One-step implementation of a hybrid Fredkin gate with quantum memories and single superconducting qubit in circuit QED and its applications

In a recent remarkable experiment [R. B. Patel et al., Science advances 2, e1501531 (2016)], a 3-qubit quantum Fredkin (i.e., controlled-SWAP) gate was demonstrated by using linear optics. Here we propose a simple experimental scheme by utilizing the dispersive interaction in superconducting quantum circuit to implement a hybrid Fredkin gate with a superconducting flux qubit as the control qubit and two separated quantum memories as the target qudits. The quantum memories considered here are prepared by the superconducting coplanar waveguide resonators or nitrogen-vacancy center ensembles. In particular, it is shown that this Fredkin gate can be realized using a single-step operation and more importantly, each target qudit can be in an arbitrary state with arbitrary degrees of freedom. Furthermore, we show that this experimental scheme has many potential applications in quantum computation and quantum information processing such as generating arbitrary entangled states (discrete-variable states or continuous-variable states) of the two memories, measuring the fidelity and the entanglement between the two memories. With state-of-the-art circuit QED technology, the numerical simulation is performed to demonstrate that two-memory NOON states, entangled coherent states, and entangled cat states can be efficiently synthesized.


Introduction
The Fredkin gate is a three-qubit controlled-SWAP gate. Conditioned on the state of the control qubit, the gate can enable the two target qubits swap their quantum states [1]. It has played an important role in quantum computation and quantum information processing (QCQIP) such as error correction [2,3], quantum cloning [4], quantum fingerprinting [5,6], and quantum digital signatures [7]. Any multiqubit gate can in principle be decomposed into a sequence of singlequbit and two-qubit basic quantum gates, and there have been many proposals to implement the Fredkin gate which required at least six [8], five [9] and four [10] two-qubit gates. In particular, the Fredkin gate has been experimentally implemented with linear optics [11] on qubit systems. However, the multiqubit and even the high-dimensional quantum gates could be indispensable for the large scale quantum network and quantum processors. It is usually a significant challenge to construct such multiqubit or high-dimensional quantum gates with the increase of the state space. The reason is mainly attributed to not only the experimental complications added but also the possibility of the errors caused by decoherence. Thus, it would be desirable to seek for efficient schemes based on a reliable physical platform to directly construct the Fredkin gate so as to reduce the operation time and experimental complications.
In analogy to cavity quantum electrodynamics (QED), the circuit QED system studying the light-matter interaction, is a specially suited platform to realize QCQIP due to its flexibility, scalability, and tunability [12][13][14][15]. The strong [16,17], ultrastrong [18,19], and beyond the ultrastrong coupling regimes [20] with a superconducitng qubit coupled to a microwave resonator have been experimentally achieved in a series of experiments, and the strong coupling of an nitrogen-vacancy center ensemble (NV ensemble) to a superconducting resonator [21,22] or flux qubit [23] has been experimentally realized in circuit QED. In addition, quantum memory is also indispensable in QCQIP such as quantum repeater and quantum computing [24]. A distinct feature of quantum memory is that it has the relatively large state space. In recent years, the solid-state devices (such as NV ensembles and superconducting resonators) have been considered as the good memory elements in QCQIP [24,25]. Up to now, the superconducting resonator lifetimes between 1 and 10 ms have been reported [26][27][28] and a lifetime of 1 s for an NV ensemble has been experimentally achieved [29]. These experimental achievements directly lead to the further breakthrough in QCQIP. In particular, as the important physical resource, quantum entanglement in the context of circuit QED has attracted a great many of interest such as the preparation of a variety of entangled states (e.g., Bell states, NOON states, and entangled coherent states) of two superconducting resonators [30][31][32][33][34][35][36][37][38][39][40] or NV ensembles [41,42]. Experimentally, the photon NOON states of two superconducting resonators have been produced [43], and a two-mode entangled coherent state of microwave fields in two superconducting resonators has been prepared [44]. Thus it is natural to consider how we can construct the Fredkin gate in high-dimensional systems by utilizing the well developed experimental technology in the superconducting quantum system.
