N Two-Transmon-Qubit Quantum Logic Gates Realized in a Circuit QED System

Home , Quantum cryptography, Quantum logic

Appl. Math. Inf. Sci. 13, No. 5, 839-846 (2019) 839 Applied Mathematics & Information Sciences An International Journal

http://dx.doi.org/10.18576/amis/130518

1, 1 1,2 T. Said ∗, A. Chouikh and M. Bennai 1 Laboratoire de Physique de la Matiere Condensee, Equipe Physique Quantique et Applications, Faculte des Sciences Ben Msik, B.P. 7955, Universite Hassan II, Casablanca, Maroc 2 LPHE-Modelisation et Simulation, Faculte des Sciences Rabat, Universite de Rabat, Maroc

Received: 2 Jan. 2019, Revised: 10 Apr. 2019, Accepted: 12 Apr. 2019 Published online: 1 Sep. 2019

Abstract: We theoretically present an effective method to realize NiSWAP, N(sqrt(iSWAP)), NSWAP, N(sqrt(SWAP)) and NTQ-NOT gates based on the qubit-qubit interaction in a circuit QED using N+1 transmon qubits driven by a strong microwave field. In this system, the interaction between the qubits and the resonator field can be achieved by turning the gate voltage and the external flux. The operation time is independent of the number of qubits involved in the scheme, and the gates operations are insensitive to the initial state of the resonator. These quantum logic gates can be realized in a time (nanosecond-scale) much smaller than decoherence time (microsecond-scale), and it is more immune to the 1/f charge noise and has longer dephasing time due to the favorable properties of the transmon qubits in the circuit QED. Numerical simulation under the influence of the gates operations shows that the scheme can be implemented with high fidelity. We also propose a detailed procedure and experimentally analyze its feasibility. Moreover, the scheme might be experimentally achieved efficiently within current state-of-the-art technology.

Keywords: Transmon qubit, NiSWAP gate, NSWAP gate, NTQ-NOT gate, N√iSWAPgate, N√SWAP gate, circuit QED.

1 Introduction currently the promising candidate for future quantum computer. Recently, Yang et al. [15] proposed a novel scheme Quantum computation has attracted much attention since for implementing multi-qubit quantum controlled-phase a quantum computer has an ability to solve hard gate with one superconducting qubit simultaneously computational problems with high efficiency compared to controlling n qubits selected from N qubits (1 < n < N). a classical computer [1,2,3]. Also, it is well known that it In ref. [16] the authors proposed a scheme to realize is difficult to simulate the behaviour of a quantum NTCP gate in a circuit QED by using the system of mechanical system with a classical computer. The (N + 1) transmon qubits. The method of implementing n difficulty arises because quantum systems are not qubits SWAP gates simultaneously between one confined to their eigenstates but can in general exist in superconducting charge qubit with n qubits coupled to a any superposition of them, thus the vector space needed cavity was firstly proposed by Song et al. [17]. to describe the system is extremely large. Quantum logic gates are the basic building blocks of a quantum computer In recent works, We have proposed another type of [4,5]. A universal quantum computer can be built on a multiqubit controlled gate that is a NTCP gate by using series of two-qubit logic gates [6,7]. Up to now, a large qubit-qubit interaction in a circuit QED [18]. Also, we number of theoretical proposals have been introduced to have presented a method for realizing NTCP-NOT and implement various gates, such as two-qubit NTQ-NOT gates with one control superconducting qubit Controlled-NOT (CNOT) gate [8], SWAP gate [9], simultaneously controlling N target qubits in a cavity iSWAP gate [10], √iSWAP gate [8] and √SWAP gate QED [19]. In the present study, we present and [11] in different systems, such as cavity QED system [13] demonstrate a method for realizing N two-transmon-qubit and circuit QED system [14]. The circuit QED is quantum logic gates (NiSWAP, N√iSWAP, NSWAP,

