Abstract
Cavitybased large scale quantum information processing (QIP) may involve multiple cavities and require performing various quantum logic operations on qubits distributed in different cavities. Geometricphasebased quantum computing has drawn much attention recently, which offers advantages against inaccuracies and local fluctuations. In addition, multiqubit gates are particularly appealing and play important roles in QIP. We here present a simple and efficient scheme for realizing a multitargetqubit unconventional geometric phase gate in a multicavity system. This multiqubit phase gate has a common control qubit but different target qubits distributed in different cavities, which can be achieved using a singlestep operation. The gate operation time is independent of the number of qubits and only two levels for each qubit are needed. This multiqubit gate is generic, e.g., by performing singlequbit operations, it can be converted into two types of significant multitargetqubit phase gates useful in QIP. The proposal is quite general, which can be used to accomplish the same task for a general type of qubits such as atoms, NV centers, quantum dots and superconducting qubits.
Introduction
Multiqubit gates are particularly appealing and have been considered as an attractive building block for quantum information processing (QIP). In parallel to Shor algorithm^{1}, Grover/Long algorithm^{2,3}, quantum simulations, such as analogue quantum simulation^{4} and digital quantum simulation^{5}, are also important QIP tasks where controlled quantum gates play important roles. There exist two kinds of significant multiqubit gates, i.e., multiqubit gates with multiple control qubits acting on a single target qubit^{6,7,8,9,10,11,12,13,14} and multiqubit gates with a single qubit simultaneously controlling multiple target qubits^{15,16,17}. These two kinds of multiqubit gates have many applications in QIP such as quantum algorithms^{1,18,19,20}, quantum Fourier transform^{19}, error correction^{21,22,23}, quantum cloning^{24} and entanglement preparation^{25}.
A multiqubit gate can in principle be constructed by using singlequbit and twoqubit basic gates. However, when using the conventional gatedecomposition protocols to construct a multiqubit gate^{26,27,28}, the number of basic gates increases and the procedure usually becomes complicated as the number of qubits increases. Hence, building a multiqubit gate may become very difficult since each basic gate requires turning on and off a given Hamiltonian for a certain period of time and each additional basic gate adds experimental complications and the possibility of more errors. Thus, the study of reducing the operation time and the number of switching Hamiltonians is crucial in multiqubit gates^{29,30,31}. Proposals have been presented for directly realizing both multicontrolqubit gates^{6,7,8,9,10,11,12,13,14} and multitargetqubit gates^{15,16,17} in various physical systems. However, note that the gate implementation using these previous proposals^{6,7,8,9,10,11,12,13,14,15,16,17} was based on nongeometric dynamical evolution.
During the past years, there is much interest in faulttolerant geometric quantum computing based on Abelian geometric phases^{32,33,34,35} and Holonomic quantum computing based on nonAbelian holonomies^{36}. The construction of conventional geometric phase gates usually requires to remove the dynamical phase by choosing the adiabatic cyclic evolution^{37} or employing multiloop schemes (the evolution is driven by a Hamiltonian along several closed loops)^{38,39}. In recent years, attention has been shifted to unconventional geometric phases introduced in^{40}, which can be used as an alternative resource for geometric quantum computation without the need to remove the dynamic phase. According to^{40}, an unconventional geometric phase gate is characterized by a unitary operator U({γ}), where γ is the total phase, which consists of a geometric phase and a dynamic phase (see^{40}). Thus, additional operations are not needed to cancel the dynamical phase, because the total phase is dependent only on global geometric features and independent of initial states of the system. In this paper, we mainly focus on the construction of multiqubit gates based on unconventional geometric phases.
A number of proposals have been presented for realizing both conventional and unconventional geometric phase gates^{37,38,39,40,41,42,43,44,45,46,47,48,49,50,51}. Some approaches also combine the geometric computation with other theories in order to improve the robustness (e.g., combined with decoherence free subspace or dynamical decoupling)^{50,51}. Moreover, highfidelity geometric phase gates have been experimentally demonstrated in several physical systems^{52,53,54,55,56,57}. For instances, Jones et al.^{52} experimentally demonstrated a conditional Berry phase shift gate using NMR and Leibfried et al.^{53} realized a twoqubit geometric phase gate in a trapped ion system. On the other hand, much progress has been achieved in Holonomic quantum computing. Experimentally, Abdumalikov Jr et al.^{54} realized singlequbit Holonomic gates in a superconducting transmon, Feng et al.^{55} implemented onequbit and twoqubit Holonomic gates in a liquidstate NMR quantum information processor and two groups^{56,57} demonstrated singlequbit or twoqubit Holonomic gates using the NV centers at room temperature, respectively. However, we note that previous works mainly focus on constructing single or twoqubit geometric phase gates/Holonomic gates^{37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57}, or implementing a multicontrolqubit gate^{6,7,8,9,10,11,12,13,14} and a multitargetqubit gate^{15,16,17} based on nongeometric dynamical evolution.
