Multiqubit gates are particularly appealing and have been considered as an attractive building block for quantum information processing (QIP). In parallel to Shor algorithm[@b1], Grover/Long algorithm[@b2][@b3], quantum simulations, such as analogue quantum simulation[@b4] and digital quantum simulation[@b5], 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[@b6][@b7][@b8][@b9][@b10][@b11][@b12][@b13][@b14], and multiqubit gates with a single qubit simultaneously controlling multiple target qubits[@b15][@b16][@b17]. These two kinds of multiqubit gates have many applications in QIP such as quantum algorithms[@b1][@b18][@b19][@b20], quantum Fourier transform[@b19], error correction[@b21][@b22][@b23], quantum cloning[@b24], and entanglement preparation[@b25].

A multiqubit gate can in principle be constructed by using single-qubit and two-qubit basic gates. However, when using the conventional gate-decomposition protocols to construct a multiqubit gate[@b26][@b27][@b28], 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[@b29][@b30][@b31]. Proposals have been presented for directly realizing both multi-control-qubit gates[@b6][@b7][@b8][@b9][@b10][@b11][@b12][@b13][@b14] and multi-target-qubit gates[@b15][@b16][@b17] in various physical systems. However, note that the gate implementation using these previous proposals[@b6][@b7][@b8][@b9][@b10][@b11][@b12][@b13][@b14][@b15][@b16][@b17] was based on non-geometric dynamical evolution.

During the past years, there is much interest in fault-tolerant geometric quantum computing based on Abelian geometric phases[@b32][@b33][@b34][@b35] and Holonomic quantum computing based on non-Abelian holonomies[@b36]. The construction of conventional geometric phase gates usually requires to remove the dynamical phase by choosing the adiabatic cyclic evolution[@b37] or employing multi-loop schemes (the evolution is driven by a Hamiltonian along several closed loops)[@b38][@b39]. In recent years, attention has been shifted to unconventional geometric phases introduced in[@b40], which can be used as an alternative resource for geometric quantum computation without the need to remove the dynamic phase. According to[@b40], 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[@b40]). 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[@b37][@b38][@b39][@b40][@b41][@b42][@b43][@b44][@b45][@b46][@b47][@b48][@b49][@b50][@b51]. 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)[@b50][@b51]. Moreover, high-fidelity geometric phase gates have been experimentally demonstrated in several physical systems[@b52][@b53][@b54][@b55][@b56][@b57]. For instances, Jones *et al*.[@b52] experimentally demonstrated a conditional Berry phase shift gate using NMR, and Leibfried *et al*.[@b53] realized a two-qubit 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*.[@b54] realized single-qubit Holonomic gates in a superconducting transmon, Feng *et al*.[@b55] implemented one-qubit and two-qubit Holonomic gates in a liquid-state NMR quantum information processor, and two groups[@b56][@b57] demonstrated single-qubit or two-qubit Holonomic gates using the NV centers at room temperature, respectively. However, we note that previous works mainly focus on constructing single- or two-qubit geometric phase gates/Holonomic gates[@b37][@b38][@b39][@b40][@b41][@b42][@b43][@b44][@b45][@b46][@b47][@b48][@b49][@b50][@b51][@b52][@b53][@b54][@b55][@b56][@b57], or implementing a multi-control-qubit gate[@b6][@b7][@b8][@b9][@b10][@b11][@b12][@b13][@b14] and a multi-target-qubit gate[@b15][@b16][@b17] based on non-geometric dynamical evolution.

In this work, we consider how to implement a multi-target-qubit 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 *n*-target-qubit 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)](#eq1){ref-type="disp-formula"} 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)](#eq79){ref-type="disp-formula"} for a two-qubit phase gate on control qubit *A* and target qubit *j*, and (ii) the state transformation described by Eq. [(21)](#eq89){ref-type="disp-formula"} below for a three-qubit phase gate on control qubit *A* and two target qubits (1, 2). Note that the multiqubit phase gate described by Eq. [(1)](#eq1){ref-type="disp-formula"} is equivalent to such *n* two-qubit phase gates, i.e., each of them has a common control qubit *A* but a different target qubit 1, 2, \..., or *n*, and the two-qubit phase gate acting on the control qubit *A* and the target qubit *j* (*j* = 1, 2, \..., *n*) is described by Eq. [(18)](#eq79){ref-type="disp-formula"}.

