Abstract

We present a novel method to realize a multi-target-qubit controlled phase gate with one microwave photonic qubit simultaneously controlling n − 1 target microwave photonic qubits. This gate is implemented with n microwave cavities coupled to a superconducting flux qutrit. Each cavity hosts a microwave photonic qubit, whose two logic states are represented by the vacuum state and the single photon state of a single cavity mode, respectively. During the gate operation, the qutrit remains in the ground state and thus decoherence from the qutrit is greatly suppressed. This proposal requires only a single-step operation and thus the gate implementation is quite simple. The gate operation time is independent of the number of the qubits. In addition, this proposal does not need applying classical pulse or any measurement. Numerical simulations demonstrate that high-fidelity realization of a controlled phase gate with one microwave photonic qubit simultaneously controlling two target microwave photonic qubits is feasible with current circuit QED technology. The proposal is quite general and can be applied to implement the proposed gate in a wide range of physical systems, such as multiple microwave or optical cavities coupled to a natural or artificial Λ-type three-level atom.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Multiple qubit gates play important roles and are a crucial element in quantum information processing (QIP). A multiqubit gate can in principle be decomposed into a series of two-qubit and single-qubit gates, and thus can be constructed by using these basic gates. However, it is commonly recognized that building a multiqubit gate is difficult via the conventional gate-decomposition protocol. This is because the number of basic gates, required for constructing a multiqubit gate, increases drastically as the number of qubits increases. As a result, the gate operation time would be quite long and thus the gate fidelity would be significantly decreased by decoherence. Hence, it is worthwhile to seek efficient approaches to realize multiqubit quantum gates. Many efficient schemes have been presented for the direct realization of a multiqubit controlled-phase or controlled-NOT gate, with multiple-control qubits acting on one target qubit [1–14]. This type of multiqubit gate is of significance in QIP, such as quantum algorithms and error corrections.

In this work, we focus on another type of multiqubit gate, i.e., a multi-target-qubit controlled phase gate with one qubit simultaneously controlling multiple target qubits. This multi-target-qubit controlled phase gate is described by

|01|i2|i3|in|01|i2|i3|in,|11|i2|i3|in|11(1)i2(1)i3(1)in|i2|i3|in,
where i2, i3, ...in ∈ {0, 1} ; subscript 1 represents the control qubit while subscripts 2, 3, . . ., and n represent target qubits. From Eq. (1), it can be seen that when the control qubit 1 is in |1〉, a phase flip (from sign + to −) happens to the state |1〉 of each of target qubits 2, 3, . . ., and n ; however nothing happens to the states of each of target qubits 2, 3, . . ., and n when the control qubit 1 is in the state |0〉.

This multiqubit gate (1) is useful in QIP, such as entanglement preparation [15], error correction [16], quantum algorithms [17], and quantum cloning [18]. How to efficiently implement this multiqubit gate becomes necessary and important. Over the past years, based on cavity QED or circuit QED, many efficient methods have been proposed for the direct implementation of this multiqubit phase gate, by using natural atoms or artificial atoms (e.g., superconducting qubits, quantum dots, or nitrogen-vacancy center ensembles) [19–23].

Circuit QED is analogue of cavity QED, which consists of superconducting qubits and microwave resonators or cavities. It has developed fast recently and is considered as one of the most promising candidates for QIP [23–29]. Owing to the microfabrication technology scalability, individual qubit addressability, and ever-increasing qubit coherence time [30–38], superconducting qubits are of great importance in QIP. The strong and ultrastrong couplings between a superconducting qubit and a microwave cavity have been experimentally demonstrated [39,40]. For a review on the ultrastrong coupling, refer to [41]. On the other hand, a (loaded) quality factor Q ∼ 106 has been experimentally reported for a one-dimensional coplanar waveguide microwave resonator [42,43], and a (loaded) quality factor Q ∼ 3.5× 107 has also been experimentally reported for a three-dimensional microwave cavity [44]. A microwave resonator or cavity with the experimentally-reported high quality factor here can act as a good quantum data bus [45–47] and be used as a good quantum memory [48,49], because it contains microwave photons whose lifetimes are much longer than that of a superconducting qubit [50]. These good features make microwave resonators or cavities as a powerful platform for quantum computation and microwave photons as one of promising qubits for QIP. Recently, quantum state engineering and QIP with microwave fields or photons have become considerably interesting [51–72].

Motivated by the above, we will propose a method to realize the multi-target-qubit controlled phase gate (1) with microwave photonic qubits, by using n microwave cavities coupled to a superconducting flux qutrit (a Λ-type three-level artificial atom) (Fig. 1). Note that to simplify the presentation, we will use “MP qubit” to denote “microwave photonic qubit” and “MP qubits” to define “microwave photonic qubits”. This work is based on circuit QED. As shown below, this proposal has the following advantages: (i) During the gate operation, the qutrit stays in the ground state and thus decoherence from the qutrit is greatly suppressed; (ii) Because of only using one-step operation, the gate implementation is quite simple; (iii) Neither classical pulse nor measurement is required; (iv) The gate operation time is independent of the number of the qubits; and (v) This proposal is quite general and can be extended to a wide range of physical systems to realize the proposed gate, such as multiple microwave or optical cavities coupled to a natural or artificial Λ-type three-level atom. To the best of our knowledge, this work is the first to show the one-step implementation of a multi-target-qubit controlled phase gate with MP qubits, based on cavity- or circuit-QED and without any measurement.

 figure: Fig. 1

Fig. 1 (a) Diagram of n cavities (1, 2, ..., n) coupled to a superconducting flux qutrit A. A square represents a cavity, which can be a one-dimensional or three-dimensional cavity. The qutrit is capacitively or inductively coupled to each cavity. (b) Level configuration of the flux qutrit, for which the transition between the two lowest levels can be made weak by increasing the barrier between two potential wells. (c) Diagram of a flux qutrit, which consists of three Josephson junctions and a superconducting loop.

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This paper is organized as follows. In Sec. 2, we explicitly show how to realize a controlledphase gate with one MP qubit simultaneously controlling n − 1 target MP qubits. In Sec. 3, we discuss the experimental feasibility for implementing a three-qubit controlled phase gate, by considering a setup of three one-dimensional transmission line resonators coupled to a superconducting flux qutrit. We end up with a conclusion in Sec. 4.

