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Abstract
We propose a one-step scheme for implementing multi-qubit phase gates on microwave photons in multiple resonators mediated by a superconducting bus in circuit quantum electrodynamics (QED) system. In the scheme, multiple single-mode resonators carry quantum information with their vacuum and single-photon Fock states, and a multi-level artificial atom acts as a quantum bus which induces the indirect interaction among resonators. The method of pulse engineering is used to shape the coupling strength between resonators and the bus so as to improve the fidelity and robustness of the scheme. We also discuss the influence of finite coherence time for the bus and resonators on gate fidelity respectively. Finally, we consider the suppression of unwanted transitions and propose the method of optimized detuning compensation for offsetting unwanted transitions, showing the feasibility of the scheme within the current experiment technology.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Multi-qubit phase gates play an important role and are a crucial element in quantum computation and quantum information processing (QCQIP). In general, there are two types of important multi-qubit gates, which have attracted attention during the past years. One consists of multiple control qubits simultaneously controlling on a single-target qubit [1,2]. The other contains a single qubit simultaneously controlling multi-target qubits [3–5]. These two types of multi-quantum gates are important in QCQIP, such as entanglement preparation [6], error correction [7,8], quantum algorithms [9], and quantum cloning [10].
Circuit quantum electrodynamics (QED) is analogue of cavity QED, which consists of superconducting qubits and microwave resonators or cavities. It is a specially suited platform to realize QCQIP owing to its flexibility, scalability, and tunability [11–14]. It was theoretically predicted earlier that the strong coupling can be readily achieved with superconducting charge qubits [12] or flux qubits [15]. It has been experimentally demonstrated that the strong and ultrastrong couplings between a superconducting qubit and a microwave resonator [16]. QED is now moving toward multiple superconducting qubits and multiple three-dimensional (3D) cavities with greatly enhanced coherence time, making them particularly appealing for large-scale quantum computing [16]. So far, it has been reported that the life of a superconducting resonator is between 1 and 10 ms [17–19]. Superconducting devices including cooper pair boxes, Josephson junctions, and superconducting quantum interference devices (SQUIDs) have been one of the most promising candidates for quantum computing [20–22]. Many schemes have been proposed to achieve a multi-qubit phase gate in circuit QED by encoding quantum information in levels of the artificial atoms [23–28]. Ye et al. [28] proposed a multiplex-controlled phase gate of $n-1$ control qubits simultaneously controlling one target qubit, with $n$ qubits placed in $n$ different cavities. This multi-qubit gate implementation in Ref. [28] is complex by using $2n+2$ basic operations. However, a microwave resonator can act as not only a good quantum data bus [11,29] but a good quantum memory [30] with the experimentally-reported high quality factor here. Schemes for realizing quantum phase gates by encoding quantum information in zero- and one-photon Fock states in cavity QED or circuit QED have been also presented [31–35]. Alternatively, two orthogonal a cat states of the single cavity mode also represented the two logical states of cat qubit in Ref. [34,35]. For the method proposed in [34], a multi-target-qubit controlled phase gate realized by one cat-state qubit (cqubit) simultaneously controlling $n-1$ target cqubits with the cat state qubits in the cavity acting the role of the quantum information carriers. This gate operation is quite simplified because only one-step operation is needed and neither classical pulse nor measurement is required in comparison with Ref. [28].
The focus of this work is on realizing single-step multi-qubit phase gates on multiple single-mode resonators mediated by a bus of a multi-level superconducting atom in circuit QED. On the one hand, in our model the superconducting bus induces the indirect interaction among multiple resonators, and the quantum information is encoded in vacuum- and single-photon states of multiple single-mode resonators other than Refs. [25–28]. On the other hand, it is noted that we use multiple single-mode resonators which is different from Refs. [31,32] that employed multiple modes of one resonator. Therefore, we can realize the distributed quantum computation via employing multiple single-mode resonators. Moreover, the multi-qubit phase gate is constructed by only one step in comparison with Ref. [28].
