Quantum Information Processing (2018) 17:240 https://doi.org/10.1007/s11128-018-2011-x
One-step implementation of a multi-target-qubit controlled phase gate in a multi-resonator circuit QED system Tong Liu1 · Bao-Qing Guo1 · Yang Zhang2 · Chang-Shui Yu1 · Wei-Ning Zhang1 Received: 25 January 2018 / Accepted: 30 July 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Circuit quantum electrodynamics system composed of many qubits and resonators may provide an excellent way to realize large-scale quantum information processing (QIP). Because of key role for large-scale QIP and quantum computation, multi-qubit gates have drawn intensive attention recently. Here, we present a one-step method to achieve a multi-target-qubit controlled phase gate in a multi-resonator system, which possesses a common control qubit and multiple different target qubits distributed in their respective resonators. Noteworthily, the implementation of this multi-qubit phase gate does not require classical pulses, and the gate operation time is independent of the number of qubits. Besides, the proposed scheme can in principle be adapted to a general type of qubits like natural atoms, quantum dots, and solid-state qubits (e.g., superconducting qubits and NV centers). Keywords Circuit quantum electrodynamics · Large-scale QIP · Multi-target-qubit gate · Multi-resonator system
1 Introduction Circuit quantum electrodynamics (QED) is an analogue of cavity QED and describes the interaction between superconducting qubits and microwave photons stored in resonators [1–6]. Strikingly, such system exhibits an unprecedented level of flexibility, scalability, and tunability, specifically concerning its improvements in coherence times [7–18]. The simplest circuit QED system consists of a single superconducting qubit
B
Chang-Shui Yu
[email protected]
1
School of Physics, Dalian University of Technology, Dalian 116024, China
2
Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China 0123456789().: V,-vol
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coupled to a microwave resonator. This system is a specially suited platform to achieve quantum computation and quantum information processing (QIP) and has allowed the realization of strong, ultrastrong, and beyond the ultrastrong coupling between a superconducting qubit and a microwave resonator [19–24]. Implementing large-scale QIP usually requires a complex system with superconducting qubits or/and multiple superconducting microwave resonators. Based on microwave resonators combined with superconducting qubits, it allows many approaches for realizing various QIP tasks [25–42]. In addition, numerous microwave quantum optics phenomena and operations involved a multi-qubit-resonator system have been experimentally demonstrated [43–49]. For example, Ref. [43] experimentally implemented the quantum von Neumann architecture with superconducting circuits, Ref. [44] generated photon NOON states of two resonators, Ref. [45] realized full deterministic quantum teleportation with feed-forward, Ref. [46] created a twomode cat state of microwave fields in two resonators, Ref. [47] extended the lifetime of a superconducting qubit with error correction, Ref. [48] demonstrated the coherent coupling between a resonator and an ensemble of superconducting qubits, and Ref. [49] demonstrated a resonator-induced phase gate in a multi-qubit circuit QED system, respectively. The experimental results [43–49] will provide crucial information and guidance for the development of large-scale QIP and quantum computation. In addition, the multi-qubit gate is a key component for quantum computation and QIP [50–60]. Any multi-qubit gate can be decomposed into a sequence of one-qubit and two-qubit gates [61,62]. However, a disadvantage of using such universal gate decomposition protocol is that as the number of qubits increases, the procedure and the computational overhead increase. To reduce the operation time and experimental complications, an alternative approach is the direct implementation of multi-qubit phase gates. During the past years, a number of proposals have been presented for directly realizing multi-qubit gates in various physical systems [50–60]. These multiqubit gates are usually divided into two significant types: one type is multiple control qubits acting on a single target qubit [50–56], the other is one control qubit and multiple target qubits [57–60]. These two types of multi-qubit gates have played a crucial role in QIP such as quantum algorithms [63–65], error correction [66–68], quantum Fourier transform [69], entanglement preparation [70], and quantum cloning [71]. We noticed that the previous proposals [50–58] have been proposed for implementation of a multi-qubit phase gate using qubits coupled to a single cavity/resonator. However, placing all qubits in a single