2

Fig. 4. Numerical results for fidelity of the gate operations versus the ratio b. Here, b =

(53)

δ . 2Ω

Where |Ψ(t)i represents the final state the whole system after the gate operations that the initial state |Ψ(0)i followed by an ideal phase operation; and U (t) describers the overall evolution operator of the system are performed in a real situation. Our numerical calculation shows that a high fidelity ∼ 100% can be achieved when 2Ω ≫ δ (Fig. 4). 4

Possible experimental implementation

For this method to work, we discuss some issues which are relevant for future experimental implementation of our proposal. As a concrete example, let us consider the experimental feasibility of implementing a two-target-qubit controlled phase gate using superconducting


479

charge qubits with parameters Cg = 1aF , CJ0 = 300aF , Ec /h = 32GHz and EJ0 /h = 5GHz [14]. The charge qubits with these parameters are available at present. For a superconducting one dimensional standing-wave CPW (coplanar waveguide) cavity and each qubit placed at an antinode of the cavity field, the amplitude of the quantum part of the gate voltage is given by V0qu = (~ωc )1/2 (Lc0 )−1/2 , where L is the cavity length and c0 is the capacitance per unit length of the cavity. The coupling constant is then given by g = 2Ec Cg (~e)−1 V0qu , ′ showing that g does not depend on the detuning δ. Then, the condition g = g required above can be satisfied. For charge qubits with the above parameters and a cavity with ωc /2π = 10GHz, L = λ ∼ 12mm, c0 ∼ 0.22aF/µm, and εe = 6.3, where λ is the wavelength of the cavity mode and εe is the effective relative dielectric constant [14]. For first Step the deviation from the degeneracy point for each one of the qubits (1, 2, ..., N + 1) is given by ε0 = ~(Ω + g)/(4Ec ). For second Step, the deviation from the degeneracy point for dc qubits (2, 3, ..., N + 1) is ε1 = ~(Ω′ + g ′ )/(4Ec ). For third step, from ndc g,1 and ng above, we obtain ε2 = Ω1 /(4Ec ) = ~(N λ + 2Ω)/(8Ec ) and ε3 = Ωr /(4Ec ) = ~λ/(8Ec ) where ε2 is the deviation from the degeneracy point for the control qubit 1 and ε4 the deviation from the degeneracy point for target qubits (2, 3, ..., N + 1). For a five-target-qubit gate (n = 5) and Ω/2π = 600M Hz with the parameters given above, we have ε0 = ε1 ∼ 5.47 × 10−3 , ε2 ∼ 4.48 × 10−3 , ε3 ∼ 1.1 × 10−4 . So, the conditions for the qubits to work near the degeneracy point are well satisfied. The fidelity of the gate operations versus the variation in qubit-qubit couplings is given by 4 π g2 ∆Γ F = cos( ( )(1 − )) 5 ΩΓ0 Γ

(54)

Fig. 5. The fidelity versus the b′ . Here, b′ = ∆Γ . The parameters used in the numerical calculation Γ are g/2π = 100M Hz, Ω = 600M Hz and Γ0 = 0.25GHz.

Where ∆Γ = Γ − Γ0 . Figure 5 shows the fidelity versus ∆Γ Γ . From this figure, one can see that for Γ = Γ0 = 0.25GHz [47] a high fidelity 99.6% is achievable for a NTCP gate. ′′ The total operation time of the logic gates (top , t′op and top ) which is much shorter than −1 the cavity-mode lifetime κ = Q/ωc ∼ 159ns for a cavity with the quality factor of the

480


cavity Q = 104 has been demonstrated by cavity QED experiments with superconducting charge qubits [48]. 5

Conclusion

In summary, we have proposed a simple scheme to realize an N -two-qubit quantum logic gates of one superconducting qubit simultaneously controlling N target qubits in a cavity QED by introducing of the qubit-qubit interaction. In this system, logical gates are realized by driving the resonator with microwave fields. In our scheme, we have applied an inhomogeneous field to distinguish between the (N + 1) qubits. The scheme is insensitive to the thermal field. Furthermore, the gate operation time is independent of the number of qubits, where the qubit definitions are the same which makes the work easier. In addition, the operation time is only dependent of the detuning. Therefore, the time can be controlled by adjusting the frequency between the |gj i and |ej i. Thus, the gate operation is independent of the initial state of the cavity mode. However, we have presented an effective method for obtained a logical gates (NTCP gate, NTQ-NOT gate, and NTCP-NOT gate) in a cavity QED, we have calculated an evolution operator with the three steps in the case of the qubit-qubit interaction. Finally, we have applied the overall evolution operator to working basis of the qubit 1 and the qubits j (j = 2, . . . , N + 1) for find the logical gates. The essential advantage of the scheme is that this gate can be realized in a time much shorter than decoherence time and dephasing time of the superconducting qubits chosen in the system. In addition, numerical simulation of the the gate operations shows that the scheme could be achieved with high fidelity under current state-of-the-art technology. References 1. M. H. Devoret and R. J. Schoelkopf, ”Superconducting Circuits for Quantum Information: An Outlook”, Science 339, 1169 (2013). 2. H. F. Wang, X. Ji and S. Zhang, ”Improving the security of multiparty quantum secret splitting and quantum state sharing”, Phys. Lett. A 358, 11(2006). 3. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, ”Measurement of conditional phase shifts for quantum logic”, Phys. Rev. Lett. 75, 4710 (1995). 4. G. Burkard and D. D. Awschalom, ”A cavity-mediated quantum CPHASE gate between NV spin qubits in diamond”, arXiv:1402.6351 (2014). 5. X. Lin, ”Implementation of a multiqubit phase gate with one qubit simultaneously controlling n qubits in the ion-trap system”, Quant. Info. & Comput. pages 625-632 (2014). 6. T. R. Tan et al., ”Demonstration of a dressed-state phase gate for trapped ions”, Phys. Rev. Lett. 110, 263002 (2013). 7. J. Teles et al., ”Quantum information processing by nuclear magnetic resonance on quadrupolar nuclei”. Phil. Trans. Math. Phys. Eng. Sci. 370, 4770 (2012). 8. H. Chen, et al., ”Experimental demonstration of probabilistic quantum cloning”. Phys. Rev. Lett. 106, 180404 (2011). 9. X. Li et al., ”An All-Optical Quantum Gate in a Semiconductor Quantum Dot”, Science 301, 809 (2003). 10. M. Raith et al., ”Theory of single electron spin relaxation in Si/SiGe lateral coupled quantum dots”, Phys. Rev. B 83, 195318 (2011). 11. J. M. Chow et al., ”Microwave-activated conditional-phase gate for superconducting qubits”, New J. Phys. 15 115012 (2013). 12. J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319 (2007). 13. J. Koch et al, ”Charge-insensitive qubit design derived from the Cooper pair box”, Phys. Rev. A 76, 042319 (2007).


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