In addition, in order to enhance the fidelity and robustness of the scheme, the pulse engineering technique [36–40] can be used to shape the coupling strength among resonators and the bus, and then the realization of the multi-qubit phase gate loosens the strict operation time without a precise control. The tunable coupling strength plays a broad role in circuit QED system [41–52]. In experiment, the engineered coupling strength can be tunable by using controlled voltage pulses to modulate the flux threading the SQUID loop of qubit [44,53]. Besides, full tunability of coupling strength can also achieved by an magnetic flux to periodically modulate the frequency of qubits [46] or changing the coupler frequency between two qubits [52]. Finally, we consider the suppression of unwanted transitions and propose the method of optimized detuning compensation for offsetting unwanted transitions.
2. Description of the quantum system
2.1 Description of a superconducting bus
In this section, we would like to introduce the physical model of the system for the protocol, whose schematic diagram is shown in Fig. 1(a). Supposed that there are $n$ single-mode resonators capacitively coupled with a SQUID, i.e., an artificial atom named as a “bus". The Hamiltonian of this bus can be described as [15,54]
in which $C$ is the junction capacitance, $L$ the loop inductance, $Q$ the total change on the capacitor, $\Phi _0=h/2e$ the flux quantum, $E_J=I_c\Phi _0/2\pi$ the Josephson energy with $I_c$ being the critical current of the junction, $\Phi$ the magnetic flux threading the ring, and $\Phi _x$ the static external flux applied to the ring. There is an equation between $\Phi$ and $\Phi _x$, $\Phi =\Phi _x+LI_c$. In Eq. (1), the first term describes the charging energy of the Josephson junction, the second term the free energy of the Josephson junction, and the third term the electromagnetic energy stored in the loop. The potential energy part of the bus Hamiltonian can be written as [15,54]
Fig. 1. (a) Schematic diagram of the SQUID qubit defined as the bus coupled to $N$ resonators with capacitance. (b) Schematic diagram of the level configuration for the bus interacting with $n$ resonators.
2.2 A superconducting bus coupled with $n$ single-mode microwave resonator fields
Considering a superconducting bus coupled capacitively to $n$ single-mode microwave resonators. The superconducting bus holds a level structure as shown in Fig. 1(b) formed ground and excited levels, denoted by $|g_1\rangle , |g_2\rangle , |g_3\rangle \cdots |g_n\rangle$ and $|e_1\rangle , |e_2\rangle , |e_3\rangle \cdots |e_n\rangle$, respectively. The classical pulses applied to the bus drive the transitions resonantly between $|e_j\rangle$ and $|g_{j+1}\rangle$ with Rabi frequency $\Omega _j$, while $g_j$ is coupling strength which are coupled resonantly to transitions $|g_j\rangle \leftrightarrow |e_j\rangle$($j=1,2,3 \cdots n$) between $j$ resonators . $\Omega _j$ and $g_j$ are expressed by [15,54]
3. Construction of multi-qubit phase gate
In this section, we use the theoretical model to realize the multi-qubit phase gate and show the detailed derivations of effective Hamiltonian for achieving two qubit phase gate, three qubit phase gate and $n$-qubit phase gate.
3.1 Two qubit phase gate
Firstly, let us consider the case that the bus is coupled to two single-mode resonators. We denote the bus possessing a level structure as shown in Fig. 2(a), that is, two excited levels $|e_1\rangle$ and $|e_2\rangle$ and two lowest levels $|g_1\rangle$ and $|g_2\rangle$. A classical field drives the transition resonantly between $|e_1\rangle$ and $|g_2\rangle$ with Rabi frequency $\Omega _1$ , while two resonators are coupled resonantly to transitions $|g_1\rangle \leftrightarrow |e_1\rangle$ and $|g_2\rangle \leftrightarrow |e_2\rangle$ with coupling strength $g_1$ and $g_2$, respectively. The Hamiltonian of the bus interacting with two single-mode resonators can be written as
Fig. 2. (a) Schematic diagram of the level configuration for the bus interacting with two resonators. (b) Schematic diagram of the level configuration for the bus interacting with three resonators.