cavity/resonator may introduce the unwanted qubit–qubit interaction, increase the vulnerability to cavity/resonator, and decrease the qubit-cavity/resonator coupling strength. In this sense, large-scale QIP may need to place qubits in multiple cavities/resonators and thus require performing various quantum logic operations on qubits distributed in different cavities/resonators. Recently, a method has been proposed for realizing a multi-target-qubit controlled phase gate in coupled cavity arrays [59], and an approach has been presented for achieving a multi-target-qubit unconventional geometric phase gate in a multi-cavity system [60]. In this work, we propose a method for directly realization of a multi-target-qubit controlled phase gate, by using a setup (Fig. 1) of n one-dimensional superconducting transmission line resonators each hosting m j superconducting flux qubit and coupled to a flux qubit A. The qubit placed in resonator j ( j = 1, 2, . . . , n) is labeled as qubit
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(a)
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(b)
Fig. 1 a Setup of a superconducting flux qubit A (coupler) and n superconducting transmission line resonators each embedding multiple flux qubits. Blue squares represent intra-resonator qubits embedding in each resonator, while a black square represents a coupler A which is capacitively coupled to each resonator. b Each resonator is far-off resonant with |e ↔ | f transition of qubit A with coupling strength g A and detuning δ A , while each resonator is far-off resonant with |e ↔ | f transition of its intra-resonator qubit with coupling strength g and detuning δ. Here, δ = ωc − ω f e , δ A = ωc − ω f e A , and = δ − δ A . Note that the red (purple) arrow represents the |e ↔ | f transition frequency ω f e A (ω f e ) of the coupler A (intra-resonator qubit), and the green arrow indicates the frequency ωc of the resonator (Color figure online)
ji ( ji = j1 , j2 , . . . , jm j ). This multi-target-qubit controlled phase gate is given by the following transformation: |0 A
mj n
|k ji → |0 A
j=1 i=1 n mj
|1 A
j=1 i=1
mj n
|k ji
j=1 i=1 n mj
|k ji → |1 A
eiθ1|k ji |k ji ,
(1)
j=1 i=1
where subscript A represents a control qubit, subscript ji represents the i-th target qubit embedded in resonator j, and |k ji (with k ∈ {0, 1}) is the computational basis state of the ji -th target qubit. Equation (1) means that the state |1 of the target qubit ji is flipped eiθ only if the control qubit A is in the |1 state and not flipped otherwise. For θ = π, gate (1) leads to a phase shift of π for the state |1 of the target qubit ji iff the control qubit A is in the state |1. Namely, this corresponds to a sign change of the state |1 for the target qubit ji when the control qubit A is in the state |1 (i.e., |1 A |1 ji → −|1 A |1 ji ). Such a multi-qubit controlled phase gate is widely used in error correction [68], quantum algorithm [69], quantum cloning [71], and the generating entangled states of multi-qubits [70]. This proposal has the following features: (1) The multitarget-qubit controlled phase gate can be achieved by employing a single-step operation and unnecessary to use classical pulse. (2) The gate operation time is independent of the number of intra-resonator qubits. (3) Our approach is generic and can be adapted to natural atoms, quantum dots and solid-state qubits. (4) After a deep search of literature, we find that there is no theoretical or experimental study of that a multi-target-qubit phase gate given by Eq. (1).
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Our present proposal differs from Refs. [50–60]. Firstly, Refs. [50–58] used all qubits coupled to a single cavity/resonator, but our proposal using multiple resonators placed many qubits. Secondly, our setup and method are different from Refs. [59,60]. The setup of Ref. [59] consists of an array of cavities that are coupled via exchange of photons, with one target qubit in each cavity. The setup of Ref. [60] consists of multiple resonators each hosting a target qubit and coupled to a control qubit. In our setup, the multiple target qubits are embedded in each resonator that is different from Refs. [59,60]. In addition, methods [59,60] need to apply a classical pulse to control qubit and each target qubit, but that is unnecessary for our proposal. This paper is structured as follows: In Sect. 2, we introduce a theoretical model and give a detailed derivation of the effective Hamiltonian. In Sect. 3, we show a method to implement a phase gate of one qubit simultaneously controlling multiple target qubits distributed in their respective resonators. In Sect. 4, we give a brief discussion on the experimental implementation of a five-qubit controlled phase gate with current circuit QED technology. A concluding summary is given in Sect. 5.