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N√SWAP, and NTQ-NOT gates) of one transmon qubit we get Ω = g jε/δ where δ = ωr ωd is the detuning − simultaneously controlling N target transmon qubits in between the resonance frequency of the TLR ωr and the circuit QED with nearest qubit–qubit interaction by frequency of the external drive applied to the TLR ωd, 1 1 adding a strong microwave field. These type of quantum Γ = g jg ( + ) [22] which describes the effective jk k ∆ j ∆k logic gates are useful in quantum computation and interaction between qubits j and k, and ∆ j(∆k) is large quantum information processing. On the other hand, the detuning as compared to the qubit-TLR coupling g j(gk) qubit-qubit interaction could influence the evolution of [23]. σ , σ , and σ are the collective operators for the the system used in ref. [17]. Moreover, the presence of z, j j− j− (1,2,...,N + 1) qubits, which are given by both qubit-qubit interaction and Jaynes-Cummings make 1 + σz, j = ( 0 j 0 j 1 j 1 j ), σ = 1 j 0 j , our system powerful. 2 | ih | − | ih | j | ih | The paper is organized as follows: In Sec.2, we σ − = 0 j 1 j where 1 j ( 0 j ) is the excited state j | ih | | i | i concretely illustrate the way NiSWAP, N√iSWAP, (ground state) of the transmon qubit, ωr = 1/√LC is the NSWAP, N√SWAP and NTQ-NOT gates via circuit QED resonance frequency of the TLR where the transmission by using the system of N + 1 transmon qubits coupled to a line resonator can be modeled as a simple harmonic resonator with nearest qubit–qubit interaction, we have oscillator composed of the parallel combination of an calculated the evolution operator a three-step, we use the inductor L and a capacitor C, ωq is the transition overall evolution operator for obtaining these quantum frequency of the transmon qubit with + logic gates, we also give a brief discussion about our ωq1 = ωq2 = ... = ωqN+1 , a ,a are the creation and proposal. In Sec.3, we study the fidelity and then discuss annihilation of the resonator mode. In the high EJ/EC its feasibility based on current experiments in circuit limit the transition frequency between the ground state QED. A concluding summary is given in Sec.4. 0 j and excited state 1 j is given by ωq = √8EJEC/h¯, | i 2 | i where EC = e /2CΣ and EJ(Φ) = EJ0 cos(πΦ/Φ0) , 1 1 1 | | CΣ = CS + (Cg− + Cg′− )− . Here, Cg, Cg′ are the gate 2 Basic theory capacitance, CS is the additional capacitor, CΣ is the effective total capacitance, Φ is the external magnetic flux N The method presented for implementing applied to the SQUID loop, Φ0 = h/2e is the flux two-transmon-qubit quantum logic gates (NiSWAP, quantum, EJ(Φ) is the effective Josephson coupling N√iSWAP, NSWAP, N√SWAP and NTQ-NOT gates) energy, EC is the charging energy, and EJ0 is the via circuit QED is based on the qubit-qubit interaction. In Josephson coupling energy, where the qubit work at the this subsection, we show how to apply the first method to regime EJ/EC 1. It is obvious that the frequency of the ≫ implement these quantum logic gates which the transmon transmon qubit ωq can be tuned by external magnetic flux qubits are capacitively coupled to a superconducting Φ. When we work with a large amplitude driving field, transmission line resonator (TLR) driven by a strong quantum fluctuations in the drive are very small with microwave field. The physical setup considered in this respect to the drive amplitude and the drive can be study is schematically illustrated in Fig.1. The operation considered, for all practical purposes, as a classical field. time of these gate is shorten because the coupling constant between the transmon qubit and the quantum For convince, we assume that we can tune the coupling date bus (QDB) is large enough in the case of TLR has constant g j and the coupling strength Γjk at the same time 2g2 been chosen as QDB. The similar architectures have been to have g j = gk = g and Γjk = Γ = ∆ (in the case ∆ j = used in refs. [20]. ∆k = ∆). By assuming that ωd = ωq, we have the following Let us now consider an (N + 1) transmon qubits Hamiltonian HI in the interaction picture [18] which are capacitively coupled to a TLR driven by a strong microwave field [20] as depicted in Fig.1. A microwave field of frequency ωd is applied to the input N+1 N+1 + iδt + iδt + wire of the TLR. The (N + 1) transmon qubits each HI = Ω ∑ σ j + σ j− + g ∑ e aσ j + e− a σ j− j=1 j=1 having two-level subspaces driven by a conventional field added, these transmon qubits are capacitively coupled to N+1 + it. Moreover, the qubit-qubit interaction should be +Γ ∑ σ j σk−, (2) j,k=1, j=k included in the circuit QED [21], where the direct 6 interaction between the qubits, in circuit QED, is virtually realized in the dispersive regime [22]. The Hamiltonian of the whole system (assuming h¯ = 1) given by [18] where δ = ωr ωq is the detuning between the frequency − N+1 N+1 of the qubit ωq and the frequency of the resonator ωr. + + iωdt iωdt When Ω δ,g,Γ and δ g,Γ we can get the H = ωq ∑ σz, j + ωra a + Ω ∑ (σ j e− + σ j−e ) ≫ ≫ j=1 j=1 Hamiltonian HI in the interaction picture [15,24,25,26] N+1 N+1 + + + + ∑ g j(a σ j− + aσ j )+ ∑ Γjkσ j σk−, (1) j=1 j,k=1, j=k HI = H0 + He f f , (3) 6