If two single-mode resonators are in $|1\rangle _{R1}|0\rangle _{R2}$, the evolution of the system will occur in a finite subspace {$|\phi _1\rangle =|g_1\rangle |1\rangle _{R1}|0\rangle _{R2}, |\phi _2\rangle =|e_1\rangle |0\rangle _{R1}|0\rangle _{R2}, |\phi _3\rangle =|g_2\rangle |0\rangle _{R1}|0\rangle _{R2}$}. In the presentation of the dressed states (i.e., eigenstates) of $H_{\Omega }=\Omega _1|\phi _2\rangle \langle \phi _3|+\textrm {H.c.}$, the Hamiltonian of the whole system becomes
When $|1\rangle _{R1}|1\rangle _{R2}$ is the initial state, the system evolves in the finite subspace {$|\varphi _1\rangle =|g_1\rangle |1\rangle _{R1}|1\rangle _{R2}$, $|\varphi _2\rangle =|e_1\rangle |0\rangle _{R1}|1\rangle _{R2}$, $|\varphi _3\rangle =|g_2\rangle |0\rangle _{R1}|1\rangle _{R2}$, $|\varphi _4\rangle =|e_2\rangle |0\rangle _{R1}|0\rangle _{R2}$}. Then in the presentation with respect to the dressed states of $H^{'}_{\Omega }=\Omega _1|\varphi _2\rangle \langle \varphi _3|+\textrm {H.c.}$, the Hamiltonian of the system reads
3.2 Three qubit phase gate
Now, we consider the quantum system with the bus interacting with three single-mode resonators. The Hamiltonian of the system in the interaction picture becomes
In the following we consider the time evolution under different initial states by setting the bus being $|g_1\rangle$ initially.
(1) If the initial state of the resonators is $|0\rangle _{R1}|0\rangle _{R2}|0\rangle _{R3}$, $|0\rangle _{R1}|1\rangle _{R2}|0\rangle _{R3}$, $|0\rangle _{R1}|0\rangle _{R2}|1\rangle _{R3}$ or $|0\rangle _{R1}|1\rangle _{R2}|1\rangle _{R3}$. There is no interaction between the resonator modes and bus due to no photon in the resonator 1 being absorbed initially by the bus, so the whole system has no evolution.
(2) For the initial state of the resonators being $|1\rangle _{R1}|0\rangle _{R2}|0\rangle _{R3}$, $|1\rangle _{R1}|1\rangle _{R2}|0\rangle _{R3}$ or $|1\rangle _{R1}|0\rangle _{R2}$
$\otimes |1\rangle _{R3}$, the system has the similar process of dynamic evolution with the case that the initial state of the two single-mode resonators is in $|1\rangle _{R1}|0\rangle _{R2}$ when only two resonators are considered. Therefore, the system is still not evolving.
(3) When the system is in $|g_1\rangle |1\rangle _{R1}|1\rangle _{R2}|1\rangle _{R3}$ initially, the system evolves in the finite subspace { $|\varphi ^{'}_1\rangle =|g_1\rangle |1\rangle _{R1}|1\rangle _{R2}|1\rangle _{R3}$, $|\varphi ^{'}_2\rangle =|e_1\rangle |0\rangle _{R1}|1\rangle _{R2}|1\rangle _{R3}$, $|\varphi ^{'}_3\rangle =|g_2\rangle |0\rangle _{R1}|1\rangle _{R2}|1\rangle _{R3}$, $|\varphi ^{'}_4\rangle =|e_2\rangle |0\rangle _{R1}|0\rangle _{R2}|1\rangle _{R3}$, $|\varphi ^{'}_5\rangle =|g_3\rangle |0\rangle _{R1}|0\rangle _{R2}|1\rangle _{R3}$, $|\varphi ^{'}_6\rangle =|e_3\rangle |0\rangle _{R1}|0\rangle _{R2}|0\rangle _{R3}$} . Considering the condition ${|\Omega _{1}|,|\Omega _{2}|}\gg {|g_1|, |g_2|, |g_3|}$, the interaction Hamiltonian in the presentation with respect to the dressed states of $H^{''}_{\Omega }=\Omega _1|\varphi ^{'}_2\rangle \langle \varphi ^{'}_3|+\Omega _2|\varphi ^{'}_4\rangle \langle \varphi ^{'}_5|+\textrm {H.c.}$ with the dressed states $|\psi _{\pm }\rangle =(|\varphi ^{'}_2\rangle \pm |\varphi ^{'}_3\rangle )/\sqrt 2$ and $|\psi ^{'}_{\pm }\rangle =(|\varphi ^{'}_4\rangle \pm |\varphi ^{'}_5\rangle )/\sqrt 2$ becomes