2 Model and Hamiltonian Consider a circuit setup consisting of n one-dimensional transmission line resonators each hosting m j flux qubits and coupled to a flux qubit (coupler) A (Fig. 1). Let the intra-resonator flux qubits are placed around the current antinode of the resonator. The qubit–resonator coupling constant is approximately homogeneous because the magnetic fields are the same in these positions. On the other hand, the dimension of the qubits is much smaller than the wavelength of electromagnetic mode. Thus, we assume that the distance between any two nearest qubits is so large that their direct interaction can be negligible [31,57,72]. Superconducting qubits based on Josephson junctions are mesoscopic element circuits like artificial atoms, with multiple discrete energy levels whose spacings can be rapidly adjusted by varying external control parameters [10,73–75]. Typically, a transmon [7] or an Xmon qubit [10] has weakly anharmonic multi-level structure, and the transition between non-adjacent levels is forbidden or very weak, while for a flux qubit (e.g., C-shunted flux qubit [16–18]), the transition frequency between two neighbor levels is typically 1 to 20 GHz. Compared with the transmon/Xmon, the flux qubit has a larger anharmonicity. Thus, the |g ↔ |e transition frequency of flux qubit can be highly detuned from the resonator frequency. On the other hand, the |g ↔ |e dipole transition rate of the flux qubit can be made much smaller than the |e ↔ | f dipole transition rate (e.g., |g ↔ |e transition is prohibited because the dipole selection rules do not hold [76] or it can be suppressed by increasing the potential barrier between the levels |g and |e [1]) [2]. Accordingly, our proposal uses the flux qubits so that the coupling effect of the resonator with the |g ↔ |e transition is negligible. We consider each qubit has three levels |g, |e and | f . Here, the |g and | f (|g and |e) of qubit A (each intra-resonator qubit) are two logical states, while the |e (| f ) for coupler (each intra-resonator qubit) is an auxiliary level. For simplicity, we consider the identical intra-resonator qubits. As shown in Fig. 1b, each resonator is off-resonantly coupled to the |e ↔ | f transition of coupler A with a coupling
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constant g A , while each resonator is off-resonantly coupled to the |e ↔ | f transition of its respective intra-resonator qubits with the coupling constant g. The system Hamiltonian in the interaction picture reads (in units of = 1) HI =
n
g(eiδt a †j S − j + h.c.) +
j=1
n
g A (eiδ A t a †j σ A− + h.c.),
(2)
j=1
m j − where a †j is the photon creation operator for the mode of resonator j, S − i=1 σ ji j = − (with σ ji = |e ji f |), σ A = |e A f | , δ = ωc − ω f e and δ A = ωc − ω f e A . Here, σ j−i is the lowering operator of intra-resonator qubit ji , σ A− is the lowering operator for qubit A, ωc is the frequency of each resonator, and ω f e A and ω f e are the |e ↔ | f transition frequencies of qubit A and intra-resonator qubits. Consider the large-detuning conditions δ g and δ A g A , Hamiltonian (2) changes to [77] mj mj n g2 † † He = |e ji e| − a j a j | f ji f | ajaj δ j=1