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We now show how to utilize the above model (Eq.6) to implement an NiSWAP, N√iSWAP, NSWAP, N√SWAP and NTQ-NOT gates. We note that: (i) for each one of qubits 1,2,...,N + 1 of each step of the dc operations, the dc gate voltage Vg = e/Cg, such that Ez = 0 for each qubit, and (ii) the resonator mode frequency ωr is fixed during the entire operation. We now consider two special cases: δ > 0 as well as δ < 0. The results from the unitary evolution, obtained for these two special cases, are employed below for the gate implementation. The detailed procedure is given as follows: ac Step 1. Let us begin with the case δ > 0, set Vg = Fig. 1: (Color online) Schematic diagram of a TLR and several V0 cos(ωt) for each qubit, adjust the transition frequencies (N + 1) transmon qubits are coupled in a circuit QED (blue) of (N + 1) qubits by external flux Φ j, then ωq = ωd is driven by a strong microwave field of frequency ωd (this field satisfied, with the pulse Rabi frequency Ω = gε/δ of the is applied to the input wire of the TLR). The TLR is connected amplitude V0 where the detuning δ = ωr ωq > 0. So, ∼ − to the input wiring with a capacitor Cin, and output wiring with a after a period of evolution time τ1 = 2π/δ, the evolution capacitor Cout . In addition, the coupling between qubits i and j is operator for the qubit system corresponding to this step via inductances coupling, where this coupling can be interpreted would be the UI(t) of Eq.6. as an energy coupling between qubits. N+1 2 N+1 g + + U(τ1) = exp i2Ωτ ∑ σx, j i τ ∑ (σ j σ −j + a aσz, j) (− j=1 − δ " j=1 N+1 N+1 + with + σ σ − iΓτ σx jσ ∑ j k − ∑ , x,k N+1 j,k=1, j=k # j,k=1, j=k 6 6 H0 = 2Ω σx, j (4) 1 ∑ + σ +σ + σ σ + , (8) j=1 4 j k− −j k 2 N+1 N+1 ac g + + + Step 2. Let us now consider the case δ < 0, set Vg = 0 He f f = ∑ (σ j σ j− + a aσz, j)+ ∑ σ j σk− (5) δ " j=1 j,k=1, j=k # for qubit 1, adjust the level spacings of qubit 1 such that 6 the resonator mode is coupled to qubits (2,3,...,N + 1), so N+1 1 + + ωq = ωd is satisfied, with the pulse Rabi frequency is Ω ′ +Γ σx, jσx,k + σ j σ − + σ j−σ , ∑ k k ′ j,k=1, j=k 4 of the amplitude V0, where the detuning 6 δ ′ = ωr ωq′ = δ > 0. Then, after a period of time 1 + − − where σx, j = 2 σ j + σ j− . From Eq.3, we can easily τ2 = 2π/δ ′, the evolution operator of the system corresponding to this step can be described by obtain the corresponding evolution operator U t as N+1 2 N+1 ( ) g + + follows [15,25,27] U(τ2) = exp i2Ωτ ∑ σx, j i τ ∑ (σ j σ −j + a aσz, j) (− j=2 − δ " j=2 N+1 2 N+1 g + + N+1 N+1 U(t) = exp i2Ωt σx, j i t σ σ − + a aσz, j + ∑ ∑ j j + σ σ − iΓτ σx, jσx k (− j=1 − δ " j=1 ∑ j k ∑ , j,k=2, j=k # − j,k=2, j=k N+1 N+1 6 6 + 1 1 + σ σ − iΓ t σx, jσx,k σ +σ σ σ + , (9) ∑ j k − ∑ 2 + j k− + −j k j,k=1, j=k # j,k=1, j=k 4 6 6 1 The combined time evolution operator of the whole + σ +σ + σ σ + . 4 j k− −j k system, after these two steps, is arrived at (6) 2 g + + U(τ1 + τ2) = exp i2Ωτσx 1 i τ(σ σ − + a aσz 1) − , − δ 1 1 , 2 N+1 N+1 For a charge qubit 1 coupled to the resonator, assumed g + + i τ (σ σ − + σ −σ ) iΓτ σx 1σx j δ g . Then, the corresponding evolution operator can − δ ∑ 1 j 1 j − ∑ , , ≫ | | j=2 j=2 be expressed as [17,28] 1 σ +σ + σ σ + . (10) 2 2 1 −j 1− j g + + UI1(t)= exp 2iΩtσx 1 i t(σ σ − + a aσz 1) . − , − δ 1 1 , Step 3. We adjust the transition frequency of qubits (7) 2,3,...,N + 1 such that the resonator mode is largely