The multiqubit gate described by Eq. [(1)](#eq1){ref-type="disp-formula"} is generic. For example, by performing a single-qubit operation such that and but nothing to and the transformation (1) becomeswhich 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](#f1){ref-type="fig"}). For *θ*~*j*~ = *π*/2, the state transformation (2) corresponds to a multi-target-qubit 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[@b15]), can also be achieved using this multiqubit phase gate combined with two Hadamard gates on the control qubit[@b15]. 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[@b20]), the gate plays a key role in quantum cloning[@b24] and error correction[@b23], and it can be used to generate multiqubit entangled states such as Greenberger-Horne-Zeilinger states[@b25]. This multiqubit gate can be combined with a set of universal single- or two-qubit quantum gates to construct quantum circuits for implementing quantum information processing tasks[@b20][@b23][@b24][@b25]. In addition, for *θ*~*j*~ = *π*/2^*j*^, the state transformation (2) corresponds to a multi-target-qubit 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 multi-target-qubit gate is equivalent to a multiqubit gate with different control qubits acting on the same target qubit (see [Fig. 2](#f2){ref-type="fig"}), which is a key element in quantum Fourier transform[@b1][@b19].

In what follows, our goal is propose a simple method for implementing a generic unconventional geometric (UG) multi-target-qubit gate described by Eq. [(1)](#eq1){ref-type="disp-formula"}, 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. Large-scale QIP usually involves a number of qubits. Placing many qubits in a single cavity may cause some fundamental problems such as introducing the unwanted qubit-qubit interaction, increasing the cavity decay, and decreasing the qubit-cavity coupling strength. In this sense, large-scale 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 spatially-separated 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 multi-target-qubit UG phase gate can be implemented using a single-step 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 multi-target-qubit 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 three-qubit 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 highly-fidelity implementation of a three-qubit (i.e., two-target-qubit) 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* \[[Fig. 3(a)](#f3){ref-type="fig"}\]. 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.[@b58][@b59][@b60][@b61]); 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[@b62][@b63]; and the level spacings of atoms/quantum dots can be adjusted by changing the voltage on the electrodes around each atom/quantum dot[@b64]. 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* \[[Fig. 3(b,c)](#f3){ref-type="fig"}\]. 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 asIn a new interaction picture under the Hamiltonian , one obtains from Eq. [(4)](#eq48){ref-type="disp-formula"}In the strong driving regime , one can apply a rotating-wave approximation and eliminate the terms that oscillate with high frequencies. Thus, the Hamiltonian (5) becomesFor simplicity, we setThe 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 towithwhere 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 two-qubit UG phase gate with qubit *A* controlling the target qubit *j*. We then discuss how to use the effective Hamiltonian (8) to construct a multi-qubit 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 asand thus the time evolution operator *U*~*Aj*~(*t*) corresponding to the Hamiltonian can be expressed aswhere and are given bywithwhere *pp* ∈ {++, − −}, *p* ∈ {+, −}, *ε*~++~ = −*ε*~−−~ = 1, *D* is the displacement operator (for details, see Methods below), is the time ordering operator and Δ*τ* = *t*/*N* is the time interval. From Eq. [(12)](#eq67){ref-type="disp-formula"} and Eq. [(31)](#eq157){ref-type="disp-formula"} below, one obtains and . Thus, one haswhere *T*~*j*~ is the evolution time required to complete a closed path.