2. Multi-Target-Qubit controlled phase gate

Consider n microwave cavities (1, 2, ..., n) coupled to a superconducting flux qutrit [Fig. 1(a)]. The three levels of the qutrit are denoted as |g〉, |e〉 and |f〉, as shown in Fig. 1(b). In general, there exists the transition between the two lowest levels |g〉 and |e〉, which, however, can be made to be weak by increasing the barrier between the two potential wells. Suppose that cavity 1 is dispersively coupled to the |g〉 ↔ |f〉 transition of the qutrit with coupling constant g1 and detuning δ1 but highly detuned (decoupled) from the |e〉 ↔ |f〉 transition of the qutrit. In addition, assume that cavity l (l = 2, 3, ..., n) is dispersively coupled to the |e〉 ↔ |f〉 transition of the qutrit with coupling constant gl and detuning δl but highly detuned (decoupled) from the |g〉 ↔ |f〉 transition of the qutrit (Fig. 2). Note that these conditions can be satisfied by prior adjustment of the qutrit’s level spacings or/and the cavity frequency. For a superconducting qutrit, the level spacings can be rapidly (within 1–3 ns) tuned by varying external control parameters (e.g., magnetic flux applied to the loop of a superconducting phase, transmon [73], Xmon [33], or flux qubit/qutrit [74]). In addition, the frequency of a microwave cavity or resonator can be rapidly adjusted with a few nanoseconds [75,76].

 figure: Fig. 2

Fig. 2 Cavity 1 is dispersively coupled to the |g〉 ↔ |f〉 transition of the qutrit with coupling strength g1 and detuning δ1, while cavity l (l = 2, 3, ..., n) is dispersively coupled to the |e〉 ↔ |f〉 transition of the qutrit with coupling strength gl and detuning δl. The purple vertical line represents the frequency ωc1 of cavity 1, while the blue, green, ..., and red vertical lines represent the frequency ωc2 of cavity 2, the frequency ωc3 of cavity 3,..., and the frequency ωcn of cavity n, respectively.

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Under the above assumptions and by considering the ideal cavities, the Hamiltonian of the whole system, in the interaction picture and after making the rotating-wave approximation (RWA), can be written as (in units of ħ = 1)

HI=g1(eiδ1ta^1+σfg+h.c.)+l=2ngl(eiδlta^l+σfe+h.c.),
where σfg=|gf|, σfe=|ef|, δ1 = ωfgωc1 > 0, and δl = ωfeωcl > 0. The detunings δ1 and δl have a relationship δl = δ1 + Δ1l, with Δ1l = ωc1ωclωeg > 0 (Fig. 2). Here, â1 (âl) is the photon annihilation operator of cavity 1 (l), ωcl is the frequency of cavity l (l = 2, 3, ..., n); while ωfg, ωfe, and ωeg are the |f〉 ↔ |g〉, |f〉 ↔ |e〉, and |e〉 ↔ |g〉 transition frequencies of the qutrit, respectively.

Under the large-detuning conditions δ1g1 and δlgl, the Hamiltonian (2) becomes [77]

He=λ1(a^1+a^1|gg|a^1a^1+|ff|)l=2nλl(a^l+a^l|ee|a^la^l+|ff|)l=2nλ1l(eiΔ1lta^1+a^lσeg+h.c.)+kl;k,l=2nλkl(eiΔklta^k+a^l+h.c.)(|ff||ee|),
where λ1=g12/δ1, λl=gl2/δl, λ1l = (g1gl/2) (1/δ1 + 1/δl), λkl = (gkgl/2) (1/δk + 1/δl), Δ1l = δlδ1 = ωc1ωclωeg, Δkl = δlδk = ωckωcl, and σeg=|ge|. In Eq. (3), the terms in the first two lines describe the photon number dependent stark shifts of the energy levels |g〉, |e〉 and |f〉; the terms in the third line describe the |e〉 ↔ |g〉 coupling caused due to the cooperation of cavities 1 and l; while the terms in the last line describe the coupling between cavities k and l. For Δ1l ≫ {λ1, λl, λ1l, λkl}, the effective Hamiltonian He changes to [77]
He=λ1(a^1+a^1|gg|a^1a^1+|ff|)l=2nλl(a^l+a^l|ee|a^la^l+|ff|)l=2nχ1l(a^1+a^1a^la^l+|gg|a^1a^1+a^l+a^l|ee|)+kl;k,l=2nλkl(eiΔklta^k+a^l+h.c.)(|ff||ee|),
where χ1l=λ1l2/Δ1l. From this equation, one can see that each term is associated with the level |g〉, |e〉, or |f〉. When the levels |e〉 and |f〉 are initially not occupied, they will remain unpopulated because neither |g〉 → |e〉 nor |g〉 → |f〉 is induced under the Hamiltonian (4). Hence, the Hamiltonian (4) reduces to
He=λ1a^1+a^1|gg|l=2nχ1la^1+a^1a^la^l+|gg|.
Note that [al,al+]=1, i.e., a^la^l+=1+a^l+a^l. Thus, the Hamiltonian (5) can be rewritten as
He=λ1n^1|gg|l=2nχ1ln^1|gg|l=2nχ1ln^1n^l|gg|,
where n^1=a^1+a^1 and n^l=a^l+a^l are the photon number operators for cavities 1 and l, respectively.

Assume that the qutrit is initially in the ground state |g〉. It will remain in this state because the Hamiltonian (6) cannot induce any transition for the qutrit. Therefore, the Hamiltonian He reduces to

H˜e=ηn^1χl=2nn^1n^l,
where η = λ1 + (n − 1) χ. Here, we have set χ1l = χ (l = 2, 3, ..., n). Note that the e is the effective Hamiltonian governing the dynamics of the n cavities (1, 2, ..., n).

The unitary operator U = eiH̃et can be written as

U=U1[l=2nU1l],
with
U1=exp(iηn^1t),
U1l=exp(iχn^1n^lt),
where l=2nU1l=U12U13U1n. Here, U1 is a unitary operator on cavity 1, while U1l is a unitary operator on cavities 1 and l.

Let us now consider n MP qubits 1, 2, ..., and n, which are hosted by cavities 1, 2, ..., and n, respectively. The two logical states of MP qubit l′ are represented by the vacuum state |0〉 and the single-photon state |1〉 of cavity l′ (l′ = 1, 2, ..., n). Based on Eq. (10), one can easily see that for χt = π, the unitary operation U1l leads to the following state transformation

U1l|010l=|010l,U1l|011l=|011l,U1l|110l=|110l,U1l|111l=|111l,
which implies that the operator U1l implements a universal controlled-phase gate on two qubits 1 and l. Eq. (11) can be expressed as
U1l|01il|g=|01il|gU1l|11il|g=(1)i1|11il|g,
where il ∈ {0, 1}.

Based on Eq. (12), one can easily obtain the following state transformation

l=2nU1l|01|i2|i3|in=|01|i2|i3|in,l=2nU1l|11|i2|i3|in=|11(1)i2(1)i3(1)in|i2|i3|in.

According to (9), one can see that for ηt = 2 (m is a positive integer), the unitary operator U1 leads to

U1|01=|01,U1|11=|11.
Combining Eq. (13) and Eq. (14), we have
U1[l=2nU1l]|01|i2|i3|in=|01|i2|i3|in,U1[l=2nU1l]|11|i2|i3|in=|11(1)i2(1)i3(1)in|i2|i3|in,
which shows that when the control qubit 1 is in the state |1〉, a phase flip (from sign + to −) happens to the state |1〉 of each of target qubits (2, 3, ..., n), while nothing happens to the states of each of target qubit (2, 3, ..., n) when the control qubit 1 is in the state |0〉. From Eq. (8), it can be seen that the jointed unitary operators U1[l=2nU1l] involved in Eq. (15) is equivalent to the unitary operator U. By comparing Eq. (15) with Eq. (1), one can see that a multi-target-qubit controlled phase gate, described by Eq. (1), is realized with n MP qubits (1, 2, ..., n), after the above operation, described by the unitary operator U.