Here, we propose a method for the direct realization of a general hybrid tripartite Fredkin gate by using superconducting resonators as two quantum memories coupled to a superconducting flux qubit. This gate can be expressed as where γ and η are the normalized complex numbers, |ψ and |ϕ are arbitrary pure states of target qudits encoded in two quantum memories 1 and 2, and |g and |e are the states of the control qubit. Eq. (1) shows that if and only if the control qubit is in the state |g , the two target qudits will swap their states, otherwise they remain in their initial states. Considering the experimental progress made in the NV center, we also propose an experimental scheme to realize the same aim as above by utilizing the NV ensembles as the quantum memories. The two proposals have the following distinct advantages: (i) The Fredkin gate can be realized by employing a single unitary operation without need of any microwave pulse; (ii) Our method and experimental setup are simple because only a single qutrit and two target quantum memories are used; (iii) The experimental scheme is based on the superconducting resonator or the NV ensemble which has a long coherence time; (iv) Each controlled target qudit of this gate can be in an arbitrary state (discrete-variable or continuous-variable state) which can further lead to the wide applications such as (a) preparing an arbitrary entangled state of two superconducting resonators or NV ensembles, (b) directly measuring the fidelity between the two quantum memories as well as the entanglement between them without any information on the initial states required. This paper is organized as follows. In Sec. 2, we explicitly show how to implement the hybrid Fredkin gate of a single superconducting flux qubit simultaneously controlling two target qudits  (2) is far-off resonant with |g ↔ |a transition of coupler with coupling strength g 1 (g 2 ) and detuning δ 1 (δ 2 ). Here, the detunings δ 1 = ω ag − ω a 1 and δ 2 = ω ag − ω a 2 , the ω ag is the |g ↔ |a transition frequency of coupler and the ω a 1 (ω a 2 ) is the frequency of resonator 1 (2). encoded in two superconducting coplanar waveguide resonators or NV ensembles. In Sec. 3, we discuss the applications and the possible experimental implementation of our proposal and numerically calculate the operational fidelity for creating NOON states, entangled coherent states, and entangled cat states of two resonators or NV ensembles. A concluding summary is given in Sec. 4.

Hybrid Fredkin gate between a single superconducting qubit and two quantum memories
Superconducting resonators as quantum memories-We first consider such a system that consists of two superconducting microwave coplanar waveguide resonators coupled to a three-level superconducting flux qutrit (coupler) [ Fig. 1(a)]. As shown in Fig. 1(b), the resonators 1 and 2 are off-resonantly coupled to the |g ↔ |a transition of coupler with the coupling constants g 1 and g 2 , respectively. In the interaction picture, after making the rotating-wave approximation, the Hamiltonian of the whole system reads (in units of = 1) where a 1 (a 2 ) is the photon annihilation operator for the resonator 1 (2), σ + ag = |a g|, δ 1 = ω ag − ω a 1 and δ 2 = ω ag − ω a 2 . Here, ω ag is the frequency of coupler related to the transition |g ↔ |a and ω a 1 (ω a 2 ) is the frequency of resonator 1 (2).
Considering the large-detuning conditions δ 1 ≫ g 1 and δ 2 ≫ g 2 , the Hamiltonian (2) can be written as [45] where λ = g 1 g 2 2 ( 1 δ 1 + 1 δ 2 ) and δ ′ = δ 2 − δ 1 . The first line of Eq. (3) describes Stark shifts of the level |a (|g ) of the coupler; which the second line describes the interaction between the resonators 1 and 2 when the coupler is in the state |a or |g .
For simplicity, we set and assume that the level |a of coupler is not occupied. Thus, the effective Hamiltonian (3) is reduced to with where ω = g 2 1 /δ 1 = g 2 2 /δ 2 . The Hamiltonian H i describes the interaction between the resonators when the coupler is in the state |g .