If *t* = *T*~*j*~ is equal to 2*m*~*j*~*π*/\|*δ*~*j*~\| with a positive integer *m*~*j*~, we have ∫~*c*~*α*~*pp*,*j*~ = 0 according to Eq. [(14)](#eq72){ref-type="disp-formula"}, 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)](#eq67){ref-type="disp-formula"} that we haveHere *θ*~*pp*,*j*~ is the total phase given by Eq. [(14)](#eq72){ref-type="disp-formula"}, 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](#eq64){ref-type="disp-formula"}) and ([15](#eq73){ref-type="disp-formula"}) that the cyclic evolution is described byEq. [(14)](#eq72){ref-type="disp-formula"} shows that *θ*~*pp*,*j*~ is independent of index *pp*. Thus, we have *θ*~++,*j*~ = *θ*~−−,*j*~ ≡ *θ*~*j*~. Further, according to Eq. [(14)](#eq72){ref-type="disp-formula"}, after an integration for *T*~*j*~ = 2*m*~*j*~*π*/\|*δ*~*j*~\| (set above), we havewhich can be adjusted by varying the coupling strength *g*~*j*~ and detuning *δ*~*j*~. Note that a negative detuning *δ*~*j*~ \< 0 \[see [Fig. 3(b,c)](#f3){ref-type="fig"}\] has applied to the last equality of Eq. [(17)](#eq75){ref-type="disp-formula"}. The unitary operator (16) describes a two-qubit UG phase gate operation. For *θ*~*j*~ ≠ 2*nπ* with an i*n*teger *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 aswhere we have set Ω*T*~*j*~ = *kπ* (*k* is a positive integer). For *T*~*j*~ = 2*m*~*j*~*π*/\|*δ*~*j*~\|, we have 2Ω = *k*\|*δ*~*j*~\|/*m*~*j*~. The result (18) shows that a two-qubit UG phase gate was achieved after a single-step 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 \[[Fig. 3(a)](#f3){ref-type="fig"}\]. From Eqs [(8](#eq54){ref-type="disp-formula"}) and ([9](#eq55){ref-type="disp-formula"}), one can see that: (i) each term of *H*~*eff*~ acts on a different intra-cavity 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 cavity-qubit system is described by the following unitary operatorwhere *U*~*Aj*~(*T*~*j*~) is the unitary operator given in Eq. [(16)](#eq74){ref-type="disp-formula"}, which characterizes the cyclic evolution of a two-qubit 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 havewhich 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)](#eq75){ref-type="disp-formula"}. Hence, one can easily find from Eqs [(18](#eq79){ref-type="disp-formula"}) and ([19](#eq81){ref-type="disp-formula"}) that after a common evolution time *T*, the *n* two-qubit UG phase gates characterized by a jointed unitary operator *U*(*T*) of Eq. [(19)](#eq81){ref-type="disp-formula"}, which have a common control qubit *A* but different target qubits (1, 2, \..., *n*), are simultaneously implemented. As discussed in the introduction, the *n* two-qubit UG phase gates here are equivalent to a multiqubit UG phase gate described by Eq. [(1)](#eq1){ref-type="disp-formula"}. 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 three-qubit (two-target-qubit) UG phase gate. For three qubits, there are a total number of eight computational basis states, denoted by . According to Eqs [(18](#eq79){ref-type="disp-formula"}) and ([19](#eq81){ref-type="disp-formula"}), one can obtain a three-qubit UG phase gate, which is described byAs discussed in the introduction, by applying single-qubit operations, this three-qubit UG phase gate described by Eq. [(21)](#eq89){ref-type="disp-formula"} can be converted into a three-qubit phase gate which is illustrated in the above-mentioned [Fig. 1](#f1){ref-type="fig"} or [Fig. 2](#f2){ref-type="fig"} for *n* = 2. In the next section, as an example, we will give a discussion on the experimental implementation of this three-qubit UG phase gate for the case of *θ*~1~ = *θ*~2~ = *π*/2. Based on Eq. [(17)](#eq75){ref-type="disp-formula"} 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[@b58][@b65][@b66][@b67][@b68][@b69]. Circuit QED is analogue of cavity QED with solid-state devices coupled to a microwave cavity on a chip and is considered as one of the most promising candidates for QIP[@b65][@b66][@b67][@b68][@b69][@b70][@b71][@b72]. 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 one-dimensional transmission line resonator (TLR). We consider a setup in [Fig. 4](#f4){ref-type="fig"} for achieving a three-qubit 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 followswhere *H*~*I*~ is the needed interaction Hamiltonian in Eq. [(3)](#eq156){ref-type="disp-formula"} for *n* = 2, while Θ~*I*~ is the unwanted interaction Hamiltonian, given bywhere and The first term describes the unwanted off-resonant coupling between cavity *j* and the transition of qutrit *j*, with coupling constant and detuning \[[Fig. 5(a,b)](#f5){ref-type="fig"}\], while the second term is the unwanted off-resonant coupling between cavity *j* and the transition of qutrit *A*, with coupling constant and detuning \[[Fig. 5(c)](#f5){ref-type="fig"}\]. The third term of Eq. [(23)](#eq96){ref-type="disp-formula"} describes the intercavity crosstalk between the two cavities, where is the detuning between the two-cavity frequencies and *g*~12~ is the intercavity coupling strength between the two cavities. The last two terms of Eq. [(23)](#eq96){ref-type="disp-formula"} describe unwanted off-resonant 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](#f5){ref-type="fig"}).