We stress that the gate is realized through a single unitary operator U, which was obtained by starting with the original Hamiltonian (2). In this sense, the gate is implemented with only a single operation. In addition, it is noted that the qutrit remains in the ground state |g〉 during the gate operation. Hence, decoherence from the qutrit is greatly suppressed.

In above, we have set χ1l = χ, which turns out into

g12gl24Δ1l(1δ1+1δl)2=χ.
In addition, we have set χt = π and ηt = 2, from which we obtain
g12δ1=(2mn+1)χ.
Given g1, δ1, m, and n, the value of χ can be calculated based on Eq. (17). In addition, given g1, δ1, and χ, Eq. (16) can be satisfied by varying gl or δl or both. Note that the detuning δl can be adjusted by varying the frequency of cavity l, and the coupling strength gl can be adjusted by a prior design of the sample with appropriate capacitance or inductance between the qutrit and cavity l [13,78].

As shown above, the Hamiltonian (5) was obtained from the Hamiltonian (4) when the levels |e〉 and |f〉 are initially not occupied. This derivation has nothing to do with Δkl. In this sense, one can have Δkl ≠ 0 or Δkl = 0. Note that Δkl = δlδk = ωckωcl. Thus, the frequencies of cavities (2, 3, ..., n) can be chosen to be different or the same. However, it is suggested that for circuit QED, the frequencies of cavities should be different in order to suppress the unwanted inter-cavity crosstalk.

3. Possible experimental implementation

In this section, we briefly discuss the experimental feasibility of realizing a three-qubit controlled phase gate with one MP qubit simultaneously controlling two target MP qubits, by considering a setup of three microwave cavities (1, 2, 3) coupled to a superconducting flux qutrit (Fig. 3). Each cavity considered in Fig. 3 is a one-dimensional transmission line resonator (TLR).

 figure: Fig. 3

Fig. 3 Setup for three one-dimensional transmission line resonators capacitively coupled to a superconducting flux qutrit.

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In reality, there exist the inter-cavity crosstalk between cavities [79], the unwanted coupling of cavity 1 with the |e〉 ↔ |f〉 transition, and the unwanted coupling of cavities 2 and 3 with the |g〉 ↔ |f〉 transition of the qutrit (Fig. 4). After taking these factors into account, the Hamiltonian (2) is modified as

H˜I=HI+δH+ε,
with
δH=g˜1(eiδ˜1ta^1+σfe+h.c.)+l=23g˜l(eiδ˜lta^l+σfg+h.c.),
ε=kl;k,l=13gkl(eiΔ˜klta^k+a^l+h.c.).
Here, HI is the Hamiltonian (2) for n = 3. δH is the Hamiltonian, which describes the unwanted coupling between cavity 1 and the |e〉 ↔ |f〉 transition with coupling strength 1 and detuning δ̃1 = ωfeωc1, as well as the unwanted coupling between cavity l and the |g〉 ↔ |f〉 transition with coupling strength l and detuning δ̃l = ωfgωcl (l = 2, 3) (Fig. 4). In addition, ε represents the inter-cavity crosstalk, with the coupling strength gkl between cavities k and l, as well as the frequency difference Δ̃kl = ωckωcl of cavities k and l (kl; k, l ∈ {1, 2, 3}).

 figure: Fig. 4

Fig. 4 Illustration of the unwanted coupling between cavity 1 and the |e〉 ↔ |f〉 transition of the qutrit (with coupling strength 1 and detuning δ̃1) as well as the unwanted coupling between cavity l and the |g〉 ↔ |f〉 transition of the qutrit (with coupling strength l and detuning δ̃l) (l = 2, 3). Note that the coupling of each cavity with the |g〉 ↔ |e〉 transition of the qutrit is negligible because of the weak |g〉 ↔ |f〉 transition.

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When the dissipation and dephasing are included, the dynamics of the lossy system is determined by

dρdt=i[H˜I,ρ]+l=13κl[al]+γeg[σeg]+γfe[σfe]+γfg[σfg]+j=e,f{γφj(σjjρσjjσjjρ/2ρσjj/2)},
where I is the above full Hamiltonian; σeg=|ge|, σjj = |j〉〈j |(j = e, f); and [ξ]=ξρξξξρ/2ρξξ/2, with ξ = al, σeg, σfe, σfg. In addition, κl is the photon decay rate of cavity l (l = 1, 2, 3), γeg is the energy relaxation rate for the level |e〉 of the qutrit, γfe(γfg) is the energy relaxation rate of the level |f〉 of the qutrit for the decay path |f〉 → |e〉(|g〉), and γφj is the dephasing rate of the level |j〉(j = e, f) of the qutrit.

The fidelity of the operation is given by

=ψid|ρ|ψid,
where |ψid〉 is the output state of an ideal system without dissipation, dephasing and crosstalk; while ρ is the final practical density operator of the system when the operation is performed in a realistic situation. For simplicity, we consider the three qubits are initially in the following state
|ψin=122(|000+|001+|010+|011+|100+|101+|110+|111).
Thus, the ideal output state of the whole system is
|ψid=122(|000+|001+|010+|011+|100|101|110+|111)|g.

For a flux qutrit, the typical transition frequency between neighboring levels can be made as 1 to 20 GHz. As an example, we consider ωeg/2π = 5.0 GHz, ωfe/2π = 7.5 GHz, and ωfg/2π = 12.5 GHz. By choosing δ1/2π = 1.5 GHz, δ2/2π = 1.51 GHz, and δ3/2π = 1.53 GHz, we have Δ12/2π = 10 MHz, Δ13/2π = 30 MHz, ωc1/2π = 11 GHz, ωc2/2π = 5.99 GHz, and ωc3/2π = 5.97 GHz, for which we have Δ̃12/2π = 5.01 GHz, Δ̃23/2π = 0.02 GHz, and Δ̃13/2π = 5.03 GHz. With the transition frequencies of the qutrit and the frequencies of the cavities given here, we have δ̃1/2π = −3.5 GHz, δ̃2/2π = 6.51 GHz, and δ̃3/2π = 6.53 GHz. Other parameters used in the numerical simulation are: (i) γeg1=5Tμs, γfe1=2Tμs, γfg1=Tμs, (ii) γϕe1=γϕf1=Tμs, and (iii) g1/2π = 150 MHz. According to Eqs. (16) and (17), one can calculate the g2 and g3, which are g2/2π ∼ 86.89 MHz and g3/2π ∼ 151.49 MHz. For a flux qutrit, one has 1g1, g2g2, and g3g3. Note that the coupling constants chosen here are readily available because a coupling constant ∼ 2π × 636 MHz has been reported for a flux device coupled to a one-dimensional transmission line resonator [40]. We set gkl = 0.01gmax, where gmax = max{g1, g2, g3} ∼ 2π × 151.49 MHz, which can be achieved in experiments [56,69]. In addition, assume κ1 = κ2 = κ3 = κ for simplicity.