In this work, we consider how to implement a multitargetqubit unconventional geometric phase gate, which is described by the following transformation:
where subscript A represents a control qubit, subscripts (1, 2, ..., n) represent n target qubits (1, 2, ..., n) and (with i_{j} ∈ {+, −}) is the ntargetqubit computational basis state. For n target qubits, there are a total number of 2^{n} computational basis states, which form a set of complete orthogonal bases in a 2^{n}dimensional Hilbert space of the n qubits. Equation (1) shows that when the control qubit A is in the state , a phase shift happens to the state but nothing happens to the state of the target qubit j (j = 1, 2, ..., n). For instance, under the transformation (1), one has: (i) the state transformation described by following Eq. (18) for a twoqubit phase gate on control qubit A and target qubit j and (ii) the state transformation described by Eq. (21) below for a threequbit phase gate on control qubit A and two target qubits (1, 2). Note that the multiqubit phase gate described by Eq. (1) is equivalent to such n twoqubit phase gates, i.e., each of them has a common control qubit A but a different target qubit 1, 2, ..., or n and the twoqubit phase gate acting on the control qubit A and the target qubit j (j = 1, 2, ..., n) is described by Eq. (18).
The multiqubit gate described by Eq. (1) is generic. For example, by performing a singlequbit operation such that and but nothing to and the transformation (1) becomes
which implies that when and only when the control qubit A is in the state , a phase shift happens to the state of the target qubit j but nothing otherwise (see Fig. 1). For θ_{j} = π/2, the state transformation (2) corresponds to a multitargetqubit phase gate, i.e., if and only if the control qubit A is in the state , a phase flip from the sign + to − occurs to the state of each target qubit. Note that a CNOT gate of one qubit simultaneously controlling n qubits, (see Fig. 1(b) in^{15}), can also be achieved using this multiqubit phase gate combined with two Hadamard gates on the control qubit^{15}. Such a multiqubit phase or CNOT gate is useful in QIP. For instance, this multiqubit gate is an essential ingredient for implementation of quantum algorithm (e.g., the discrete cosine transform^{20}), the gate plays a key role in quantum cloning^{24} and error correction^{23} and it can be used to generate multiqubit entangled states such as GreenbergerHorneZeilinger states^{25}. This multiqubit gate can be combined with a set of universal single or twoqubit quantum gates to construct quantum circuits for implementing quantum information processing tasks^{20,23,24,25}. In addition, for θ_{j} = π/2^{j}, the state transformation (2) corresponds to a multitargetqubit phase gate, i.e., if and only if the control qubit A is in the state , a phase shift θ_{j} = π/2^{j} happens to the state of each target qubit. It is noted that this multitargetqubit gate is equivalent to a multiqubit gate with different control qubits acting on the same target qubit (see Fig. 2), which is a key element in quantum Fourier transform^{1,19}.
In what follows, our goal is propose a simple method for implementing a generic unconventional geometric (UG) multitargetqubit gate described by Eq. (1), with one qubit (qubit A) simultaneously controlling n target qubits (1, 2, ..., n) distributed in n cavities (1, 2, ..., n). We believe that this work is also of interest from the following point of view. Largescale QIP usually involves a number of qubits. Placing many qubits in a single cavity may cause some fundamental problems such as introducing the unwanted qubitqubit interaction, increasing the cavity decay and decreasing the qubitcavity coupling strength. In this sense, largescale QIP may need to place qubits in multiple cavities and thus require performing various quantum logic operations on qubits distributed in different cavities. Hence, it is important and imperative to explore how to realize multiqubit gates performed on qubits that are spatiallyseparated and distributed in different cavities.