Now we can show that the effective Hamiltonian (5) can be used to construct a hybrid Fredkin gate with the flux qubit simultaneously controlling the two target qudits encoded by two quantum memories. We suppose the quantum memory is prepared by a superconducting coplanar waveguide resonator or an NV ensemble. Let the initial state of the quantum memory 1 (2) be an arbitrary pure state |0 2 ) and the flux qutrit be an arbitrary superposition state |φ q = γ|g + η|e . Note that only the two levels |g and |e of the flux qutrit are encoded here and will be used as the control qubit in the Fredkin gate. Here, c n and d m (γ and η) represent arbitrary complex amplitudes satisfying the normalization condition, |0 1 (|0 2 ) denotes the vacuum state of the quantum memory 1 (2). From Eq. (6), one can see that [H 0 , H i ] = 0. Thus, the time-evolution operator for the Hamiltonian (5) can be defined as U = e −iH 0 t · e −iH i t . With the Hamiltonian (5), the state |φ q |ψ 1 |ϕ 2 of the qubit-memory system evolves into where , and g|e = 0 has been used. It should be noted that the Hamiltonian H ′ 0 and H ′ i are different from H 0 and H i because the Hamiltonian H 0 and H i contain a qubit operator |g g|.
We would like to mention that various schemes for achieving the three-qubit Fredkin gates have been previously suggested in [46][47][48][49][50][51][52], in which the two target qubits are encoded in two natural/artificial atoms [46][47][48][49][50][51] or photons [52]. Unlike the previous proposals, the target qubits of our gate are encoded in two superconducting resonators or NV ensembles which have the long coherence times. In contrast to [51] where the target qubits are encoded by the discretevariable state, in our proposal the target qubits also can be encoded by the continuous-variable state. In addition, the proposals [46][47][48][49][50][51][52] require several operational steps and the microwave pulses, however, our proposal is much improved because our gate can be realized only using a single-step operation and no microwave pulses are needed.
NV ensembles as quantum memories-Now we would like to consider another system composing of a superconducting flux qubit coupled to two NV ensembles to realize the above Fredkin gate. The energy-level of an NV center consists of a ground state 3 A, an excited state 3 E and a metastable state 1 A. Both 3 A and 3 E are spin triplet states while the metastable 1 A is a spin singlet state [53,54]. The NV center has an electronic spin triplet ground state with zero-field splitting D gs /(2π) ≈ 2.878 GHz between the |m s = 0 and |m s = ±1 levels. By applying an external magnetic field along the crystalline axis of the NV center [55,56], an additional Zeeman splitting between |m s = ±1 sublevels occurs [ Fig. 2(a)].
We first consider a system consisting of a superconducting flux qubit coupled to an NV ensemble. The NV center is usually regarded as a spin while an NV ensemble is generally considered as a spin ensemble. We choose the |g ↔ |a transition of qubit is coupled to the transition between the ground level |m s = 0 and the excited level |m s = +1 of the spins in the ensemble, but decoupled from the transition between the two levels |m s = 0 and |m s = −1 . In the interaction picture, after making the rotating-wave approximation, the Hamiltonian of the flux qubit and the NV ensemble system is where ∆ = ω ag − ω 0,+1 , τ − k = |m s = 0 k m s = +1| and τ + k = |m s = +1 k m s = 0| are the lowering and raising operators of the kth spin, and µ k is the coupling constant between the kth spin and the |g ↔ |a transition of qubit. Here, ω 0,+1 is the transition frequency between the two levels |m s = 0 and |m s = +1 . We then introduce a collective operator where µ 2 = N k=1 | µ k | 2 /N with µ the root mean square of the individual couplings.