By solving the master equation (21), we numerically calculate the fidelity versus T and κ−1, as depicted in Fig. 5. From Fig. 5, one can see that when T ⩾ 5 μs and κ−1 ⩾ 10 μs, fidelity exceeds 0.9909, which implies that a high fidelity can be obtained for the gate being performed in a realistic situation.

 figure: Fig. 5

Fig. 5 Fidelity versus T and κ−1. The parameters used in the numerical simulation are referred to the text.

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To investigate the effect of the detuning errors on the fidelity, we consider a small deviation for δ1, δ2, and δ3. Thus, we modify δ1, δ2, and δ3 as δ1 + , δ2 + , and δ3 + . With this modification, we numerically calculate the fidelity for T = 5 μs and κ−1 = 10 μs and plot Fig. 6 showing the fidelity versus . From Fig. 6, one can see that the fidelity can reach 0.98 or greater for −75 MHz ≤ /2π ≤ 75 MHz.

 figure: Fig. 6

Fig. 6 Fidelity versus . Here, is the detuning error, which applies to each of detunings δ1, δ2, and δ3. The figure is plotted for T = 5 μs and κ−1 = 10 μs. Other parameters used in the numerical simulation are the same as those used in Fig. 5.

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The gate operational time is estimated as ∼ 66.7 ns for the parameters chosen above, which is much shorter than the decoherence times of the qutrit (5 μs – 75 μs) and the cavity decay times (5 μs – 20 μs) considered in Fig. 5. Here, we consider a rather conservative case for decoherence time of the flux qutrit because decoherence time 70 μs to 1 ms has been experimentally reported for a superconducting flux device [32,36,38]. For the cavity frequencies given above and κ−1 = 10 μs, one has Q1 ∼ 6.9 × 105 for cavity 1, Q2 ∼ 3.76 × 105 for cavity 2, and Q3 ∼ 3.75 × 105 for cavity 3, which are available because TLRs with a (loaded) quality factor Q ∼ 106 have been experimentally demonstrated [42,43]. The analysis here implies that high-fidelity realization of a quantum controlled phase gate with one MP qubit simultaneously controlling two target MP qubits is feasible with the present circuit QED technology.

In above, we have provided the specific implementation of the three qubits case. For the gate with more than three qubits, the extension is straightforward. From Fig. 7, one can see that each of the multiple cavities can in principle be coupled to a single superconducting flux qutrit via a capacitor. However, it should be pointed out that in the solid-state setup scaling up to many cavities coupled to one qutrit will introduce new challenges. For instance, the cavity crosstalk may become worse as the number of cavities increases, which will decrease the operation fidelity.

 figure: Fig. 7

Fig. 7 Schematic diagram for n cavities coupled by a superconducting flux qutrit. Each cavity here is a one-dimensional transmission line resonator, which is coupled to the qutrit via a capacitor.

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4. Conclusion

We have presented a one-step approach to realize an n-qubit controlled phase gate with one microwave photonic qubit simultaneously controlling n − 1 target microwave photonic qubits, based on circuit QED. As shown above, this proposal has the following advantages: (i) During the gate operation, the qutrit remains in the ground state; thus decoherence from the qutrit is greatly suppressed; (ii) Because only one-step operation is needed and neither classical pulse nor measurement is required, the gate implementation is simple; (iii) The gate operation time is independent of the number of the qubits; and (iv) This proposal is quite general and can be applied to realize the proposed gate with a wide range of physical systems, such as multiple microwave or optical cavities coupled to a single Λ-type three-level natural or artificial atom. Furthermore, our numerical simulations demonstrate that high-fidelity implementation of a three-qubit controlled phase gate with one microwave photonic qubit simultaneously controlling two target microwave photonic qubits is feasible with present circuit QED technology. We hope that this work will stimulate experimental activities in the near future.

Funding

NKRDP of China (2016YFA0301802); National Natural Science Foundation of China (11074062, 11374083,11774076).

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29. X. Gu, A. F. Kockum, A. Miranowicz, Y. X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718, 1–102 (2017). [CrossRef]  

30. J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. F. David, G. Cory, Y. Nakamura, J. S. Tsai, and W. D. Oliver, “Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,” Nat. Phys. 7, 565–570 (2011). [CrossRef]  

31. H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011). [CrossRef]  

32. C. Rigetti, S. Poletto, J. M. Gambetta, B. L. T. Plourde, J. M. Chow, A. D. Corcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, “Superconducting qubit in waveguide cavity with coherence time approaching 0.1 ms,” Phys. Rev. B 86, 100506(R) (2012). [CrossRef]  

33. R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent Josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013). [CrossRef]   [PubMed]  

34. Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014). [CrossRef]   [PubMed]  

35. M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett. 113, 123601 (2014). [CrossRef]   [PubMed]  

36. I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, “Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles,” Nature (London) 508, 369–372 (2014). [CrossRef]  

37. M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, “Coherence and decay of higher energy levels of a superconducting transmon qubit,” Phys. Rev. Lett. 114, 010501 (2015). [CrossRef]   [PubMed]  

38. F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, “The Flux Qubit Revisited to Enhance Coherence and Reproducibility,” Nat. Commun. 7, 12964 (2016) [CrossRef]  

39. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature (London) 431, 162–167 (2004). [CrossRef]  

40. T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nat. Phys. 6, 772–776 (2010). [CrossRef]  

41. A. F. Kockum, A. Miranowicz, S. D. Liberato, S. Savasta, and F. Nori, “Ultrastrong coupling between light and matter,” arXiv:1807.11636.

42. W. Chen, D. A. Bennett, V. Patel, and J. E. Lukens, “Substrate and process dependent losses in superconducting thin film resonators,” Supercond. Sci. Technol. 21, 075013 (2008). [CrossRef]  

43. P. J. Leek, M. Baur, J. M. Fink, R. Bianchetti, L. Steffen, S. Filipp, and A. Wallraff, “Cavity quantum electrodynamics with separate photon storage and qubit readout modes,” Phys. Rev. Lett. 104, 100504 (2010). [CrossRef]   [PubMed]  

44. M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “A quantum memory with near-millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016). [CrossRef]  

45. C. P. Yang, S. I. Chu, and S. Han, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003). [CrossRef]  

46. J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68, 064509 (2003). [CrossRef]  

47. A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits:An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004). [CrossRef]  

48. M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature (London) 459, 546–549 (2009). [CrossRef]  

49. H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, “Improving the Coherence Time of Superconducting Coplanar Resonators,” Appl. Phys. Lett. 95, 233508 (2009). [CrossRef]  

50. M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: an outlook,” Science 339, 1169–1174 (2013). [CrossRef]   [PubMed]  

51. Y. X. Liu, L. F. Wei, and F. Nori, “Generation of nonclassical photon states using a supercon ducting qubit in a microcavity,” Europhys. Lett. 67, 941–947 (2004). [CrossRef]  

52. M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of fock states in a superconducting quantum circuit,” Nature (London) 454, 310–314 (2008). [CrossRef]  