As shown below, this proposal has the following features and advantages: (i) The gate operation time is independent of the number of qubits; (ii) The proposed multitargetqubit UG phase gate can be implemented using a singlestep operation; (iii) Only two levels are needed for each qubit, i.e., no auxiliary levels are used for the state coherent manipulation; (iv) The proposal is quite general and can be applied to accomplish the same task with a general types of qubits such as atoms, superconducting qubits, quantum dots and NV centers. To the best of our knowledge, this proposal is the first one to demonstrate that a multitargetqubit UG phase gate described by (1) can be achieved with one qubit simultaneously controlling n target qubits distributed in n cavities.
In this work we will also discuss possible experimental implementation of our proposal and numerically calculate the operational fidelity for a threequbit gate, by using a setup of two superconducting transmission line resonators each hosting a transmon qubit and coupled to a coupler transmon qubit. Our numerical simulation shows that highlyfidelity implementation of a threequbit (i.e., twotargetqubit) UG phase gate by using this proposal is feasible with rapid development of circuit QED technique.
Results
Model and Hamiltonian
Consider a system consisting of n cavities each hosting a qubit and coupled to a common qubit A [Fig. 3(a)]. The coupling and decoupling of each qubit from its cavity can be achieved by prior adjustment of the qubit level spacings. For instance, the level spacings of superconducting qubits can be rapidly adjusted by varying external control parameters (e.g., magnetic flux applied to the superconducting loop of a superconducting phase, transmon, Xmon or flux qubit; see, e.g.^{58,59,60,61}); the level spacings of NV centers can be readily adjusted by changing the external magnetic field applied along the crystalline axis of each NV center^{62,63}; and the level spacings of atoms/quantum dots can be adjusted by changing the voltage on the electrodes around each atom/quantum dot^{64}. The two levels of coupler qubit A are denoted as and while those of intracavity qubit j as and (j = 1, 2, ···, n). A classical pulse is applied to qubit A and each intracavity qubit j [Fig. 3(b,c)]. For identical qubits, we have , where ω is the pulse frequency and is the transition frequency of qubit A (qubit j). The system Hamiltonian in the interaction picture reads (in units of ħ = 1)
where is the photon creation operator for the mode of cavity j, and are the raising and lowering operators for qubit A (qubit j), and are detunings (with being the frequency of cavity j), Ω is the Rabi frequency of the pulse applied to each qubit, (g_{j}) is the coupling constant of qubit A (j) with cavity j. We choose and as the rotated basis states of qubit j and qubit A, respectively.
In a rotated basis , one has and , where , and . Here, l = 1, 2, 3, ···n, A. Hence, the Hamiltonian (3) can be expressed as
In a new interaction picture under the Hamiltonian , one obtains from Eq. (4)
In the strong driving regime , one can apply a rotatingwave approximation and eliminate the terms that oscillate with high frequencies. Thus, the Hamiltonian (5) becomes
For simplicity, we set
The first term of condition (7) can be achieved by adjusting the position of qubit j in cavity j and second term can be met for identical qubits. Thus, the Hamiltonian (6) changes to
with
where is the effective Hamiltonian of a subsystem, which consists of qubit A, intracavity qubit j and cavity j. In the next section, we first show how to use the Hamiltonian (9) to construct a twoqubit UG phase gate with qubit A controlling the target qubit j. We then discuss how to use the effective Hamiltonian (8) to construct a multiqubit UG phase gate with qubit A simultaneously controlling n target qubits distributed in n cavities.
Implementing multiqubit UG phase gates
Consider a system consisting of the coupler qubit A and an intracavity qubit j, for which are eigenstates of the operator with eigenvalues ±1. In the rotated basis , the Hamiltonian (9) can be rewritten as
and thus the time evolution operator U_{Aj}(t) corresponding to the Hamiltonian can be expressed as
where and are given by
with
where pp ∈ {++, − −}, p ∈ {+, −}, ε_{++} = −ε_{−−} = 1, D is the displacement operator (for details, see Methods below), is the time ordering operator and Δτ = t/N is the time interval. From Eq. (12) and Eq. (31) below, one obtains and . Thus, one has
where T_{j} is the evolution time required to complete a closed path.