Under the conditions of the large N and the low excitations, b † behaves as a bosonic operator and the spin ensemble behaves as a bosonic mode. Thus, one has [b, b † ] ≈ 1, and b † b|n b = n|n b [56,57], where |n b = 1 √ n! (b † ) n |0 b with |0 b = |m s = 0 1 |m s = 0 2 · · · |m s = 0 N . Accordingly, one has the frequency of the bosonic mode ω b = ω 0,+1 .Therefore, the Hamiltonian (9) can be further rewritten as with µ = √ N µ. We then consider a system consisting of a flux qubit coupled to two NV ensembles [ Fig. 2(b)]. As depicted in Fig. 2(c), NV ensembles 1 and 2 are off-resonantly coupled to the |g ↔ |a transition of qubit with coupling constants µ 1 and µ 2 , respectively. Based on Eq. (11), the Hamiltonian of the whole system is where b 1 and b 2 are the corresponding annihilation operators for the NV ensembles 1 and 2, Here, ω ag is the |g ↔ |a transition frequency of qubit and ω b 1 (ω b 2 ) is the frequency of NV ensemble 1 (2). Let's assume the large-detuning conditions ∆ 1 ≫ µ 1 and ∆ 2 ≫ µ 2 with (i) ∆ 1 = ∆ 2 , µ 1 = µ 2 , (ii) the level |a of coupler qubit not occupied. One can find that the final effective Hamiltonian [for details, see Eqs. (3)-(5)] can be given by with where . When the qubit is in the state |g , the Hamiltonian H i describes the interaction between the NV ensembles. It is obvious that the Hamiltonian Eq. (13) has the same form as Eq. (5), so the above demonstrated Fredkin gate can also be prepared with the NV ensembles as memories.

Applications and possible experimental implementation
The Fredkin gate has played a vital role and has many useful applications in QCQIP, such as error correction [2,3], quantum cloning [4], quantum fingerprinting [5,6], and quantum digital signatures [7]. In the following subsections, we will discuss some additional applications and the possible experimental implementation of our proposed Fredkin gate.
Preparation of entanglement-Entanglement as a physical phenomenon, is one of the most fundamental features of quantum mechanics [58]. Furthermore, entanglement is also an important physical resource to achieve many quantum information processing and communication tasks. For instance, the NOON states play the central role in quantum metrology [59], quantum optical lithography [60], and precision measurement [61]; the entangled coherent states can serve as an important resource for quantum networks [62], quantum teleportation [63], quantum cryptography [64], and quantum metrology [65]. Thus, the generation of entangled states is one of the key goals of QCQIP. Over the past two decades, many experiments have been reported for the generation of multiple-particle entangled states of photons [66,67], ions [68,69], natural atoms [70,71], NV centers [72,73], and superconducting qubits [74][75][76].
Our Fredkin gate can be used as an efficient quantum generator to produce the entanglement between two resonators or NV ensembles with the arbitrary (discrete variable or continuous variable) initial state. For example, we apply a microwave pulse to control qubit such that the pulse is resonant with the |g ↔ |e transition of the control qubit. Making the rotating-wave approximation, the Hamiltonian in the interaction picture is written as H I , 2 = Ω(e iθ |g e| + h.c.), where Ω and θ are the Rabi frequency and the initial phase of the pulse. We choose t ′ = π/(4Ω) and θ = −π/2 to pump the state |e to (|e − |g )/ √ 2 and |g to (|e + |g )/ √ 2. Accordingly, the state (8) changes to 1 √ 2 (|ψ + |e + |ψ − |g ), where |ψ ± are the entangled states of two quantum memories, given by |ψ ± = γ|ϕ 1 |ψ 2 ± η|ψ 1 |ϕ 2 . Now if a von Neumann measurement is performed on flux qubit along a measurement basis {|g , |e }, one can see that the entangled state of two quantum memories is prepared in the |ψ + or |ψ − . It should be noted here that |ψ and |ϕ are arbitrary nonsymmetric states.