3.3 Multi-qubit phase gate
Multi-qubit gates can achieve large-scale quantum computing which have many applications in QCQIP. Thus, it is meaningful to construct the multi-qubit phase gate. In the following, we use the bus with the configuration of levels shown in Fig. 1(b) to couple to $n$ single-mode resonators. The interaction Hamiltonian of the whole system can be written as
By using the similar conditions mentioned above from Eq. (10) to Eq. (15), an effective Hamiltonian can be expressed as
where $g_\textrm {eff,n}=g_1g_2 \cdots g_n/\Omega _1\Omega _2 \cdots \Omega _{n-1}$, $|\psi _{n}\rangle =|e_{n}\rangle _{bus}|0\rangle _{R1}|0\rangle _{R2} \cdots |0\rangle _{Rn}$ and $|\psi _{1}\rangle =|g_{1}\rangle _{bus}|1\rangle _{R1}$ $\otimes |1\rangle _{R2} \cdots |1\rangle _{Rn}$. By choosing $t_n=\pi /g_\textrm {eff,n}$, we can obtain a multi-phase gate. Only if the system is in $|\psi _{1}\rangle$, then will the system have a $\pi$-phase flip while the other states remain unchanged.4. Simulation and analysis
4.1 Application of pulse engineering in shaping coupling strength
In order to verify the validity of the quantum gate constructed in the present scheme, we take an example of conducting the three-qubit phase gate. We numerically plot the evolution of population and phase of the state $|\varphi ^{'}_1\rangle$ with the full Hamiltonian Eq. (10). The value of coupling strength between the bus and resonators is chosen as approximate $2\pi \times 10$ MHz that can be easily achieved in experiment [56]. For convenience, we set $g_1/2\pi =g_2/2\pi =g_3/2\pi =10~$MHz. In realistic situation, learning from Eq. (3) we can realize this condition by adjusting the magnetic components of three resonators, even when three resonators are of largely different mode frequencies. For the condition $\{|\Omega _1|, |\Omega _2|\gg |g_1|, |g_2|, |g_3|\}$, we set $\Omega _1/2\pi =\Omega _2/2\pi =200~$MHz. As expected, the population of $|\varphi ^{'}_1\rangle$ is close to unity at the time $t^{'}_{II}$ ($20~\mu s$) with a perfect $\pi$-phase flip on $|\varphi ^{'}_1\rangle$. The results in Figs. 3(a) and 3(b) show that above theoretical analysis is correct. This implementation of the three-qubit phase gate on three resonators is based on the constant rectangular coupling strength. As shown in Fig. 3(a), the population of $|\varphi ^{'}_1\rangle$ reaches nearly $1$ and has a peak at the strict time of $t^{'}_{II}$ ($20~\mu s$) which means that the operating time requirement is very demanding. Besides, oscillation of the line is conspicuous by enlarging the range of $t\in [19,21]~\mu s$. There is a obvious decline for population from $99.8\%$ down to $97.2\%$ in respect to the operation time of $20 \mu s$ and $21 \mu s$ respectively.
Fig. 3. (a) Time evolution of population for $|\varphi ^{'}_1\rangle$. (b) Time evolution of phase for $|\varphi ^{'}_1\rangle$. Parameters: $g_1/2\pi =g_2/2\pi =g_3/2\pi =10~$MHz and $\Omega _1/2\pi =\Omega _2/2\pi =200~$MHz.