53. F. W. Strauch, K. Jacobs, and R. W. Simmonds, “Arbitrary control of entanglement between two superconducting resonators,” Phys. Rev. Lett. 105, 050501 (2010). [CrossRef]   [PubMed]  

54. Q. P. Su, C. P. Yang, and S. B. Zheng, “Fast and simple scheme for generating NOON states of photons in circuit QED,” Sci. Rep. 4, 3898 (2014). [CrossRef]   [PubMed]  

55. M. Hua, M. J. Tao, and F. G. Deng, “Universal quantum gates on microwave photons assisted by circuit quantum electrodynamics,” Phys. Rev. A 90, 18824 (2014). [CrossRef]  

56. S. J. Xiong, Z. Sun, J. M. Liu, T. Liu, and C. P. Yang, “Efficient scheme for generation of photonic NOON states in circuit QED,” Opt. Lett. 40, 2221–2224 (2015). [CrossRef]   [PubMed]  

57. M. Hua, M. J. Tao, and F. G. Deng, “Quantum state transfer and controlled-phase gate on one-dimensional superconducting resonators assisted by a quantum bus,” Sci. Rep. 6, 22037 (2016). [CrossRef]   [PubMed]  

58. A. N. Korotkov, “Flying microwave qubits with nearly perfect transfer efficiency,” Phys. Rev. B 84, 014510 (2011). [CrossRef]  

59. E. A. Sete, E. Mlinar, and A. N. Korotkov, “Robust quantum state transfer using tunable couplers,” Phys. Rev. B 91, 144509 (2015). [CrossRef]  

60. H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011). [CrossRef]   [PubMed]  

61. S. J. Srinivasan, N. M. Sundaresan, D. Sadri, Y. Liu, J. M. Gambetta, T. Yu, S. M. Girvin, and A. A. Houck, “Time-reversal symmetrization of spontaneous emission for quantum state transfer,” Phys. Rev. A 89, 033857 (2014). [CrossRef]  

62. J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency,” Phys. Rev. Lett. 112, 210501 (2014). [CrossRef]  

63. C. P. Yang, Q. P. Su, and S. Y. Han, “Generation of Greenberger-Horne-Zeilinger entangled states of photons in multiple cavities via a superconducting qutrit or an atom through resonant interaction,” Phys. Rev. A 86, 022329 (2012). [CrossRef]  

64. C. P. Yang, Q. P. Su, S. B. Zheng, and S. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013). [CrossRef]  

65. S. E. Nigg, “Deterministic hadamard gate for microwave cat-state qubits in circuit QED,” Phys. Rev. A 89, 022340 (2014). [CrossRef]  

66. R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, “Implementing a universal gate set on a logical qubit encoded in an oscillator,” arXiv:1608.02430 (2016).

67. C. P. Yang, Q. P. Su, S. B. Zheng, F. Nori, and S. Han, “Entangling two oscillators with arbitrary asymmetric initial states,” Phys. Rev. A 95, 052341 (2017). [CrossRef]  

68. C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrodinger Cat Living in Two Boxes,” Science 352, 1087–1091 (2016). [CrossRef]   [PubMed]  

69. Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96, 052317 (2017). [CrossRef]  

70. H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Deterministic CNOT gate and entanglement swapping for photonic qubits using a quantum-dot spin in a double-sided optical microcavity,” Phys. Lett. A 377, 2870 (2013). [CrossRef]  

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2018 (1)

B. Ye, Z. F. Zheng, and C. P. Yang, “Multiplex-controlled phase gate with qubits distributed in a multicavity system,” Phys. Rev. A 97, 062336 (2018).
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2017 (5)

Q. Wei, X. Wang, A. Miranowicz, Z. Zhong, and F. Nori, “Heralded quantum controlled-PHASE gates with dissipative dynamics in macroscopically distant resonators,” Phys. Rev. A 96, 012315 (2017).
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X. Gu, A. F. Kockum, A. Miranowicz, Y. X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718, 1–102 (2017).
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C. P. Yang, Q. P. Su, S. B. Zheng, F. Nori, and S. Han, “Entangling two oscillators with arbitrary asymmetric initial states,” Phys. Rev. A 95, 052341 (2017).
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Y. Zhang, X. Zhao, Z. F. Zheng, L. Yu, Q. P. Su, and C. P. Yang, “Universal controlled-phase gate with cat-state qubits in circuit QED,” Phys. Rev. A 96, 052317 (2017).
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Qi-Ping Su, H. H. Zhu, L. Yu, Y. Zhang, S. J. Xiong, J. M. Liu, and C. P. Yang, “Generating double NOON states of photons in circuit QED,” Phys. Rev. A 95, 022339 (2017).
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2016 (7)

C. P. Yang, Q. P. Su, S. B. Zheng, and F. Nori, “Crosstalk-insensitive method for simultaneously coupling multiple pairs of resonators,” Phys. Rev. A 93, 042307 (2016).
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C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrodinger Cat Living in Two Boxes,” Science 352, 1087–1091 (2016).
[Crossref] [PubMed]

C. H. Bai, D. Y. Wang, S. Hu, W. X. Cui, X. X. Jiang, and H. F. Wang, “Scheme for implementing multitarget qubit controlled-NOT gate of photons and controlled-phase gate of electron spins via quantum dot-microcavity coupled system,” Quantum. Inf. Process 15, 1485–1498 (2016).
[Crossref]

M. Hua, M. J. Tao, and F. G. Deng, “Quantum state transfer and controlled-phase gate on one-dimensional superconducting resonators assisted by a quantum bus,” Sci. Rep. 6, 22037 (2016).
[Crossref] [PubMed]

F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, “The Flux Qubit Revisited to Enhance Coherence and Reproducibility,” Nat. Commun. 7, 12964 (2016)
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M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “A quantum memory with near-millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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T. Liu, X. Z. Cao, Q. P. Su, S. J. Xiong, and C. P. Yang, “Multi-target-qubit unconventional geometric phase gate in a multicavity system,” Sci. Rep. 6, 21562 (2016).
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2015 (4)

M. Hua, M. J. Tao, and F. G. Deng, “Fast universal quantum gates on microwave photons with all-resonance operations in circuit QED,” Sci. Rep. 5, 9274 (2015).
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M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, “Coherence and decay of higher energy levels of a superconducting transmon qubit,” Phys. Rev. Lett. 114, 010501 (2015).
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S. J. Xiong, Z. Sun, J. M. Liu, T. Liu, and C. P. Yang, “Efficient scheme for generation of photonic NOON states in circuit QED,” Opt. Lett. 40, 2221–2224 (2015).
[Crossref] [PubMed]

E. A. Sete, E. Mlinar, and A. N. Korotkov, “Robust quantum state transfer using tunable couplers,” Phys. Rev. B 91, 144509 (2015).
[Crossref]

2014 (13)

S. J. Srinivasan, N. M. Sundaresan, D. Sadri, Y. Liu, J. M. Gambetta, T. Yu, S. M. Girvin, and A. A. Houck, “Time-reversal symmetrization of spontaneous emission for quantum state transfer,” Phys. Rev. A 89, 033857 (2014).
[Crossref]