The previous schemes for synthesizing an arbitrary two-resonator entangled state through two three-level superconducting qutrits coupled to three resonators assisted by a sequence of microwave pulses applied to the two qubits [31], or a four-level superconducting qudit [32] coupled to two resonators and driven by a microwave pulse. Compared with [31,32], our experimental setup is greatly simplified and the experimental difficulty is reduced because only a single three-level qubit is employed and no microwave pulse is used. In addition, our proposal is based on a first-order large detuning, while the [32] was based on a second-order large detuning. This makes our operation faster than the one proposed by [32]. In recent years, several statesynthesis algorithms have been proposed to generate entangled states of two superconducting resonators [37][38][39][40]. These methods [37][38][39][40] depend on the maximum photon number and the number of operational steps required. While our proposal requires only a single unitary operation, which can significantly reduce the number of steps and the preparation time.   , one can obtain that a high fidelity 96.0%, 98.5% and 96.5% for D = 16, 10 and 22, respectively. In the following analysis, we will choose D = 16, 10 and 22 for the cases of (i) (ii) and (iii), respectively. For D = 16, 10 and 22, one has the resonator-qubit or NV ensemble-qubit frequency detunings δ/2π = 1.12 GHz, 0.7 GHz and 1.54 GHz. The numerical simulation shows that the high-fidelity generation of above entangled states of two quantum memories is feasible with the state-of-the-art circuit QED technology.
In a realistic situation, the inhomogeneous broadening of quantum memories may induce the inhomogeneous memory-qubit coupling and unequal memory-qubit frequency detuning. Thus, we numerically calculate the fidelity by setting δ 1 /2π = δ, δ 2 /2π = c δ, g 1 = g, and g 2 = d g, with c ∈ [0.9995, 1.0005] and d ∈ [0.95, 1.05]. The other parameters used in the numerical simulation for Fig. 4 are the same as those used in Fig. 3. Figures 4(a)-4(c) display the fidelity versus c and d, which are plotted by choosing D = 16, 10 and 22, respectively. Figure 4(a) shows that for c ∈ [0.9995, 1.0003] and d ∈ [0.98, 1.05], the fidelity can be greater than 90%.
As illustrated in Fig. 4(b) Fig. 4, one can see that the high-fidelity generation of entangled states of two quantum memories can be achieved for small errors in memory-qubit coupling and detuning.

Conclusion
We have proposed a method to implementing a hybrid Fredkin gate between a single superconducting flux qubit and two resonators or NV ensembles in any discrete-variable or continuousvariable states. Due to the usage of only one single three-level qubit, the experimental setup is simplified and the experimental difficulty is greatly reduced. In addition, the protocol requires only a single unitary operation, thus the operation procedure is greatly simplified. The proposal can also be applied to other kinds of superconducting qubits (e.g., superconducting charge qubits, transmon qubits, Xmon qubits, phase qubits) coupled to two 1D resonators or two 3D cavities. Furthermore, our scheme can be used to generate arbitrary entangled states of two quantum memories, such as NOON states, entangled coherent states, and entangled cat states. It is also shown that our scheme can be used to measure the fidelity and entanglement between the two memories. Numerical simulation shows that these entangled states can be high-fidelity created with circuit QED of the existing technology.
Finally, we would like to emphasize that our scheme can also be used to realize a controlledshift operation with a qudit as a control state. If the control qudit includes two particular energy levels |up and |down with the transition frequency much less than the other pairs of energy levels, the two target memories will swap their states conditioned on the control state |down . Similarly, if the highest energy level is much farther off resonant with the other energy levels, one could use the other energy levels as a whole to control the swapping of the states of the two memories. All the realizations need us to carefully choose the various parameters covered in the system. It should be noted that the introduction of the multiple energy levels usually lead to the complex undesired couplings which could be small to some good approximation, but could be a little larger than the case of less energy levels under the same condition. All the above can be easily demonstrated following the similar procedures as our main text. Our finding provides a new way for realizing the Fredkin gate or quantum entanglement between two quantum memories, which may have many potential applications in quantum information processing based on circuit QED.