Seeking for robust dynamics and steady final quantum state, many efforts have been devoted to pulse engineering [36–40]. Inspired by pulse engineering, we replace the constant coupling strength by the shaped time-dependent coupling strength in order to strengthen the robustness against a control error. Tunable coupling strength has been widely used in circuit QED system [41–52]. Recently, experiments demonstrated that the coupling strength tunability is suitable for quantum simulation of many body physics [44,46,52]. For example, an experiment was performed by using two qubits and one coplanar waveguide (CPW) resonator on a microchip [44]. The coupling strength between the CPW resonator and qubits can be tunable through adopting controlled voltage pulses generated by an arbitrary waveform generator (AWG) to tune the flux threading the SQUID loop of each qubit individually using flux bias line [53]. In addition, the tunable coupling strength between two qubits can be also achieved by two different ways in [46,52]. In the first scheme, full tunability of the coupling strength can be achieved by parametrically modulating the qubits, that is, one of the qubit is biased by an ac magnetic flux to periodically modulate its frequency [46]. In the second scheme, the strength of the indirect coupling between a pair of nearest-neighor qubits is adjusted by changing the coupler frequency with an additional on-chip bias line, giving a net zero qubit-qubit coupling at a specific flux bias [52].
For the present scheme, the shape of the coupling strength can be engineered by using an AWG to tune the flux pulses threading the resonators corresponding to $\mathbf {B}^{j}_r(\vec {r},t)$ which is the magnetic components of the $j$th resonator mode in Eq. (3). Here, as long as the coupling strength is satisfied with $\int ^{t^{'}_{II}}_{0}g_\textrm {eff,3}dt=\pi$, we choose coupling strength as a single-period $\cos$-like function [setting $g_1=g_2=g_3=g(t)$]
where $g_m$ is the maximum amplitude.Based on $g(t)$ in Eq. (19), the operation time duration can be derived, as $t^{'}_{II}=16\Omega _1\Omega _2\pi /5g^{3}_m$. Then, we plot the time evolutions of population and phase for the state $|\varphi ^{'}_1\rangle$ by using the single-period $\cos$-like coupling strength, for which we pick up $g_m/2\pi =10~$MHz in Figs. 4(a) and 4(b). Obviously, the population not only reaches unity at the time $t^{'}_{II}$ but also keeps stationary during the time $t\in [45,65]~\mu s$. In addition, there is no oscillation once the population of $|\varphi ^{'}_1\rangle$ reaches nearly unity. By designing the coupling strength, the scheme of realizing the three-qubit phase gate loosens the demand of operation time.
Fig. 4. (a) Time evolution of population for $|\varphi ^{'}_1\rangle$ with single-period $\cos$-like function $g(t)$. (b) Time evolution of phase for $|\varphi ^{'}_1\rangle$ with $g(t)$. Parameters: $g_m/2\pi =10~$MHz and $\Omega _1/2\pi =\Omega _2/2\pi =200~$MHz.
For the sake of illustrating the validity and the robustness of the scheme by engineering the coupling strength, we define the fidelity of $\pi$-phase gate as $F(t)=|\langle \Psi _t|\Psi (t)\rangle |^2$, where $|\Psi _t\rangle$ is the target state after the $\pi$-phase gate on a general initial state $|\Psi (0)\rangle =|g_1\rangle _{bus}\otimes (\cos \alpha |0\rangle _{R1}+\sin \alpha |1\rangle _{R1})\otimes (\cos \beta |0\rangle _{R2}+\sin \beta |1\rangle _{R2})\otimes (\cos \gamma |0\rangle _{R3}+\sin \gamma |1\rangle _{R3})$ with $\alpha , \beta , \gamma \in [0,2\pi )$, and $|\Psi (t)\rangle$ is the state of the system at any time by solving the Schrödinger equation with the initial state $|\Psi (0)\rangle$. We plot the time evolution of the gate fidelity in Fig. 5 with two different cases of coupling strength by choosing five different initial states. As shown in Fig. 5(a), the $\pi$-phase gate can be achieved with the fidelity near unity at the final operating time with the constant coupling strength for the five different initial states. By magnifying the time range $t \in [18,20]~\mu s$, there are obvious oscillations of fidelity from $0.95$ to $1.00$. The operation time of the scheme is extremely rigorous and the robustness of the scheme by using constant coupling strength needs to improve. In Fig. 5(b), it is evident that the fidelity can remain over $0.999$ when the operation time $t \geq 39~\mu s$ by engineering the coupling strength. Here, we also plot the fidelity of five different initial states by setting the $y$-coordinate as $\log _{10}(1-F)$ and enlarging the time range $t \in [40, 42]~\mu s$. The fidelity fluctuation for the five initial states can be limited in the range approximately between $0.999$ and $0.9999$, which proves the validity and the robustness of the scheme and loosens the requirement of the strict operation time. In order not to lose the generality, alternatively we set unequal coupling strengths {$g_1/2\pi =10$ MHz, $g_2/2\pi =11$ MHz, $g_3/2\pi =15$ MHz} in Fig. 5(c) and {$g_1/2\pi =10$ MHz, $g_2/2\pi =11$ MHz, $g_m/2\pi =15$ MHz, $g_3/2\pi =g(t)$} in Fig. 5(d). It’s obvious that Figs. 5(c) and 5(d) have the same climate as Figs. 5(a) and 5(b).