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decoupled from each qubit, but we leave the transition the evolution operator of the whole system (Eq.12). 2 frequency unchanged for qubit 1 such that g / δ g According to the evolution operator Ugate(1, j) above | ′| ≪ | | (i.e. δ g where δ < 0) and Ω = gε/δ . When the (from Eq.12) on the basis ( 01 , 11 ) for qubit 1, so the | ′| ≫ | | ′ ′ ′ | i | i condition δ ′ = δ is satisﬁed, the time evolution operato basis ( 0 j , 1 j ) for qubit (2,3,....N + 1), we can obtain after a period of− time τ is [17,28] following| i evolutions| i

2 Ugate(1, j) 01 0 j = cos χτ 01 0 j isinχτ 11 1 j g + + | i| i | i| i− | i| i UI1(τ3)= exp 2iΩτσx,1 + i τ(σ σ − + a aσz,1) . iηπ δ 1 1 Ugate(1, j) 01 1 j = e [[cos(γ + χ)τ 01 1 j ] | i| i | i| i (11) isin(γ + χ)τ 11 0 j ] − | i| i iηπ Ugate(1, j) 11 0 j = e [[cos(γ + χ)τ 11 0 j ] After the three-step process, a phase ﬂip (i.e., | i| i | i| i 1 1 ) on the state 1 of each target transmon qubit isin(γ + χ)τ 01 1 j ] | i → −| i | i − | i| i is achieved when the control transmon qubit 1 is initially Ugate(1, j) 11 1 j = cos χτ 11 1 j isinχτ 01 0 j , | i| i | i| i− | i| i in the state 1 , but nothing happens to the states 0 and (15) 1 of each| i target transmon qubit when the| controli transmon| i qubit 1 is initially in the state 0 . When the 2 | i A phase factor ηπ in the previous evolutions can be conditions Γ + g = γ and Γ = 4χ, the evolution operator 2 δ produced by several different proposals [31]. By setting of the system, after the above three steps operation, is χτ = 2kπ (with k being an integer), η = 2m (with m being given by 3 N+1 an integer) and γτ = 2n + 2 π (with n being an integer), U τ U 1 j , (12) we obtain N-two-qubit iSWAP operations as (from Eq.15) s( )= ∏ gate( , ) j=2 UiSWAP(1, j) 01 0 j = 01 0 j | i| i | i| i where UiSWAP(1, j) 01 1 j = i 11 0 j + + Ugate(1, j)= exp[ iχτ4σx,1σx, j iγτ(σ σ − + σ −σ )], | i| i | i| i − − 1 j 1 j UiSWAP(1, j) 11 0 j = i 01 1 j j = 2,3,...,N + 1. In the next section, we will use this | i| i | i| i UiSWAP(1, j) 11 1 j = 11 1 j . (16) evolution operator of the whole system (Eq.12) for | i| i | i| i demonstrate how NiSWAP, N√iSWAP, NSWAP, N√SWAP and NTQ-NOT gates can be realized. By setting χτ = 2kπ (with k being an integer), 5 η = 2m (with m being an integer) and γτ = 2n + 4 π (with n being an integer), we obtain N-two-qubit 3 Preparation of quantum logic gates and discussion √iSWAP operations as

U (1, j) 01 0 j = 01 0 j 3.1 Implementation of NiSWAP and N√iSWAPgates √iSWAP | i| i | i| i 1 U (1, j) 01 1 j = [ 01 1 j + i 11 0 j ] It is demonstrated that more complex quantum gates can √iSWAP | i| i √2 | i| i | i| i be created by the iSWAP and √iSWAPgates [29]. 1 √ U√ (1, j) 11 0 j = [ 11 0 j + i 01 1 j ] Furthermore, the iSWAP and iSWAP gates forms a iSWAP | i| i √2 | i| i | i| i universal gate set, a CNOT gate for example can be U (1, j) 11 1 j = 11 1 j . (17) obtained from two iSWAP or two √iSWAP gates and √iSWAP | i| i | i| i single-qubit rotations [4]. Recently, it has been shown that the iSWAP and √iSWAP gates can be very useful for Then, the N two qubit iSWAP and √iSWAP applications in quantum information process QIP and operations are simultaneously performed on N qubit pairs quantum computing [30]. The purely quantum iSWAP (1,2), (1,3), ..., (1,N + 1). Each gate operation (iSWAP and iSWAP gates are an appropriate elementary two-qubit and √iSWAP gate) includes the same control qubit and a gates, these quantum logic gates can be represented by different target qubit. Thus, the NiSWAP and N√iSWAP the the following input–output operators: gates are implemented simultaneously between the ﬁrst qubit and the N qubits. Finally, we give a brief discussion iSWAP 00 00 + i 01 10 + i 10 01 + 11 11 . about our proposal. We notice that δ satisﬁes the equation ≡ | ih | | ih | | ih | | ih (13)| 2 3 (g/δ1) = n + 4 of the NiSWAP gate and the equation 2 5 √ 10 00 (g/δ2) = n + 8 of the iSWAP gate, where n is an 1 i 2√3 0 0 integer. So, when n = 0, δ takes maximum δ1 = g and √iSWAP  √2 √2 . (14) 3 ≡ 0 i 1 0 2√10 √2 √2 δ2 = 5 g. Then, the operation time 00 01 3√3   top = 3τ = 2π (NiSWAP gate) and   1 × 2g 3√10 The goal of this subsection is to demonstrate how the top2 = 3τ = 2π (√iSWAP gate) are independent × 4g NiSWAP and N√iSWAP gates can be realized based on of the number of target qubits N. Then, the direct