J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency,” Phys. Rev. Lett. 112, 210501 (2014).
[Crossref]

J. R. Johansson, N. Lambert, I. Mahboob, H. Yamaguchi, and F. Nori, “Entangled-state generation and Bell inequality violations in nanomechanical resonators,” Phys. Rev. B 90, 174307 (2014).
[Crossref]

S. E. Nigg, “Deterministic hadamard gate for microwave cat-state qubits in circuit QED,” Phys. Rev. A 89, 022340 (2014).
[Crossref]

Q. P. Su, C. P. Yang, and S. B. Zheng, “Fast and simple scheme for generating NOON states of photons in circuit QED,” Sci. Rep. 4, 3898 (2014).
[Crossref] [PubMed]

M. Hua, M. J. Tao, and F. G. Deng, “Universal quantum gates on microwave photons assisted by circuit quantum electrodynamics,” Phys. Rev. A 90, 18824 (2014).
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H. W. Wei and F. G. Deng, “Scalable quantum computing based on stationary spin qubits in coupled quantum dots inside double-sided optical microcavities,” Sci. Rep. 4, 7551 (2014).
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M. Hua, M. J. Tao, and F. G. Deng, “Universal quantum gates on microwave photons assisted by circuit quantum electrodynamics,” Phys. Rev. A 90, 012328 (2014).
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H. F. Wang, A. D. Zhu, and S. Zhang, “One-step implementation of multiqubit phase gate with one control qubit and multiple target qubits in coupled cavities,” Opt. Lett. 39, 1489–1492 (2014).
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C. P. Yang, Q. P. Su, F. Y. Zhang, and S. B. Zheng, “Single-step implementation of a multiple-target-qubit controlled phase gate without need of classical pulses,” Opt. Lett. 39, 3312–3315 (2014).
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Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett. 113, 123601 (2014).
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I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, “Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles,” Nature (London) 508, 369–372 (2014).
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2013 (8)

R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent Josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
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S. B. Zheng, “Implementation of Toffoli gates with a single asymmetric Heisenberg XY interaction,” Phys. Rev. A 87, 042318 (2013).
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H. R. Wei and F. G. Deng, “Universal quantum gates for hybrid systems assisted by quantum dots inside double-sided optical microcavities,” Phys. Rev. A 87, 022305 (2013).
[Crossref]

M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: an outlook,” Science 339, 1169–1174 (2013).
[Crossref] [PubMed]

C. P. Yang, Q. P. Su, S. B. Zheng, and S. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
[Crossref]

H. F. Wang, A. D. Zhu, S. Zhang, and K. H. Yeon, “Deterministic CNOT gate and entanglement swapping for photonic qubits using a quantum-dot spin in a double-sided optical microcavity,” Phys. Lett. A 377, 2870 (2013).
[Crossref]

Z. L. Wang, Y. P. Zhong, L. J. He, H. Wang, J. M. Martinis, A. N. Cleland, and Q. W. Xie, “Quantum state characterization of a fast tunable superconducting resonator,” Appl. Phys. Lett. 102, 163503 (2013).
[Crossref]

2012 (2)

C. P. Yang, Q. P. Su, and S. Y. Han, “Generation of Greenberger-Horne-Zeilinger entangled states of photons in multiple cavities via a superconducting qutrit or an atom through resonant interaction,” Phys. Rev. A 86, 022329 (2012).
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C. Rigetti, S. Poletto, J. M. Gambetta, B. L. T. Plourde, J. M. Chow, A. D. Corcoles, J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, “Superconducting qubit in waveguide cavity with coherence time approaching 0.1 ms,” Phys. Rev. B 86, 100506(R) (2012).
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2011 (6)

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. F. David, G. Cory, Y. Nakamura, J. S. Tsai, and W. D. Oliver, “Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,” Nat. Phys. 7, 565–570 (2011).
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H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
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I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
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J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature (London) 474, 589–597 (2011).
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H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

A. N. Korotkov, “Flying microwave qubits with nearly perfect transfer efficiency,” Phys. Rev. B 84, 014510 (2011).
[Crossref]

2010 (6)

F. W. Strauch, K. Jacobs, and R. W. Simmonds, “Arbitrary control of entanglement between two superconducting resonators,” Phys. Rev. Lett. 105, 050501 (2010).
[Crossref] [PubMed]

P. J. Leek, M. Baur, J. M. Fink, R. Bianchetti, L. Steffen, S. Filipp, and A. Wallraff, “Cavity quantum electrodynamics with separate photon storage and qubit readout modes,” Phys. Rev. Lett. 104, 100504 (2010).
[Crossref] [PubMed]

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nat. Phys. 6, 772–776 (2010).
[Crossref]

C. P. Yang, Y. X. Liu, and F. Nori, “Phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 81, 062323 (2010).
[Crossref]

C. P. Yang, S. B. Zheng, and F. Nori, “Multiqubit tunable phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 82, 062326 (2010).
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W. L. Yang, Z. Q. Yin, Z. Y. Xu, M. Feng, and J. F. Du, “One step implementation of multi-qubit conditional phase gating with nitrogen-vacancy centers coupled to a high-Q silicamicro sphere cavity,” Appl. Phys. Lett. 96, 241113 (2010).
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2009 (4)

T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Realization of the Quantum Toffoli Gate with Trapped Ions,” Phys. Rev. Lett. 102, 040501 (2009).
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M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature (London) 459, 546–549 (2009).
[Crossref]

H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, “Improving the Coherence Time of Superconducting Coplanar Resonators,” Appl. Phys. Lett. 95, 233508 (2009).
[Crossref]

P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
[Crossref]

2008 (5)

M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Phys. 4, 523–526 (2008).
[Crossref]

M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon life time,” Appl. Phys. Lett. 92, 203501 (2008).
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2007 (1)

D. F. James and J. Jerke, “Effective Hamiltonian theory and its applications in quantum information,” Can. J. Phys. 85, 625–632 (2007).
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2006 (2)

X. Zou, Y. Dong, and G. C. Guo, “Implementing a conditional z gate by a combination of resonant interaction and quantum interference,” Phys. Rev. A 74, 032325 (2006).
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C. P. Yang and S. Han, “Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED,” Phys. Rev. A 73, 032317 (2006).
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2005 (3)

L. M. Duan, B. Wang, and H. J. Kimble, “Robust quantum gates on neutral atoms with cavity-assisted photonscattering,” Phys. Rev. A 72, 032333 (2005).
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C. P. Yang and S. Han, “n-qubit-controlled phase gate with superconducting quantum-interference devices coupled to a resonator,” Phys. Rev. A 72, 032311 (2005).
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J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58, 42–47 (2005).
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2004 (3)

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature (London) 431, 162–167 (2004).
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A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits:An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004).
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Y. X. Liu, L. F. Wei, and F. Nori, “Generation of nonclassical photon states using a supercon ducting qubit in a microcavity,” Europhys. Lett. 67, 941–947 (2004).
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2003 (2)