When the dissipation and dephasing are included, the dynamics of the lossy system is determined by the following master equationwhere 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 bywhere 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 , wherewhich is obtained based on Eq. [(21)](#eq89){ref-type="disp-formula"} 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~ = 2*δ*~1~ because of Eq. [(20)](#eq86){ref-type="disp-formula"}, which corresponds to for *θ*~1~ = *θ*~2~. In order to satisfy the relation 2Ω ≫ \|*δ*~2~\| and 2Ω = *k*\|*δ*~2~\|/2, we set *k* = 18. In addition, we have , (*j* = 1, 2), and for the transmon qutrits[@b73]. For a transmon qutrit, the anharmonicity *α*/2*π* = 720 MHZ between the transition frequency and the transition frequency is readily achieved in experiments[@b74]. 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[@b75][@b76]. Therefore, we choose GHz. Other parameters used in the numerical calculation are as follows: *μ*s, *μ*s, *μ*s, *μ*s, *μ*s (*l* = 1, 2, *A*), and *μ*s (*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 inter-cavity crosstalk affects the gate fidelity, we plot [Fig. 6](#f6){ref-type="fig"} for *g*~12~ = 0, 0.01*g*~1~, 0.1*g*~1~, which shows the fidelity versus *δ*~1~/2*π*. For simplicity, the dissipation and dephasing of the system are not considered in [Fig. 6](#f6){ref-type="fig"}. As depicted in [Fig. 6](#f6){ref-type="fig"}, the effect of the inter-cavity coupling is negligible as long as *g*~12~ ≤ 0.01*g*~1~.

[Figure 7](#f7){ref-type="fig"} shows the fidelity versus *δ*~1~/2*π*, which is plotted by setting *g*~12~ = 0.01*g*~1~ and now taking the systematic dissipation and dephasing into account. From [Fig. 7](#f7){ref-type="fig"}, one can see that for *δ*~1~/2*π* ≈ −1.8 MHz, a high fidelity \~99.1% is achievable for a three-qubit UG phase gate. For *δ*~1~/2*π* ≈ −1.8 MHz, we have *T* = *T*~1~ = *T*~2~ = 0.556 *μ*s, *g*~1~/2*π* = 0.9 MHz, and *g*~2~/2*π* = 1.273 MHz. The values of *g*~1~ and *g*~2~ here are readily available in experiments[@b77].

The condition *g*~12~ = 0.01*g*~1~ is easy to satisfy with the cavity-qutrit capacitive coupling shown in [Fig. 4](#f4){ref-type="fig"}. When the cavities are physically well separated, the inter-cavity crosstalk strength is , where *C*~Σ~ = *C*~1~ + *C*~2~ + *C*~*q*~ (*C*~*q*~ is the qutrit's self-capacitance)[@b78][@b79]. For *C*~1~, *C*~2~\~ 1 fF and *C*~Σ~\~ 100 fF (typical values in experiments), one has *g*~12~ \~ 0.01*g*~1~. Thus, the condition *g*~12~ = 0.01*g*~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 *μ*s for state-of-the-art transom devices coupled to a one-dimensional TLR[@b80] and the order of 20--80 *μ*s for a transom coupled to a three-dimensional microwave resonator[@b81][@b82]. 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*~*c*1~/2*π* \~ 6.5018 GHz and *ω*~*c*2~/2*π* \~ 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[@b83][@b84] and planar superconducting resonators with internal quality factors above one million (*Q* \> 10^7^) have also been reported[@b85]. We have numerically simulated a three-qubit circuit QED system, which shows that the high-fidelity implementation of a three-qubit 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* spatially-separated 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 three-level or four-level 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 highly-fidelity implementation of a two-target-qubit 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 multi-target-qubit 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[@b86][@b87]. The displacement operator is expressed aswhere *a*^†^ and *a* are the creation and annihilation operators of an harmonic oscillator, respectively. The displacement operators satisfyFor a path consisting of *N* short straight sections Δ*α*~*j*~, the total operator isAn arbitrary path *c* can be approached in the limit *N* → ∞. Therefore, Eq. [(29)](#eq155){ref-type="disp-formula"} can be rewritten aswithFor a closed path, we havewhere Θ is the total phase which consists of a geometric phase and a dynamical phase[@b35]. In above, equations [(27](#eq153){ref-type="disp-formula"}, [28](#eq154){ref-type="disp-formula"}, [29](#eq155){ref-type="disp-formula"}, [30](#eq156){ref-type="disp-formula"}, [31](#eq157){ref-type="disp-formula"}, [32)](#eq158){ref-type="disp-formula"} 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*. Multi-target-qubit unconventional geometric phase gate in a multi-cavity system. *Sci. Rep.* **6**, 21562; doi: 10.1038/srep21562 (2016).

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 Hangzhou-City Quantum Information and Quantum Optics Innovation Research Team.