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calculation shows that the operations times required to and N √SWAP gates are implemented simultaneously implement the NiSWAP and √iSWAP gates with between one transmon qubit and N transmon qubits. √ transmon qubits are top1 = 13ns and top2 = 11.86ns.In the Hence, it is clear that the NSWAP and N SWAP gates next, we use the above evolutions (Eq.15) for can be realized after the three-step process. Now, we give implementing NSWAP, NTQ-NOT and N√SWAP gates. a brief discussion about quantum NSWAP and N √SWAP gates, we notice that δ satisﬁes the equation 2 1 (g/δ3) = n + 4 of the NSWAP gate and the equation 3.2 Implementation of NSWAP and N√SWAP gates 2 1 √ (g/δ) = n + 8 of the N SWAP, where n is an integer, when n = 0, δ takes maximum δ = 2g and δ = 2√2g. The SWAP and √SWAP gates are an appropriate 3 4 Then, the operations times required to implement the elementary two-qubit gates, where they are useful in NSWAP and N√SWAP gates are t = 2π 3 and quantum computation and quantum information op3 2g 3 × processing [4], such as establishing the universality of top = 2π respectively. So, the direct calculation 4 × 2g two-qubit gates [32], programmable gate arrays [33], and shows that the operations times are top3 = 7.5ns and constructing quantum circuits [34]. On the other hand, the top4 = 15ns. SWAP gate is equivalent to three CNOT gates and the √SWAP gate constitutes a universal set of quantum gates together with single qubit rotations around an arbitrary 3.3 Implementation of NTQ-NOT gate axis [11]. these type of quantum logic gates can be represented by the operators: In this subsection, we focus on how to realize NTQ-NOT gate (N two-transman qubit-NOT gate) with one SWAP 00 00 + 01 10 + 10 01 + 11 11 (18) transmon qubit simultaneously controlling N target qubits ≡ | ih | | ih | | ih | | ih | by introducting the qubit-qubit interaction. The quantum NTQ-NOT operation on two-qubit computational basis is 10 00 deﬁned through the following input–output relations: 0 1+i 1 i 0 √ 2 −2 SWAP  1 i 1+i . (19) UNOT 0i 0 j = 1i 1 j ; UNOT 0i 1 j = 1i 0 j , ≡ 0 −2 2 0 | i| i | i| i | i| i | i| i 00 01 UNOT 1i 0 j = 0i 1 j ; UNOT 1i 1 j = 0i 0 j . (22)   | i| i | i| i | i| i | i| i   The main objective of this subsection is to realize The method presented so far for implementing the NSWAP and √SWAP gates using the evolutions (Eq.15). NTQ-NOT gate is based on the evolutions of the By selecting χτ = 2kπ (with k being an integer), subsubsection 1 (see Eq.15) where an overall phase factor η = 2m + 1 (with m being an integer) and iηπ 1 2 e− is omitted. By setting χτ = (2k + 2 )π (with k being γτ = 2n + 1 π (with n being an integer), an N two qubit 3 2 an integer), η = 2m + 2 (with m being an integer) and SWAP operations are simultaneously performed on N γτ 2n 1 π (with n being an integer), we can easily = ( + ) qubit pairs (1,2), (1,3), ..., (1,N + 1) as (from Eq.15) verify that

USWAP(1, j) 01 0 j = 01 0 j UNOT (1, j) 01 0 j = 11 1 j | i| i | i| i | i| i | i| i USWAP(1, j) 01 1 j = 11 0 j UNOT (1, j) 01 1 j = 11 0 j | i| i | i| i | i| i | i| i USWAP(1, j) 11 0 j = 01 1 j UNOT (1, j) 11 0 j = 01 1 j | i| i | i| i | i| i | i| i USWAP(1, j) 11 1 j = 11 1 j . (20) UNOT (1, j) 11 1 j = 01 0 j . (23) | i| i | i| i | i| i | i| i