If t = T_{j} is equal to 2m_{j}π/δ_{j} with a positive integer m_{j}, we have ∫_{c}α_{pp,j} = 0 according to Eq. (14), which shows that when cavity j is initially in the vacuum state, then cavity j returns to its initial vacuum state after the time evolution completing a closed path. Thus, it follows from Eq. (12) that we have
Here θ_{pp,j} is the total phase given by Eq. (14), which is acquired during the time evolution from t = 0 to t = T_{j}. Note that θ_{pp,j} consists of a geometric phase and a dynamical phase.
It follows from Eqs (11) and (15) that the cyclic evolution is described by
Eq. (14) shows that θ_{pp,j} is independent of index pp. Thus, we have θ_{++,j} = θ_{−−,j} ≡ θ_{j}. Further, according to Eq. (14), after an integration for T_{j} = 2m_{j}π/δ_{j} (set above), we have
which can be adjusted by varying the coupling strength g_{j} and detuning δ_{j}. Note that a negative detuning δ_{j} < 0 [see Fig. 3(b,c)] has applied to the last equality of Eq. (17). The unitary operator (16) describes a twoqubit UG phase gate operation. For θ_{j} ≠ 2nπ with an integer n, the phase gate is nontrivial. After returning to the original interaction picture by performing a unitary transformation , we obtain the following state transformations: and , which can be further written as
where we have set ΩT_{j} = kπ (k is a positive integer). For T_{j} = 2m_{j}π/δ_{j}, we have 2Ω = kδ_{j}/m_{j}. The result (18) shows that a twoqubit UG phase gate was achieved after a singlestep operation described above.
Now we expand the above procedure to a multiqubit case. Consider qubit A and n qubits (1, 2, ···, n) distributed in n cavities [Fig. 3(a)]. From Eqs (8) and (9), one can see that: (i) each term of H_{eff} acts on a different intracavity qubit but the same coupler qubit A and (ii) any two terms of H_{eff}, corresponding to different j, commute with each other: . Thus, it is straightforward to show that the cyclic evolution of the cavityqubit system is described by the following unitary operator
where U_{Aj}(T_{j}) is the unitary operator given in Eq. (16), which characterizes the cyclic evolution of a twoqubit subsystem (i.e., qubit A and intracavity qubit j) in the rotated basis and .
By changing the detunings δ_{j} (e.g., via prior design of cavity j with an appropriate frequency), one can have
which leads to T_{1} = T_{2} = , ···, = T_{n} ≡ T, i.e., the evolution time for each of qubit pairs (A, 1), (A, 2), ··· and (A, n) to complete a cyclic evolution is identical. For the setting here, we have resulting from Eq. (17). Hence, one can easily find from Eqs (18) and (19) that after a common evolution time T, the n twoqubit UG phase gates characterized by a jointed unitary operator U(T) of Eq. (19), which have a common control qubit A but different target qubits (1, 2, ..., n), are simultaneously implemented. As discussed in the introduction, the n twoqubit UG phase gates here are equivalent to a multiqubit UG phase gate described by Eq. (1). Hence, after the above operation, the proposed multiqubit UG phase gate is realized with coupler qubit A (control qubit) simultaneously controlling n target qubits (1, 2, ···, n) distributed in n cavities.
To see the above more clearly, consider implementing a threequbit (twotargetqubit) UG phase gate. For three qubits, there are a total number of eight computational basis states, denoted by . According to Eqs (18) and (19), one can obtain a threequbit UG phase gate, which is described by
As discussed in the introduction, by applying singlequbit operations, this threequbit UG phase gate described by Eq. (21) can be converted into a threequbit phase gate which is illustrated in the abovementioned Fig. 1 or Fig. 2 for n = 2. In the next section, as an example, we will give a discussion on the experimental implementation of this threequbit UG phase gate for the case of θ_{1} = θ_{2} = π/2. Based on Eq. (17) and for T_{1} = T_{2} (see above), one can see that the θ_{1} = θ_{2} corresponds to , which can be met by adjusting g_{j} (e.g., varying the position of qubit j in cavity j) or detuning δ_{j} (e.g., prior adjustment of the frequency of cavity j) (j = 1, 2).