C. P. Yang, S. I. Chu, and S. Han, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003).
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J. Q. You and F. Nori, “Quantum information processing with superconducting qubits in a microwave field,” Phys. Rev. B 68, 064509 (2003).
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2001 (3)

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M. Šašura and V. Buzek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001).
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M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature (London) 459, 546–549 (2009).
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H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, “Improving the Coherence Time of Superconducting Coplanar Resonators,” Appl. Phys. Lett. 95, 233508 (2009).
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M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Phys. 4, 523–526 (2008).
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Z. L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems,” Rev. Mod. Phys. 85, 623–653 (2013).
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I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
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M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “A quantum memory with near-millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrodinger Cat Living in Two Boxes,” Science 352, 1087–1091 (2016).
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M. J. Peterer, S. J. Bader, X. Jin, F. Yan, A. Kamal, T. J. Gudmundsen, P. J. Leek, T. P. Orlando, W. D. Oliver, and S. Gustavsson, “Coherence and decay of higher energy levels of a superconducting transmon qubit,” Phys. Rev. Lett. 114, 010501 (2015).
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C. H. Bai, D. Y. Wang, S. Hu, W. X. Cui, X. X. Jiang, and H. F. Wang, “Scheme for implementing multitarget qubit controlled-NOT gate of photons and controlled-phase gate of electron spins via quantum dot-microcavity coupled system,” Quantum. Inf. Process 15, 1485–1498 (2016).
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J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency,” Phys. Rev. Lett. 112, 210501 (2014).
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Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent Josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Bauch, T.

M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon life time,” Appl. Phys. Lett. 92, 203501 (2008).
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Baur, M.

P. J. Leek, M. Baur, J. M. Fink, R. Bianchetti, L. Steffen, S. Filipp, and A. Wallraff, “Cavity quantum electrodynamics with separate photon storage and qubit readout modes,” Phys. Rev. Lett. 104, 100504 (2010).
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P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
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W. Chen, D. A. Bennett, V. Patel, and J. E. Lukens, “Substrate and process dependent losses in superconducting thin film resonators,” Supercond. Sci. Technol. 21, 075013 (2008).
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M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett. 113, 123601 (2014).
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H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
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H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, “Improving the Coherence Time of Superconducting Coplanar Resonators,” Appl. Phys. Lett. 95, 233508 (2009).
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M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum states in a superconducting resonator,” Nature (London) 459, 546–549 (2009).
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M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of fock states in a superconducting quantum circuit,” Nature (London) 454, 310–314 (2008).
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M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Phys. 4, 523–526 (2008).
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P. J. Leek, M. Baur, J. M. Fink, R. Bianchetti, L. Steffen, S. Filipp, and A. Wallraff, “Cavity quantum electrodynamics with separate photon storage and qubit readout modes,” Phys. Rev. Lett. 104, 100504 (2010).
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P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations in superconducting circuits,” Phys. Rev. B 79, 180511 (2009).
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Bienfait, A.

M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett. 113, 123601 (2014).
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F. Yan, S. Gustavsson, A. Kamal, J. Birenbaum, A. P. Sears, D. Hover, T. J. Gudmundsen, J. L. Yoder, T. P. Orlando, J. Clarke, A. J. Kerman, and W. D. Oliver, “The Flux Qubit Revisited to Enhance Coherence and Reproducibility,” Nat. Commun. 7, 12964 (2016)
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H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature (London) 431, 162–167 (2004).
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A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum electrodynamics for superconducting electrical circuits:An architecture for quantum computation,” Phys. Rev. A 69, 062320 (2004).
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Blatt, R.

T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Realization of the Quantum Toffoli Gate with Trapped Ions,” Phys. Rev. Lett. 102, 040501 (2009).
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Blumoff, J.

M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “A quantum memory with near-millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrodinger Cat Living in Two Boxes,” Science 352, 1087–1091 (2016).
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Braunstein, S. L.

S. L. Braunstein, V. Bužek, and M. Hillery, “Quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit,” Phys. Rev. A 63, 052313 (2001).
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Buluta, I.

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
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M. Šašura and V. Buzek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64, 012305 (2001).
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S. L. Braunstein, V. Bužek, and M. Hillery, “Quantum network for symmetric and asymmetric cloning in arbitrary dimension and continuous limit,” Phys. Rev. A 63, 052313 (2001).
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Bylander, J.

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. F. David, G. Cory, Y. Nakamura, J. S. Tsai, and W. D. Oliver, “Noise spectroscopy through dynamical decoupling with a superconducting flux qubit,” Nat. Phys. 7, 565–570 (2011).
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Campbell, B.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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Cao, X. Z.

T. Liu, X. Z. Cao, Q. P. Su, S. J. Xiong, and C. P. Yang, “Multi-target-qubit unconventional geometric phase gate in a multicavity system,” Sci. Rep. 6, 21562 (2016).
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Catelani, G.

M. Stern, G. Catelani, Y. Kubo, C. Grezes, A. Bienfait, D. Vion, D. Esteve, and P. Bertet, “Flux qubits with long coherence times for hybrid quantum circuits,” Phys. Rev. Lett. 113, 123601 (2014).
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I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, “Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles,” Nature (London) 508, 369–372 (2014).
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H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107, 240501 (2011).
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Chen, W.

W. Chen, D. A. Bennett, V. Patel, and J. E. Lukens, “Substrate and process dependent losses in superconducting thin film resonators,” Supercond. Sci. Technol. 21, 075013 (2008).
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Chen, Y.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency,” Phys. Rev. Lett. 112, 210501 (2014).
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R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent Josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Chen, Z.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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Chiaro, B.

Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency,” Phys. Rev. Lett. 112, 210501 (2014).
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R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, “Coherent Josephson qubit suitable for scalable quantum integrated circuits,” Phys. Rev. Lett. 111, 080502 (2013).
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Chou, K.

M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland, C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret, L. Jiang, and R. J. Schoelkopf, “A quantum memory with near-millisecond coherence in circuit QED,” Phys. Rev. B 94, 014506 (2016).
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C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A Schrodinger Cat Living in Two Boxes,” Science 352, 1087–1091 (2016).
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C. P. Yang, S. I. Chu, and S. Han, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67, 042311 (2003).
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T. Monz, K. Kim, W. Hänsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Realization of the Quantum Toffoli Gate with Trapped Ions,” Phys. Rev. Lett. 102, 040501 (2009).
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J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature 453, 1031–1042 (2008).
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Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J. Y. Mutus, P. J. J. O’Malley, C. M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Michael R. Geller, A. N. Cleland, and J. M. Martinis, “Qubit architecture with high coherence and fast tunable coupling,” Phys. Rev. Lett. 113, 220502 (2014).
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J. Wenner, Y. Yin, Y. Chen, R. Barends, B. Chiaro, E. Jeffrey, J. Kelly, A. Megrant, J. Y. Mutus, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, T. C. White, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Catching Time-Reversed Microwave Coherent State Photons with 99.4% Absorption Efficiency,” Phys. Rev. Lett. 112, 210501 (2014).
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Figures (7)