By setting χτ = 2kπ (with k being an integer), η = It’s worth to point out that N two-transmon 1 1 2m+ 4 (with m being an integer) and γτ = 2n + 4 π (see qubit-NOT operations are simultaneously performed on N Eq.15), we obtain N-two-qubit √SWAP operations as qubit pairs (1,2),(1,3),...,(1,N + 1), Each SWAP operation includes the same control qubit and a different U (1, j) 01 0 j = 01 0 j √SWAP | i| i | i| i target qubit. Then, the N two-transmon qubit-NOT gates 1 + i 1 i are implemented simultaneously between one transmon U (1, j) 01 1 j = 01 1 j + − 11 0 j √SWAP | i| i 2 | i| i 2 | i| i qubit and N qubits. Hence, it is clear that the NTQ-NOT 1 + i 1 i gate can be realised in circuit QED. Now, we give a brief U (1, j) 11 0 j = 11 0 j + − 01 1 j √SWAP | i| i 2 | i| i 2 | i| i discussion about quantum NTQ-NOT gate. We notice that 2 1 U (1, j) 11 1 j = 11 1 j . (21) δ satisﬁes the equation (g/δ5) = n + , where n is an √SWAP | i| i | i| i 2 integer, when n = 0, δ takes maximum δ5 = √2g. Then, 3√2 By this way, one can see that NSWAP and N √SWAP the operation time is top = 2π . So, the direct 5 × 2g gates are simultaneously performed on the qubit pairs calculation shows that the operation time required to

(1,2), (1,3), ..., (1,N + 1), respectively. So, the NSWAP implement the NTQ-NOT gate will be top5 = 10.61ns.

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to be on the order of 20 100 µs for transmon qubits − (when EJ/EC = 50) [38,39]. In addition, the coupling strength is g = 2π 200MHz [22,16], which is experimentally available.× Furthermore, the operation time required to implement the NiSWAP, N√iSWAP, NSWAP, N√SWAP and NTQ-NOT gates with transmon qubits is independent of the number of target qubits N. It is clear from the above calculation that the total operation time

(top1 , top2 , top3 , top4 and top5 ) of the quantum logic gates (NiSWAP, N√iSWAP, NSWAP, N√SWAP and NTQ-NOT gates) which is much shorter than the Fig. 2: Numerical results for fidelity of the gates operations decoherence times T1 and dephasing time T2, which versus the force qubit-qubit coupling Γ . satisfies our experimental requirement. Also, we note that the decoherence time of field state inside a resonator depends on the initial field state. However, the gate operation is independent of the initial state of the 4 Possible experimental implementation resonator because of the operator Ugate does not incluse the photon operator a and a+ of the circuit QED. Let us now study the fidelity of the gate operation. 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) deviate in amplitude and 5 Conclusion phase from the ideal logical| gatei transformation for the different input states [35]

2 In Summary, We have used the system in which the F = Ψ(t) U(t) Ψ(0) , (24) |h | | i| transmon qubits are capacitively coupled to a TLR driven where Ψ(t) is the final state of the whole system after by a strong microwave field. As shown above, we have the gate| operations,i Ψ(0) is the initial state followed by presented and demonstrated a method for realizing an an ideal phase operation| andi U(t) is the overall evolution NiSWAP, N√iSWAP, NSWAP, N√SWAP and operator of the system are performed in a real situation. NTQ-NOT gates in circuit QED by introducing the We numerically simulate the relationship between the qubit-qubit interaction. The operation time is independent fidelity of the system and the force qubit-qubit coupling of the number of qubits involved in the scheme, and the Γ . Our numerical calculation shows that a high fidelity gates operations are insensitive to the initial state of the can be achieved when Γ = 100 Hz (see Fig. 2). resonator. The system requires no disagreement between For this method to work, we discuss some issues the qubits and the resonator. In addition, the operation which are relevant for future experimental time is only dependent of the detuning and the time can implementation of our proposal. Compared with the usual be controlled by adjusting the frequency between the 0 j | i charge qubit, the transmon qubit is immune to the 1/ f and 1 j . However, we have calculated an evolution | i charge noise and it has much longer dephasing time as a operator in the case of the qubit-qubit interaction. Finally, result of the transmon qubits chosen in the system. For we have applied the overall evolution operator to working the sake of definiteness, let us consider the experimental basis of the qubit 1 and the qubits j ( j = 2,...,N + 1) for parameters of the transmon qubits as Cg = 1aF, find the NiSWAP, N√iSWAP, NSWAP, N√SWAP and CJ0 = 300aF, Ec = 2π 2GHz and EJ 2π 100GHz NTQ-NOT gates. In the our system, when we choose [15]. The transmon qubits× with these≃ parameters× are suitable detuning, the quantum logic gates can be realized available at present. Thus, the coupling three qubits with a in a time much shorter than decoherence time, and it is transmission line resonator has been experimentally more immune to the 1/ f charge noise and have longer demonstrated [36]. Our scheme may have potential dephasing time as a result of the transmon qubits chosen applications in multipartite entanglement. The charge in the system. In addition, numerical simulation of the fluctuations are principal only in low-frequency region gate operations shows that the scheme could be achieved and can be reduced by the echo technique [37] and by with high fidelity under current state-of-the-art controlling the gate voltage to the degeneracy point, but technology. Therefore, the present scheme might be an effective technique for suppressing charge fluctuations realizable using the presently available techniques, and and keep the state coherent for a longer time is highly the experimental implementation of the present scheme desired. So our proposal is realizable with presently would be a important step toward more complex quantum available circuit QED techniques. logic gates, serving to show the power of the In a recent experiments, it was showen that the transmon-qubit system for quantum information decoherence times T1 and dephasing time T2 can be made processing.