Possible experimental implementation
Superconducting qubits are important in QIP due to their ready fabrication, controllability and potential scalability^{58,65,66,67,68,69}. Circuit QED is analogue of cavity QED with solidstate devices coupled to a microwave cavity on a chip and is considered as one of the most promising candidates for QIP^{65,66,67,68,69,70,71,72}. In above, a general type of qubit, for both of the intracavity qubits and the coupler qubit, is considered. As an example of experimental implementation, let us now consider each qubit as a superconducting transmon qubit and each cavity as a onedimensional transmission line resonator (TLR). We consider a setup in Fig. 4 for achieving a threequbit UG phase gate. To be more realistic, we consider a third higher level of each transmon qubit during the entire operation because this level may be excited due to the transition induced by the cavity mode(s), which will affect the operation fidelity. From now on, each qubit is renamed “qutrit” since the three levels are considered.
When the intercavity crosstalk coupling and the unwanted transition of each qutrit are considered, the Hamiltonian (3) is modified as follows
where H_{I} is the needed interaction Hamiltonian in Eq. (3) for n = 2, while Θ_{I} is the unwanted interaction Hamiltonian, given by
where and The first term describes the unwanted offresonant coupling between cavity j and the transition of qutrit j, with coupling constant and detuning [Fig. 5(a,b)], while the second term is the unwanted offresonant coupling between cavity j and the transition of qutrit A, with coupling constant and detuning [Fig. 5(c)]. The third term of Eq. (23) describes the intercavity crosstalk between the two cavities, where is the detuning between the twocavity frequencies and g_{12} is the intercavity coupling strength between the two cavities. The last two terms of Eq. (23) describe unwanted offresonant couplings between the pulse and the transition of each qutrit, where is the pulse Rabi frequency. Note that the Hamiltonian (23) does not involves transition of each qutrit, since this transition is negligible because of (j = 1, 2) (Fig. 5).
When the dissipation and dephasing are included, the dynamics of the lossy system is determined by the following master equation
where and with Here, κ_{j} is the photon decay rate of cavity j (j = 1, 2). In addition, Γ_{l} is the energy relaxation rate of the level of qutrit l, is the energy relaxation rate of the level of qutrit l for the decay path and Γ_{l,φ e} (Γ_{l,φf}) is the dephasing rate of the level of qutrit l (l = 1, 2, A).
The fidelity of the operation is given by
where is the output state of an ideal system (i.e., without dissipation, dephasing and crosstalk considered), while ρ is the final density operator of the system when the operation is performed in a realistic physical system. As an example, we consider that qutrit l is initially in a superposition state (l = 1, 2, A) and cavity 1 (2) is initially in the vacuum state. In this case, we have , where
which is obtained based on Eq. (21) and for θ_{1} = θ_{2} = π/2.
We now numerically calculate the fidelity of the gate operation. Without loss of generality, consider identical transmon qutrits and cavities. Setting m_{1} = 1 and m_{2} = 2, we have δ_{2} = 2δ_{1} because of Eq. (20), which corresponds to for θ_{1} = θ_{2}. In order to satisfy the relation 2Ω ≫ δ_{2} and 2Ω = kδ_{2}/2, we set k = 18. In addition, we have , (j = 1, 2) and for the transmon qutrits^{73}. For a transmon qutrit, the anharmonicity α/2π = 720 MHZ between the transition frequency and the transition frequency is readily achieved in experiments^{74}. Thus, we set MHz and MHz (j = 1, 2). For transmon qutrits, the typical transition frequency between two neighbor levels is between 4 and 10 GHz^{75,76}. Therefore, we choose GHz. Other parameters used in the numerical calculation are as follows: μs, μs, μs, μs, μs (l = 1, 2, A) and μs (j = 1, 2). It is noted that for a transmon qutrit, the dipole matrix element is much smaller than that of the and transitions. Thus, .
To test how the intercavity crosstalk affects the gate fidelity, we plot Fig. 6 for g_{12} = 0, 0.01g_{1}, 0.1g_{1}, which shows the fidelity versus δ_{1}/2π. For simplicity, the dissipation and dephasing of the system are not considered in Fig. 6. As depicted in Fig. 6, the effect of the intercavity coupling is negligible as long as g_{12} ≤ 0.01g_{1}.
Figure 7 shows the fidelity versus δ_{1}/2π, which is plotted by setting g_{12} = 0.01g_{1} and now taking the systematic dissipation and dephasing into account. From Fig. 7, one can see that for δ_{1}/2π ≈ −1.8 MHz, a high fidelity ~99.1% is achievable for a threequbit UG phase gate. For δ_{1}/2π ≈ −1.8 MHz, we have T = T_{1} = T_{2} = 0.556 μs, g_{1}/2π = 0.9 MHz and g_{2}/2π = 1.273 MHz. The values of g_{1} and g_{2} here are readily available in experiments^{77}.