+
i=1
g 2A
n
δA
j=1
i=1
a †j a j |e A e| − a j a †j | f A f |
⎛ ⎞ mj mj n g2 ⎝ † † + − + −⎠ σ ji σ jk − a j a j σ ji σ jk ajaj + δ j=1
i=k=1
i=k=1
n n
† − + † + − it −it e λ a †j a j σ A− S + − a a σ S + λ a †j a j σ A+ S − + j j A j j j − a j a j σA S j e j=1
j=1
n g2 + A (a j ak† + a †j ak )(|e A e| − | f A f |) δA
(3)
j=k=1
1 A 1 where λ = g×g 2 ( δ + δ A ) and = δ − δ A . Note that the first (second) line of Eq. (3) describes Stark shifts of the level |e and | f (|e A and | f A ) of the intra-resonator qubits (coupler A) induced by the corresponding resonator modes, respectively. We have used [a j , a †j ] = 1, the third line of Eq. (3) describes the effective dipole–dipole interaction between the intra-resonator qubits embedded in the same resonators, the fourth line of Eq. (3) represents the dipole–dipole interaction between the intra-resonator qubits and the coupler A, and the last line of Eq. (3) describes the coupling between any two resonators induced by the coupler A. Because of using the large-detuning condition, there is no energy exchange between the qubits (i.e., intra-resonator qubits and qubit A) and the resonators. Suppose that each resonator is initially in the vacuum state, Hamiltonian (3) reduces to n mj ng 2 ng 2 | f ji f | − A | f A f | He = − δ δA
−
g2 δ
j=1 i=1 mj
n
j=1 i=k=1
σ j+i σ j−k
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−
n
−it it λ(σ A+ S − + σ A− S + ) j e j e
(4)
j=1
Under the large-detuning conditions || ≥ {λ,ng 2 /δ,ng 2A /δ A }, the effective Hamiltonian (4) becomes He = −
n mj ng 2 ng 2 | f ji f | − A | f A f | δ δA j=1 i=1
−
g2 δ
mj n j=1 i=k=1
mj
l λ2 |e A e| σ j+i σl−p
n
−
σ j+i σ j−k n
m
j=1 l=1 i=1 p=1 n n m j ml
+
λ2 | f A f | σ j−i σl+p ,
(5)
j=1 l=1 i=1 p=1
When the auxiliary level |e of coupler A and the auxiliary level | f of intra-resonator qubits are not occupied, the effective Hamiltonian (5) reduces to mj
ng 2 λ2 |e ji e|. He = − A | f A f | + | f A f | δA n
(6)
j=1 i=1
In the next section, we will show how this Hamiltonian (6) can be used to implement a multi-qubit controlled phase gate with qubit A simultaneously controlling m j target qubits distributed in n resonators.
3 Implementing multi-target-qubit controlled phase gate The time evolution operator U (t) corresponding to the Hamiltonian He [i.e., Eq. (6)] can be expressed as mj n −i He t = U A (t) ⊗ U A, ji (t) (7) U (t) = e j=1 i=1
with
ng 2A t U A (t) = exp i | f A f | , δA λ2 t | f A f | ⊗ |e ji e| . U A, ji (t) = exp −i
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From Eqs. (7)–(9), one can obtain U A |g A = |g A U A | f A = exp(ing 2A t/δ A )| f A U A, ji |g A |k ji = |g A |k ji U A, ji | f A |k ji = exp(−iλ2 te|k ji /)| f A |k ji ,
(10)
where |k ji ∈ {|g ji , |e ji }. It follows from Eqs. (7) and (10) that we have the following state transformations |g A
mj n
|k ji → |g A
j=1 i=1 n mj
| f A
mj n
|k ji
j=1 i=1 n mj
|k ji → | f A
j=1 i=1
exp i
j=1 i=1
ng 2A λ2 e|k ji − δA
t |k ji .