**Author Contributions** 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.

![(**a**) Schematic circuit of a phase gate with qubit *A* (a black dot) simultaneously controlling *n* target qubits (squares). (**b**) This multiqubit phase gate illustrated in (**a**) consists of *n* two-qubit phase gates, each having a shared control qubit (qubit *A*) but a different target qubit (qubit 1, 2, ···, or *n*). Here, the element 2*θ*~*j*~ represents a phase shift exp(*i*2*θ*~*j*~), which happens to the state of target qubit *j* (*j* = 1, 2, \..., *n*) when and only when the control qubit *A* is in the state but nothing happens otherwise. For 2*θ*~*j*~ = *π*, this gate corresponds to a multi-target-qubit phase gate (useful in QIP[@b20][@b23][@b24][@b25]), 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.](srep21562-f1){#f1}

![Schematic circuit of the *n* successive two-qubit phase gates in quantum Fourier transform.\
Here, each two-qubit phase gate has a shared target qubit (qubit *A*) but a different control qubit (qubit 1, 2, ···, or *n*). The element *π*/2^*j*^ represents a phase shift exp(*iπ*/2^*j*^), which happens to the state of target qubit *A* if and only if the control qubit *j* is in the state (*j* = 1, 2, \..., *n*). For any two-qubit controlled phase gate described by the transformation and , it is clear that the roles of the two qubits can be interchanged. Namely, the first qubit can be either the control qubit or the target qubit, and the same applies to the second qubit. When the second (first) qubit is a control qubit, while the first (second) qubit is a target, the phase of the state of the first (second) qubit is shifted by *e*^*iϕ*^ when the second (first) qubit is in the state , while nothing happens otherwise. Thus, the quantum circuit here is equivalent to the circuit illustrated in [Fig. 1](#f1){ref-type="fig"} for 2*θ*~*j*~ = *π*/2^*j*^ (*j* = 1, 2, \..., *n*).](srep21562-f2){#f2}

![(**a**) Diagram of a coupler qubit *A* and *n* cavities each hosting a qubit. A blue square represents a cavity while a green dot labels a qubit placed in each cavity, which can be an atom or a solid-state qubit. The coupler qubit *A* can be an atom or a quantum dot, and can also be a superconducting qubit capacitively or inductively coupled to each cavity. (**b**) Cavity *j* is dispersively coupled to qubit *j* (placed in cavity *j*) with coupling constant *g*~*j*~ and detuning *δ*~*j*~ \< 0. (**c**) The coupler qubit *A* dispersively interacts with cavity *j*, with coupling constant *g*~*Aj*~ and detuning *δ*~*Aj*~ \< 0 (*j* = 1, 2, \..., *n*). Here, *δ*~*Aj*~ = *δ*~*j*~, which holds for identical qubits *A* and *j*.](srep21562-f3){#f3}

![Setup of two cavities (1,2) connected by a superconducting transmon qubit *A*.\
Here, each cavity represents a one-dimensional coplanar waveguide transmission line resonator, qubit *A* is capacitively coupled to cavity *j* via a capacitance *C*~*j*~ (*j* = 1, 2). The two green dots indicate the two transmon qubits (1, 2) embedded in the two cavities, respectively. The interaction of qubits (1, 2) with their cavities is illustrated in [Fig. 5(a,b)](#f5){ref-type="fig"}, respectively. The interaction of qubit *A* with the two cavities is shown in [Fig. 5(c)](#f5){ref-type="fig"}. Due to three levels for each qubit considered in our analysis, each qubit is renamed as a qutrit in [Fig. 5](#f5){ref-type="fig"}.](srep21562-f4){#f4}

![Schematic diagram of qutrit-cavity interaction.\
(**a**) Cavity 1 is coupled to the transition with coupling strength *g*~1~ and detuning *δ*~1~, but far-off resonant with the transition of qutrit 1 with coupling strength and detuning . (b) Cavity 2 is coupled to the transition with coupling strength *g*~2~ and detuning *δ*~2~, but far-off resonant with the transition of qutrit 2 with coupling strength and detuning . (**c**) Cavity 1 (2) is coupled to the transition of qutrit *A* with coupling strength and detuning ; but far-off resonant with the transition of qutrit *A* with coupling strength and detuning . Here, and (*j* = 1, 2), where is the transition frequency of qutrit *j*, is the transition frequency of qutrit *A*, and is the frequency of cavity *j*.](srep21562-f5){#f5}

![Fidelity versus *δ*~1~/2*π*, plotted for different intercavity coupling strengths but without considering the systematic dissipation and dephasing for simplicity.](srep21562-f6){#f6}

![Fidelity versus *δ*~1~/2*π*, plotted for *g*~12~ = 0.01*g*~1~ and by taking the systematic dissipation and dephasing into account.\
The parameters used in the numerical simulation for [Figs 6](#f6){ref-type="fig"} and 7 are referred to the text.](srep21562-f7){#f7}