Fig. 1
Fig. 1 (a) Diagram of n cavities (1, 2, ..., n) coupled to a superconducting flux qutrit A. A square represents a cavity, which can be a one-dimensional or three-dimensional cavity. The qutrit is capacitively or inductively coupled to each cavity. (b) Level configuration of the flux qutrit, for which the transition between the two lowest levels can be made weak by increasing the barrier between two potential wells. (c) Diagram of a flux qutrit, which consists of three Josephson junctions and a superconducting loop.
Fig. 2
Fig. 2 Cavity 1 is dispersively coupled to the |g〉 ↔ |f〉 transition of the qutrit with coupling strength g1 and detuning δ1, while cavity l (l = 2, 3, ..., n) is dispersively coupled to the |e〉 ↔ |f〉 transition of the qutrit with coupling strength gl and detuning δl. The purple vertical line represents the frequency ωc1 of cavity 1, while the blue, green, ..., and red vertical lines represent the frequency ωc2 of cavity 2, the frequency ωc3 of cavity 3,..., and the frequency ωcn of cavity n, respectively.
Fig. 3
Fig. 3 Setup for three one-dimensional transmission line resonators capacitively coupled to a superconducting flux qutrit.
Fig. 4
Fig. 4 Illustration of the unwanted coupling between cavity 1 and the |e〉 ↔ |f〉 transition of the qutrit (with coupling strength 1 and detuning δ̃1) as well as the unwanted coupling between cavity l and the |g〉 ↔ |f〉 transition of the qutrit (with coupling strength l and detuning δ̃l) (l = 2, 3). Note that the coupling of each cavity with the |g〉 ↔ |e〉 transition of the qutrit is negligible because of the weak |g〉 ↔ |f〉 transition.
Fig. 5
Fig. 5 Fidelity versus T and κ−1. The parameters used in the numerical simulation are referred to the text.
Fig. 6
Fig. 6 Fidelity versus . Here, is the detuning error, which applies to each of detunings δ1, δ2, and δ3. The figure is plotted for T = 5 μs and κ−1 = 10 μs. Other parameters used in the numerical simulation are the same as those used in Fig. 5.
Fig. 7
Fig. 7 Schematic diagram for n cavities coupled by a superconducting flux qutrit. Each cavity here is a one-dimensional transmission line resonator, which is coupled to the qutrit via a capacitor.

Equations (24)

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| 0 1 | i 2 | i 3 | i n | 0 1 | i 2 | i 3 | i n , | 1 1 | i 2 | i 3 | i n | 1 1 ( 1 ) i 2 ( 1 ) i 3 ( 1 ) i n | i 2 | i 3 | i n ,
H I = g 1 ( e i δ 1 t a ^ 1 + σ f g + h . c . ) + l = 2 n g l ( e i δ l t a ^ l + σ f e + h . c . ) ,
H e = λ 1 ( a ^ 1 + a ^ 1 | g g | a ^ 1 a ^ 1 + | f f | ) l = 2 n λ l ( a ^ l + a ^ l | e e | a ^ l a ^ l + | f f | ) l = 2 n λ 1 l ( e i Δ 1 l t a ^ 1 + a ^ l σ e g + h . c . ) + k l ; k , l = 2 n λ k l ( e i Δ k l t a ^ k + a ^ l + h . c . ) ( | f f | | e e | ) ,
H e = λ 1 ( a ^ 1 + a ^ 1 | g g | a ^ 1 a ^ 1 + | f f | ) l = 2 n λ l ( a ^ l + a ^ l | e e | a ^ l a ^ l + | f f | ) l = 2 n χ 1 l ( a ^ 1 + a ^ 1 a ^ l a ^ l + | g g | a ^ 1 a ^ 1 + a ^ l + a ^ l | e e | ) + k l ; k , l = 2 n λ k l ( e i Δ k l t a ^ k + a ^ l + h . c . ) ( | f f | | e e | ) ,
H e = λ 1 a ^ 1 + a ^ 1 | g g | l = 2 n χ 1 l a ^ 1 + a ^ 1 a ^ l a ^ l + | g g | .
H e = λ 1 n ^ 1 | g g | l = 2 n χ 1 l n ^ 1 | g g | l = 2 n χ 1 l n ^ 1 n ^ l | g g | ,
H ˜ e = η n ^ 1 χ l = 2 n n ^ 1 n ^ l ,
U = U 1 [ l = 2 n U 1 l ] ,
U 1 = exp ( i η n ^ 1 t ) ,
U 1 l = exp ( i χ n ^ 1 n ^ l t ) ,
U 1 l | 0 1 0 l = | 0 1 0 l , U 1 l | 0 1 1 l = | 0 1 1 l , U 1 l | 1 1 0 l = | 1 1 0 l , U 1 l | 1 1 1 l = | 1 1 1 l ,
U 1 l | 0 1 i l | g = | 0 1 i l | g U 1 l | 1 1 i l | g = ( 1 ) i 1 | 1 1 i l | g ,
l = 2 n U 1 l | 0 1 | i 2 | i 3 | i n = | 0 1 | i 2 | i 3 | i n , l = 2 n U 1 l | 1 1 | i 2 | i 3 | i n = | 1 1 ( 1 ) i 2 ( 1 ) i 3 ( 1 ) i n | i 2 | i 3 | i n .
U 1 | 0 1 = | 0 1 , U 1 | 1 1 = | 1 1 .
U 1 [ l = 2 n U 1 l ] | 0 1 | i 2 | i 3 | i n = | 0 1 | i 2 | i 3 | i n , U 1 [ l = 2 n U 1 l ] | 1 1 | i 2 | i 3 | i n = | 1 1 ( 1 ) i 2 ( 1 ) i 3 ( 1 ) i n | i 2 | i 3 | i n ,
g 1 2 g l 2 4 Δ 1 l ( 1 δ 1 + 1 δ l ) 2 = χ .
g 1 2 δ 1 = ( 2 m n + 1 ) χ .
H ˜ I = H I + δ H + ε ,
δ H = g ˜ 1 ( e i δ ˜ 1 t a ^ 1 + σ f e + h . c . ) + l = 2 3 g ˜ l ( e i δ ˜ l t a ^ l + σ f g + h . c . ) ,
ε = k l ; k , l = 1 3 g k l ( e i Δ ˜ k l t a ^ k + a ^ l + h . c . ) .
d ρ d t = i [ H ˜ I , ρ ] + l = 1 3 κ l [ a l ] + γ e g [ σ e g ] + γ f e [ σ f e ] + γ f g [ σ f g ] + j = e , f { γ φ j ( σ j j ρ σ j j σ j j ρ / 2 ρ σ j j / 2 ) } ,
= ψ id | ρ | ψ id ,
| ψ in = 1 2 2 ( | 000 + | 001 + | 010 + | 011 + | 100 + | 101 + | 110 + | 111 ) .
| ψ id = 1 2 2 ( | 000 + | 001 + | 010 + | 011 + | 100 | 101 | 110 + | 111 ) | g .

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