(11)
For ng 2A t/δ A = 2π , transformation (11) can be further written as |g A
mj n
|k ji → |g A
j=1 i=1 n mj
| f A
j=1 i=1
mj n
|k ji
j=1 i=1 n mj
|k ji → | f A
eiθe|k ji |k ji ,
(12)
j=1 i=1
where θ = −λ2 t/. As discussed in Introduction, the multi-qubit phase gate here is equivalent to the phase gate described by Eq. (1). Here, the logical states |g A and | f A are corresponding to the state |0 A and |1 A of Eq. (1) for the control qubit A, while the logical states |g ji and |e ji are corresponding to the state |0 ji and |1 ji of Eq. (1) for the target qubit ji . The multi-qubit gate (12) implies that the state |e of each target qubit is flipped eiθ if and only if the control qubit A is in the state | f and not flipped otherwise (Fig. 2). We can see that this multi-qubit gate depends on the phase accumulation by the intra-resonator qubits where the qubit A is in the state | f . Under the evolution operator for time t = 2π δ A /ng 2A , the state |e of each intra-resonator qubit accumulates a desired phase θ . When θ = π, gate (12) leads to a phase shift of π for the state |e of each target qubit when and only when the control qubit A is in the state | f (i.e., | f A |e ji → −| f A |e ji ). In order to maintain the prepared multi-qubit gate, the level spacings of the coupler and intra-resonator qubits need to be adjusted so that the qubits are decoupled from their respective resonators after the desired gate production. Alternatively, to have the resonators coupled or decoupled from the qubits, one also can tune the frequencies of resonators that because the rapid tuning of resonator frequencies has been demonstrated in experiments [78,79]. To see the above more clearly, consider implementing a five-qubit (i.e., control qubit A and target qubits 11 , 12 , 21 , and 22 ) controlled phase gate. According to Eq. (12), one can obtain a five-qubit controlled phase gate, which is expressed as
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(a)
(b)
Fig. 2 a Circuit diagram of a multi-qubit phase gate with qubit A (a black dot) simultaneously controlling m j target qubits (squares). b Decomposition of a multi-qubit phase gate into a sequence of two-qubit phase gates. Each two-qubit gate has a shared control qubit (qubit A) and a different target qubit (qubit 11 , 12 , . . . , 21 , 22 , . . . , or n m j ). Here, the element θ represents a phase shift, which happens to the state |e of target qubit ji ( ji = 11 , 12 , . . . , 21 , 22 , . . . n m j ) iff the control qubit A is in the state | f and nothing happens otherwise. When θ = π, this gate leads to a phase shift of π for the state |e of each target qubit iff the control qubit A is in the state | f (i.e., | f A |e ji → −| f A |e ji )
|g A |k11 |k12 |k21 |k22 → |g A |k11 |k12 |k21 |k22 , ⎞ ⎛ 2 e|k ji ⎠ | f A |k11 |k12 |k21 |k22 , | f A |k11 |k12 |k21 |k22 → exp ⎝iθ j,i=1
(13) where |k ji ∈ {|g ji , |e ji } and
2
e|k ji ∈ [0, 4]. Here, j = 1, 2 and i = 1, 2. The
j,i=1
five-qubit phase gate (13) shows that when the control qubit A is in the state | f , a phase shift θ happens to the state |e of the target qubits (i.e., 11 , 12 , 21 , and 22 ) and nothing happens otherwise.