The condition g_{12} = 0.01g_{1} is easy to satisfy with the cavityqutrit capacitive coupling shown in Fig. 4. When the cavities are physically well separated, the intercavity crosstalk strength is , where C_{Σ} = C_{1} + C_{2} + C_{q} (C_{q} is the qutrit’s selfcapacitance)^{78,79}. For C_{1}, C_{2}~ 1 fF and C_{Σ}~ 100 fF (typical values in experiments), one has g_{12} ~ 0.01g_{1}. Thus, the condition g_{12} = 0.01g_{1} is readily achievable in experiments.
Energy relaxation time T_{1} and dephasing time T_{2} of the level can be made to be on the order of 55–60 μs for stateoftheart transom devices coupled to a onedimensional TLR^{80} and the order of 20–80 μs for a transom coupled to a threedimensional microwave resonator^{81,82}. For transmon qutrits, we have the energy relaxation time and dephasing time of the level which are comparable to T_{1} and T_{2}, respectively. With GHz chosen above, we have ω_{c1}/2π ~ 6.5018 GHz and ω_{c2}/2π ~ 6.5009 GHz. For the cavity frequencies here and the values of and used in the numerical calculation, the required quality factors for the two cavities are Q_{1} ~ 1.2249 × 10^{6} and Q_{2} ~ 1.2247 × 10^{6}. Note that superconducting coplanar waveguide resonators with a loaded quality factor Q ~ 10^{6} were experimentally demonstrated^{83,84} and planar superconducting resonators with internal quality factors above one million (Q > 10^{7}) have also been reported^{85}. We have numerically simulated a threequbit circuit QED system, which shows that the highfidelity implementation of a threequbit UG phase gate is feasible with rapid development of circuit QED technique.
Discussion
A simple method has been presented to realize a generic unconventional geometric phase gate of one qubit simultaneously controlling n spatiallyseparated target qubits in circuit QED. As shown above, the gate operation time is independent of the number n of qubits. In addition, only a single step of operation is needed and it is unnecessary to employ threelevel or fourlevel qubits and not required to eliminate the dynamical phase, therefore the operation is greatly simplified and the experimental difficulty is significantly reduced. Our numerical simulation shows that highlyfidelity implementation of a twotargetqubit unconventional geometric phase gate by using this proposal is feasible with rapid development of circuit QED technique. The proposed multiqubit gate is generic, which, for example, can be converted into two types of important multitargetqubit phase gates useful in QIP. This proposal is quite general and can be applied to accomplish the same task with various types of qubits such as atoms, quantum dots, superconducting qubits and NV centers.
Methods
Geometric phase
Geometric phase is induced due to a displacement operator along an arbitrary path in phase space^{86,87}. The displacement operator is expressed as
where a^{†} and a are the creation and annihilation operators of an harmonic oscillator, respectively. The displacement operators satisfy
For a path consisting of N short straight sections Δα_{j}, the total operator is
An arbitrary path c can be approached in the limit N → ∞. Therefore, Eq. (29) can be rewritten as
with
For a closed path, we have
where Θ is the total phase which consists of a geometric phase and a dynamical phase^{35}. In above, equations (27, 28, 29, 30, 31, 32) have been adopted for realizing an UG phase gate of one qubit simultaneously controlling n target qubits.
Additional Information
How to cite this article: Liu, T. et al. Multitargetqubit unconventional geometric phase gate in a multicavity system. Sci. Rep. 6, 21562; doi: 10.1038/srep21562 (2016).
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grant Nos [11074062, 11374083, 11504075] and the Zhejiang Natural Science Foundation under Grant No. LZ13A040002. This work was also supported by the funds from Hangzhou City for the HangzhouCity Quantum Information and Quantum Optics Innovation Research Team.
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T.L., S.J.X. and C.P.Y. conceived the idea. X.Z.C. carried out all calculations under the guidance of Q.P.S. and C.P.Y. All the authors discussed the results. T.L., S.J.X. and C.P.Y. contributed to the writing of the manuscript.
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Liu, T., Cao, XZ., Su, QP. et al. Multitargetqubit unconventional geometric phase gate in a multicavity system. Sci Rep 6, 21562 (2016). https://doi.org/10.1038/srep21562
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