Fig. 5. (a) and (c): Time evolution of $\pi$-gate fidelity with the constant coupling strength in five different case of initial state $|\Psi (0)\rangle$. (b) and (d): Time evolution of $\pi$-gate fidelity with the shaped coupling strength in five different cases of the initial state $|\Psi (0)\rangle$.
4.2 Effect of finite coherence time
We now give a brief discussion on the effect of finite coherence time on the scheme. We take into account the effect of the decoherence induced by the dissipation of the system on the final gate fidelity. The decohence mechanisms arise through two dominant channels: (i) Energy relaxations of excited states of the bus; (ii) Three resonators decay with their individual decay rates $\kappa _1, \kappa _2$ and $\kappa _3$. The dynamics of the lossy system is governed by the Markovian master equation
The fidelity of the gate operation is given by $F=|\langle \Psi _{id}|\rho |\Psi _{id}\rangle |$, where $|\Psi _{id}\rangle$ is the output state of an ideal system without dissipation of bus and decay of resonators. We now numerically simulate the fidelity of gate operation by solving the master equation Eq. (20). Figure 6 shows that the relationship between the fidelity $F$ and coherence time including coherence time of bus $\tau _b^{-1} \equiv \gamma _b$ and coherence time of resonators $\tau _R^{-1} \equiv \kappa$. As the coherence time increases, the fidelity of the gate also increases. Comparing the two lines, the fidelity is insensitive to the coherence time of bus ($\tau _b$) due to the invariable fidelity closed to unity when $\tau _b \geq 1$ ms. Obviously, the coherence time of resonators ($\tau _R$) has a greater influence on the fidelity. When $\tau _b=5$ ms ($\tau _R=5$ ms), the fidelity can reach $0.9983$ ($0.9836$). At the $10$ ms coherence time of the bus and resonators, the fidelities are $0.9985$ and $0.9909$, respectively. However, we have picked up the initial state whose fidelity is close to zero at the initial time in Fig. 5(b). If we consider other initial states or average fidelity, the fidelity will have a higher value. In experiment, by designing a $\pi$-phase difference across the Josephson junction in circuit in order to restrain the energy relaxation induced by quasiparticle dissipation, we can obtain a SQUID with coherence time over 1 ms [57]. Regarding to the coherence time of the resonator, the coherence time of the photons in the resonator can be much longer [58]. Up to now, the superconducting resonator lifetimes between 1 and 10 ms have been reported [17–19].
Fig. 6. Effect of the finite coherence time of the system on the final gate fidelity with the initial state. Parameters: $\alpha =0.24\pi$, $\beta =0.49\pi$ and $\gamma =0.40\pi$.
5. Suppression of unwanted transitions
The bus coupled to $n$ resonators has the complex and multiple level structure, which may cause unwanted transitions. In this section, for simplicity, we take an example of the two-qubit phase gate to show how we can suppress unwanted transitions.