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References [18] T. Said , A. Chouikh, K. Essammouni and M. Bennai. Implementing N-quantum phase gate via circuit QED [1] R. P. Feynman, Simulating physics with computers, with qubit qubit interaction. Mod. Phys. Lett. B, 30 International journal of theoretical physics, 21, 467, No. 0, 1650050, (2016). (1982). [19] T. Said , A. Chouikh, K. Essammouni and M. Bennai. [2] J. A. Smolin, G. Smith and A.Vargo. REALIZING AN N-TWO-QUBIT QUANTUM ”Oversimplifying quantum factoring”. Nature, LOGIC GATES IN A CAVITY QED WITH 499, 163, (2013). NEAREST QUBIT-QUBIT INTERACTION. [3] A. Farouk et al.. Quantum computing and Quantum Information and Computation, Vol. 16 cryptography: an overview, in book Series Studies in No. 5&6, 0465–0482, (2016). Big Data, Vol. 33, Springer, (2018). [20] C. W. Wu et al.. Fast quantum phase gate in a small- [4] M. A. Nielsen and I. L. Chuang. Quantum detuning circuit QED model. Phys Rev A, 82, 014303, Computation and Quantum Information. Cambridge (2010). [21] S. H. W. van der Ploeg et al.. Controllable coupling University Press, England, (2004). of superconducting ﬂux qubits. Phys. Rev. Lett. 98, [5] D. Paredes-Barato and C. S. Adams. All-optical 057004, (2007). quantum information processing using Rydberg gates. [22] A. Blais et al.. Quantum-information processing with Physical review letters, 112, 040501, (2014). circuit quantum electrodynamics. Phys Rev A, 75, 032329, (2007). [6] M. AbuGhanem , A. H. Homid and M. Abdel- [23] D. Zueco et al.. Qubit-oscillator dynamics in the Aty . Cavity control as a new quantum algorithms dispersive regime: Analytical theory beyond the implementation treatment. Front. Phys. 13(1), rotating-wave approximation. Physical Review A, 80, 130303, (2018). 033846, (2009). [7] X. Lin, R-C. Yang, and X. Chen. Implementation of a [24] L. Ye and G.C. Guo. Scheme for implementing quantum gate for two distant atoms trapped in separate quantum dense coding in cavity QED. Phys. Rev. A, cavities. Int. J. Quantum Inform, 13, 1550003,(2015). 71, 034304, (2005). [8] J. Ferrando-Soria et al.. A modular design of [25] Z. J. Deng, M. Feng, and K. L. Gao, ”Simple scheme molecular qubits to implement universal quantum for two-qubit Grover search in cavity QED”. Phys. gates. Nat. Commun, 7, 11377, (2016). Rev. A, 72, 034306, (2005). [9] J. C. Garcia-Escartin and P. Chamorro-Posada. [26] S.B. Zheng, G.C. Guo. Teleportation of atomic states A SWAP gate for qudits. Quantum Information within cavities in thermal states. Phys. Rev. A, 63, Processing, vol. 12, no. 12, pp. 3625–3631, (2013). 044302, (2001). [10] A. Chouikh, T. Said, K. Essammouni, M. Bennai. [27] K. Essammouni, A. Chouikh, T. Said and M. Bennai. Implementation of universal two- and three-qubit niSWAP and NTCP gates realized in a circuit QED quantum gates in a cavity QED. Opt Quant Electron, system. Int. J. Geom. Methods Mod. Phys., 14, 48:463, (2016). 1750100, (2017). [11] K. Eckert et al.. Quantum computing in optical [28] T Said, A Chouikh, M Bennai. Simultaneous microtraps based on the motional states of neutral Implementation of NiSWAP and NSWAP gates using atoms. Phys. Rev. A, 66, 042317, (2002). N + 1 qubits in a cavityor coupledto a circuit. Journal [12] K. Koshino, S. Ishizaka, and Y. Nakamura. of Experimental and Theoretical Physics, Volume Deterministic photon-photon √SWAP gate using 126, Issue 5, pp 573–578, (2018). a lambda system. Physical Review A, 82(1), 010301, [29] T. Tanamoto, Y. X. Liu, X. Hu, and F. Nori. Efﬁcient (2010). Quantum Circuits for One-Way Quantum Computing. Phys. Rev. Lett., 102(10), 100501, (2009). [13] T. Wu and L. Ye. Implementing Two-Qubit SWAP [30] S. Rips and M. J. Hartmann. Quantum information Gate with SQUID Qubits in a Microwave Cavity processing with nanomechanical qubits. Phys. Rev. via Adiabatic Passage Evolution. Int J Theor Phys, Lett., 110, 120503, (2013). 51:1076–1081, (2012). [31] M. Everitt, B. Garraway. Multiphoton resonances for [14] M. Hua, M. J. Tao, and F. G. Deng. Fast universal all-optical quantum logic with multiple cavities. Phys. quantum gates on microwave photons with all- Rev. A, 90, 012335 (2014). resonance operations in circuit QED. Sci. Rep., 5, [32] D. Deutsch, A. Barenco, A. Ekert. Universality in 9274, (2015). quantum computation. Proc. R. Soc. London A, 449, [15] C. P. Yang, Y. X. Liu, and F. Nori. Phase gate of one 669, (1995). qubit simultaneously controlling n qubits in a cavity. [33] M. A. Nielsen, I. L. Chuang. Programmable quantum Phys. Rev. A, 81, 062323, (2010). gate arrays. Phys. Rev. Lett., 79, 321, (1997). [16] G. L.Gao etal..1 N quantum controlled phase gate [34] F. Vatan and C. Williams. Optimal quantum circuits realized in a circuit→ QED system. Science China Phys, for general two-qubit gates. Phys. Rev. A, 69, 032315, 55(8), 1422-1426, (2012). (2004). [17] K. H. Song, Y. J. Zhao, Z. G. Shi, S. H. Xiang, X. [35] T. Said , A. Chouikh and M. Bennai, Two-Step W. Chen. Simultaneous implementation of n SWAP Scheme for Implementing N Two-Qubit Quantum gates using superconducting charge qubits coupled to Logic Gates Via Cavity QED. Appl. Math. Inf. Sci., a cavity. Optics Communications, 10, 1016, (2010). 12, No. 4, 699-704, (2018).