4 Possible experimental implementation As an example of experimental implementation, we consider a circuit setup in Fig. 3 for realizing a five-qubit phase gate. The setup is consisted of two (i.e., n = 2) onedimensional superconducting transmission line resonators each embedding two flux qubits (i.e., 11 , 12 and 21 , 22 ) and coupled to a flux qubit A. This proposed gate is
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Fig. 3 Setup of two resonators each embedding two flux qubits and capacitively coupled by a flux qubit A
based on Eq. (6), mainly from the large-detuning condition and by removing energy levels | f ji and |e A that are not occupied. There is the possibility of occupation for the levels | f ji or |e A . The occupation probabilities P f and Pe are approximately given by [57] Pf
4g 2 , 4g 2 + δ 2
Pe
4g 2A 4g 2A + δ 2A
,
(14)
where P f (Pe ) represents the occupation probability of the level | f (|e) for each intraresonator qubit (qubit A). One can see that the effects of the unwanted occupation are negligible as long as δ g and δ A g A . Thus, we choose δ ≥ 10g and δ A ≥ 10g A so that the effects of occupation are suppressed for sufficiently large detunings. When the dissipation and the dephasing are taken into account, the dynamics of the lossy system is determined by the following master equation dρ = −i[He , ρ] + κ j L[a j ] dt 2
j=1
+
2 2
γeg L |g ji e| + γ f e L |e ji f | + γ f g L |g ji f |
j=1 i=1
+
2 2
γϕe |e ji e| ρ |e ji e| − |e ji e| ρ/2 − ρ |e ji e| /2
j=1 i=1
+
2 2
γϕ f | f ji f | ρ | f ji f | − | f ji f | ρ/2 − ρ | f ji f | /2
j=1 i=1
+ γeg L |g A e| + γ f e L |e A f | + γ f g L |g A f | + γϕe (|e A e| ρ |e A e| − |e A e| ρ/2 − ρ |e A e| /2) + γϕ f | f A f | ρ | f A f | − | f A f | ρ/2 − ρ | f A f | /2
(15)
where L [ ] = ρ + − + ρ/2−ρ + /2, κ j is the photon decay rate of resonator j, γeg is the energy relaxation rate of the level |e of intra-resonator qubit ji or qubit A, γ f e (γ f g ) is the energy relaxation rate of the level | f of intra-resonator qubit ji or qubit A for the decay path | f → |e (|g), and γϕe (γϕ f ) is the dephasing rate of the level |e (| f ) of qubit ji or qubit A ( j = 1, 2 and i = 1, 2). For simplicity, here we assume that the intra-resonator qubit and qubit A have the same energy relaxation rates and dephasing rates.
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(a)
(b)
(c)
(d)
Fig. 4 Fidelity F versus k = g/γ for a θ = π , b θ = π/2, c θ = π/3, and d θ = π/4. The blue squares were based on the master equation without considering the inter-resonator cross talk, while the red squares were based on the master equation by taking the cross talk into consideration. The parameters used in the numerical simulation are referred in the text (Color figure online)
√ The fidelity of the operation is given by F = ψid | ρ |ψid , where |ψid is the output state of an ideal system and ρ is the final density operator of the system when the operation is performed in a realistic situation. In the case of n = 2 and√ m j = 2, we state (1/ 2)(|g ji + assume that intra-resonator qubit ji is initially in a superposition √ |e ji ), qubit A is initially in a superposition state (1/ 2)(|g A +| f A ) and resonator j is initially in the vacuum state. According to Eq. (13), we can easily obtain the output state |ψid . By solving the master Eq. (15), the fidelity of the five-qubit gate operation can be calculated. We now numerically calculate the fidelity for the operation. We assume that g/2π = 50 MHz, g A /2π = 80 MHz and δ/2π = 0.5 GHz such that δ A /2π ≈ 1.0 GHz. The values of g and g A here are available in experiments because the strong coupling of a superconducting flux qubit with a microwave resonator (e.g., coupling constant ∼ 636 MHz [22]) has been experimentally demonstrated. In addition, we set γϕe = γϕ f = γ = g/k, γeg = γ f e = γ f g = 0.5γ , κ1 = 10−5 γ , and κ2 = 1.2 × 10−5 γ . Figure 4a–d shows the fidelity F versus k = g/γ (k ∈ [10−6 , 10−4 ]) for four kinds of controlled phase gates: θ = π , θ = π/2, θ = π/3 and θ = π/4. From the blue squares depicted in Fig. 4a–d, one can see that for k ≥ 1 × 10−5 , the gate operational fidelity can be greater than 97.34%, 98.78%, 99.21% and 99.42%. For k = 1 × 10−4 , one can obtain that a high fidelity 99.73%, 99.88%, 99.92% and 99.94% for θ = π , θ = π/2, θ = π/3 and θ = π/4, respectively. Thus, one has
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(a)
(b)
(c)