5.1 Sufficiently large differences among transition frequencies
In above contents, we propose to realize multi-qubit gates by using resonant conditions. If the $n$th resonator is resonantly coupled to the $|g_n\rangle \leftrightarrow |e_n\rangle$ transition of the bus but highly decoupled from $|g_j\rangle \leftrightarrow |e_j\rangle$ ($\{j,n\}\in [1,2,3\cdots ]$, $j\neq n$) and classical fields drive resonantly transitions $|e_n\rangle \leftrightarrow |g_{n+1}\rangle$ but highly detuned from transitions between any two irrelevant levels, the probabilities of occurring unwanted transitions will be negligible. Note that these conditions can be satisfied by prior adjustment of the level spacings of the bus or/and the frequencies of resonators. For the superconducting bus, the level spacings can be readily adjusted by changing the external flux applied to the SQUID loop [59–61]. In addition, the frequency of a microwave resonator can be rapidly adjusted with a few nanoseconds [62,63].
In order to explore sufficiently large differences among transition frequencies in the four-level system given in Fig. 2(a) of the two-qubit phase gate, we set $\omega _{e_1,g_2}-\omega _{g_1,e_1}=\omega _{g_2,e_2}-\omega _{e_1,g_2}=\delta$ with $\omega _{g_1,e_1}$, $\omega _{e_1,g_2}$, and $\omega _{g_2,e_2}$ being the transition frequencies of $|g_1\rangle \leftrightarrow |e_1\rangle$, $|e_1\rangle \leftrightarrow |g_2\rangle$, and $|g_2\rangle \leftrightarrow |e_2\rangle$, respectively. For suppressing the unwanted transitions, $\delta$ is supposed to be as large as possible. In order to investigate numerically the effect of $\delta$ on the scheme, we consider the whole Hamiltonian including the unwanted transitions, as $\tilde {H}=H_\textrm {I,2}+H_\textrm {un}$ with the desired part $H_\textrm {I,2}$ Eq. (5) and the unwanted part
Fig. 7. (a) Final fidelity of the two-qubit gate with varying $\delta$ at the time $t=\pi \Omega _1/g_1g_2$. (b) Fidelity evolution of the two-qubit gate with $\delta /2\pi =25$ GHz. $g_1/2\pi =10$ MHz, $g_2/2\pi =11$ MHz, and $\Omega _1/2\pi =200$ MHz.
5.2 Optimized detuning compensation for suppressing unwanted transitions
Now we introduce the method of detuning compensation for suppressing unwanted transitions with $\delta$ being not large enough. Since the unwanted transitions induced by the quantum fields in resonators hardly effect the gate fidelity when $\delta /2\pi >1$ GHz, here we solely consider those induced by the classical field. There exist two unwanted transitions induced by the classical field, $|g_1\rangle \leftrightarrow |e_1\rangle$ and $|g_2\rangle \leftrightarrow |e_2\rangle$. When $\delta \gg \Omega _1$, these two unwanted transitions will cause second-order Stark shifts of four levels $|g_1\rangle$, $|e_1\rangle$, $|g_2\rangle$, and $|e_2\rangle$, respectively, as
Fig. 8. (a) Final fidelity of the two-qubit phase gate with varying $D_2$ and $D_3$ at the time $t=\pi \Omega _1/g_1g_2$. (b) Fidelity evolutions of the two-qubit phase gate with the detuning compensation and the optimized detuning compensation with $D_2/2\pi =-2$ MHz and $D_3/2\pi =-5.4$ MHz. $\delta /2\pi =10$ GHz. $g_1/2\pi =10$ MHz, $g_2/2\pi =11$ MHz, and $\Omega _1/2\pi =200$ MHz.
6. Conclusion
To summary, we have presented a scheme to realize multi-qubit phase gates on multiple resonators mediated by a superconducting bus in circuit QED system. Quantum information is loaded on the multiple single-mode resonators, and the superconducting bus mediates the interaction among resonators. Through introducing the shaped coupling strength, we improve the fidelity and robustness of the phase gate at the cost of a longer gate time. The effect of the decoherence induced by dissipation on the fidelity is taken into account, and the result proves that the scheme is very robust to the energy relaxation of the bus and is relatively sensitive to the decay of resonators. In addition, we propose the method of the optimized detuning compensation so as to suppress unwanted transitions.
Funding
National Natural Science Foundation of China (11675046); Program for Innovation Research of Science in Harbin Institute of Technology (A201412); Postdoctoral Scientific Research Developmental Fund of Heilongjiang Province (LBH-Q15060).
Disclosures
The authors declare no conflicts of interest.
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