c 2019 NSP Natural Sciences Publishing Cor. 846 T. Said et al.: N two-transmon-qubit quantum logic gates realized...

[36] J. M. Fink et al.. Dressed collective qubit states and Mohamed Bennai is a the Tavis-Cummings model in circuit QED. Phys. Rev. Professor of quantum physics Lett., 103, 083601, (2009). at Hassan II University [37] Y. Nakamura et al.. Charge echo in a Cooper-pair box. Casablanca Morocco. Phys. Rev. Lett., 88, 047901, (2002). He is also a visiting Professor [38] H. Paik et al.. Observation of high coherence in of Theoretical Physics at Josephson junction qubits measured in a three- Mohamed V University Rabat dimensional circuit QED architecture. Phys. Rev. Morocco. He received the Lett., 107, 240501, (2011). CEA (1990), Doctorat de 3me [39] C. Rigetti et al.. Superconducting qubit in a waveguide Cycle (1993) and the PhD cavity with a coherence time approaching 0.1 (2002) from Hassan II University in Theoretical Physics. ms.Phys. Rev. B, 86, 100506(R), (2012). He is actually a Research Director of physics and quantum technology group at the Laboratory of Condensed Matter Physics of Ben M’sik Sciences Taouﬁk Said Faculty, Casablanca, Morocco. received his master degree in Systems and materials at Ibno Zohr University. He also received his PhD (2016) in the Laboratory of Physics of Condensed Matter at Ben M’sik Sciences Faculty, Casablanca, Morocco. His research interests include modeling of quantum information processing systems, implementation of quantum algorithm in a circuit and cavity QED, realization of quantum gates in a circuit and cavity QED, quantum cryptography, information processing theory and theoretical physics.

Abdelhaq Chouikh received his Graduate degree (D.E.S) in physics of materials at Casablanca An Chock Faculty of Sciences in 1996. He also received his PhD in the Laboratory of Physics of Condensed Matter at Faculty of Ben M’sik Sciences, Casablanca, Morocco. His research interests include modeling of quantum information processing systems, implementation of quantum algorithm in a system QED, realization of quantum gates in a system QED.

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