Multi-target-qubit unconventional geometric phase gate in a multi-cavity system

Cavity-based large scale quantum information processing (QIP) may involve multiple cavities and require performing various quantum logic operations on qubits distributed in different cavities. Geometric-phase-based 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 multi-target-qubit unconventional geometric phase gate in a multi-cavity system. This multiqubit phase gate has a common control qubit but different target qubits distributed in different cavities, which can be achieved using a single-step 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 single-qubit operations, it can be converted into two types of significant multi-target-qubit 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.

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 ∏ | 〉 = i j n j 1 (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) shows that when the control qubit A is in the state + ( − ), a phase shift θ e i j 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 two-qubit phase gate on control qubit A and target qubit j, and (ii) the state transformation described by Eq. (21) 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) 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 which implies that when and only when the control qubit A is in the state − , a phase shift θ e i2 j 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 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 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 Greenberger-Horne-Zeilinger states 25 . 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 20,[23][24][25] . 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), 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) multi-target-qubit 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. 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 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 2θ j represents a phase shift exp(i2θ 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 2θ j = π, this gate corresponds to a multitarget-qubit phase gate (useful in QIP 20,23-25 ), 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. 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 + + → + + , , 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 for 2θ j = π/2 j (j = 1, 2, ..., n). 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)]. 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 Figure 3. (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.
Scientific RepoRts | 6:21562 | DOI: 10.1038/srep21562 each atom/quantum dot 64 . The two levels of coupler qubit A are denoted as g A and e A while those of intracavity qubit j as g j and e j (j = 1, 2, ···, n). A classical pulse is applied to qubit A and each intracavity qubit j [ Fig. 3 where ω is the pulse frequency and ω e g A ω ( ) e g j is the ↔ g e transition frequency of qubit A (qubit j). The system Hamiltonian in the interaction picture reads (in units of where † a j is the photon creation operator for the mode of cavity j, σ σ are the raising and lowering operators for qubit are detunings (with ω c j being the frequency of cavity j), Ω is the Rabi frequency of the pulse applied to each qubit, g A j (g j ) is the coupling constant of qubit A (j) with cavity j. We choose as the rotated basis states of qubit j and qubit A, respectively.

In a new interaction picture under the Hamiltonian
, one can apply a rotating-wave 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 where , H eff j 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.
Scientific RepoRts | 6:21562 | DOI: 10.1038/srep21562 Implementing multiqubit UG phase gates. Consider a system consisting of the coupler qubit A and an intracavity qubit j, for which ± j ( ± ) A are eigenstates of the operator σ  z j σ ( )  z A with eigenvalues ± 1. In the rotated basis + + , , the Hamiltonian (9) can be rewritten as   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 , which can be further written as where we have set Ω T j = kπ (k is a positive integer). For T j = 2m 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)]. From Eqs (8) and (9), 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 operator where U Aj (T j ) is the unitary operator given in Eq. (16), which characterizes the cyclic evolution of a two-qubit subsystem (i.e., qubit A and intracavity qubit j) in the rotated basis + + , resulting from Eq. (17). Hence, one can easily find from Eqs (18) and (19) 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), 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). 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 2 . According to Eqs (18) and (19), one can obtain a three-qubit UG phase gate, which is described by  As discussed in the introduction, by applying single-qubit operations, this three-qubit UG phase gate described by Eq. (21) can be converted into a three-qubit phase gate which is illustrated in the above-mentioned 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 three-qubit 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 δ δ / = / g g 1 2 1 2 2 2 , 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 solid-state 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 one-dimensional transmission line resonator (TLR). We consider a setup in Fig. 4 for achieving a three-qubit UG phase gate. To be more realistic, we consider a third higher level f of each transmon qubit during the entire operation because this level f may be excited due to the ↔ e f 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 ↔ e f transition of each qutrit are considered, the Hamiltonian (3)  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) describe unwanted off-resonant couplings between the pulse and the ↔ e f transition of each qutrit, where Ω ∼ is the pulse Rabi frequency. Note that the Hamiltonian (23) does not involves ↔ g f transition of each qutrit, since this transition is negligible because of ω ω ω ω , ,  c f g fg j j A (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  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), respectively. The interaction of qubit A with the two cavities is shown in Fig. 5(c). Due to three levels for each qubit considered in our analysis, each qubit is renamed as a qutrit in Fig. 5.
The condition g 12 = 0.01g 1 is easy to satisfy with the cavity-qutrit capacitive coupling shown in Fig. 4. 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) 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 e can be made to be on the order of 55-60 μs for state-of-the-art transom devices coupled to a one-dimensional TLR 80 and the order of 20-80 μs for a transom coupled to a three-dimensional microwave resonator 81 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 κ − 1 1 and κ − 2 1 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 e f transition frequency of qutrit A, and ω c j is the frequency of cavity j.
resonators with internal quality factors above one million (Q > 10 7 ) have also been reported 85 . 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 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  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.