(d)
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Fig. 5 Fidelity F versus ε = t/T for a θ = π , b θ = π/2, c θ = π/3, and d θ = π/4
−1 = γ −1 = 2 × 10−12 s, γ −1 = γ −1 = γ −1 = 1 × 10−12 s, κ −1 = 1 × 10−11 s, γϕe eg 1 ϕf fe fg
and κ2−1 = 1.2 × 10−11 s. To investigate the effect of the unwanted inter-resonator cross talk on the fidelity, we also numerically calculate the operation fidelity for four kinds of controlled phase gates with include the cross talk of two resonators, as the red squares displayed in Fig. 4a–d. The effect of the inter-resonator cross talk can be taken into account by adding a term H12 = g12 (a1† a2 + a2† a1 ) in He , where g12 is the inter-resonator crosstalk coupling strength. We set g12 = 0.1g, which can be readily achieved in experiments [60]. From the red squares depicted in Fig. 4, one can see that the effect of the inter-resonator cross talk is negligible. This is because the vacuum state of the resonator system does not evolve during the entire process with the crosstalk Hamiltonian H12 ; thus, the unwanted inter-resonator cross talk can be efficiently suppressed. Figure 5 displays the fidelity F versus ε = t/T with ε ∈ [0.95, 1.05] for four kinds of controlled phase gates: θ = π , θ = π/2, θ = π/3 and θ = π/4. Here, ε is the operation time error, T is the optimal operation time chosen by Fig. 4 and the actual operation time t is varied depending on the ε. In Fig. 5, we choose k = 1 × 10−4 and other parameters used in the numerical simulation are the same as those used in Fig. 4. Figure 5a displays that for θ = π , the fidelity has a small decrease for ε = 0.95, 1.05. When ε ∈ [0.98, 1.02], the fidelity is almost unaffected by the operation time error. Figure 5b–d shows that for θ = π/2, θ = π/3 and θ = π/4, the effect of the operation time error on the fidelity is negligible for ε ∈ [0.95, 1.05]. From Fig. 5, one can see that the controlled phase gate with high fidelity can be achieved for small errors in operation time. Recently, the energy relaxation time T1 and dephasing time T2 can be made to be on the order of 55 − 85 μs for flux qubits [16–18]. For superconducting flux qubits,
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the transition frequency between two neighbor levels is 1–20 GHz. Thus, we choose ω f e A /2π = 6.0 GHz and ω f e /2π = 6.5 GHz. Accordingly, we have ωc /2π = 7.0 = Q j /ωc , one can obtain the GHz. Because the decay time of resonator is κ −1 j 2 quality factors of two resonators Q 1 ∼ 7 × 10 and Q 2 ∼ 8.4 × 102 . Numerical simulations show that the gate operation can be high-fidelity performed assisted by the low-Q resonators, because the resonator photons are virtually excited for the entire operation time and decoherence caused by the resonator is greatly suppressed. One also can employ a longer resonator decay time in the numerical simulation because onedimensional transmission line resonators with a loaded quality factor ∼ 106 [80,81] or with internal quality factors above 107 have been previously reported [82]. The above discussion shows that the implementation of a five-qubit controlled phase gate is possible with current circuit QED technology.
5 Conclusions We have presented a scheme to realize a controlled phase gate of one qubit simultaneously controlling multiple target qubits in a multi-resonator circuit QED system. Our proposal only needs one operational step and without the requirement of classical pulses. Moreover, the gate operation time is independent of the number of intraresonator qubits, and the target qubits are distributed in their respective resonators. Our numerical simulation shows that high-fidelity implementation of a five-targetqubit controlled phase gate is feasible with current circuit QED technology. Besides, this approach is generic and can be adapted to other types of qubits such as natural atoms (e.g., Rydberg atoms), quantum dots, and NV centers. In addition, our proposal provides a way for realizing a multi-target-qubit controlled phase gate in a circuit QED system, and such phase gate may be helpful for large-scale QIP and quantum computation. Acknowledgements This work was supported by the National Natural Science Foundation of China, under Grant Nos. 11775040 and 11375036, the Xinghai Scholar Cultivation Plan, and the Fundamental Research Fund for the Central Universities under Grant No. DUT18LK45.
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