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]

$$H_{bus}=\frac{Q^2}{2C}+\frac{(\Phi-\Phi_x)^2}{2L}-E_J\cos\frac{2\pi\Phi}{\Phi_0},$$
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]
$$V=V_0\left\{\frac{1}{2}\left[\frac{2\pi(\Phi-\Phi_x)}{\Phi_0}\right]-\frac{E_J}{V_0}\cos\left(2\pi\frac{\Phi}{\Phi_0}\right)\right\},$$
with $V_0=\Phi _0^2/(4\pi ^2L)$. The height of potential barrier in the energy level structure of the bus can be adjusted by changing $E_J$, and the symmetry of potential well can also be adjusted by changing the external magnetic flux $\Phi _x$.

 

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.

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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]

$$\begin{aligned} \Omega_j&=\frac{1}{2L\hbar}\langle g_{j+1}|\Phi|e_j\rangle\int_S \mathbf{B}^{j}_{\mu\omega}(\vec{r},t)\cdot d\mathbf{S},\\ g_j&=\frac{1}{L}\sqrt{\frac{\omega_j}{2\mu_0\hbar}}\langle g_j|\Phi|e_j\rangle\int_S \mathbf{B}^{j}_r(\vec{r},t)\cdot d\mathbf{S}, \end{aligned}$$
where $j=1,2,3 \cdots n$, $S$ the surface bounded by the loop of the bus, $\omega _j$ the resonator frequencies of resonator $j$th. The magnetic component of $j$th classical microwave applied to the bus is given by $\mathbf {B}^{j}_{\mu \omega }(\vec {r},t)=\mathbf {\tilde {B}}^{j}_{\mu \omega }(\vec {r},t)\cos 2\pi \nu _{\mu m}t$, in which $\nu _{\mu m}=\omega _{\mu \omega }/2\pi$ and $\omega _{\mu \omega }$ is the frequency of the microwave pulse. Thus, $\mathbf {\tilde {B}}^{j}_{\mu \omega }(\vec {r},t)$ is the maximum amplitude of magnetic component. Accordingly, $\mathbf {B}^{j}_r(\vec {r},t)$ is the the magnetic components of the $j$th resonator mode. For a standing-wave resonators, $\mathbf {B}^{j}_r(\vec {r},t)=\mu _0\sqrt {2/V_j}\cos k_j z_j$ ($k_j$ is the wave number of $j$th resonator, $V_j$ and $z_j$ are the $j$th resonator volume and the $j$th resonator axis).

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

$$\begin{aligned} H&=H_0+H_i,\\ H_0&=\omega_1 a^\dagger_1 a_1+\omega_2 a^\dagger_2 a_2+\sum_{l=g,e}\sum_{j=1,2}\omega_{lj}|l_j\rangle\langle l_j|,\\ H_i&=\sum^{2}_{n=1}g_na_n|e_n\rangle\langle g_n|+\Omega_1|e_1\rangle\langle g_2|e^{{-}i\omega_{L1}t}+\textrm{H.c.}, \end{aligned}$$
where $\omega _{lj}$ is the frequency of state $|l_j\rangle$ for the bus, $a_1(a_2)$ and $a^\dagger _1(a^\dagger _2)$ the annihilation and creation operators of resonator $1$ (resonator $2$) respectively, and $\omega _{L1}$ the classical filed frequency. In the interaction picture with respect to the unitary transformation $\exp (-iH_0t)$, the interaction Hamiltonian can be written as
$$H_{\textrm{I},2}=\sum^{2}_{n=1}g_na_n|e_n\rangle\langle g_n|+\Omega_1|e_1\rangle\langle g_2|+{\textrm H.c.},$$
for which we have considered resonant conditions $\omega _{ej}-\omega _{gj}=\omega _{j}$ and $\omega _{e1}-\omega _{g2}=\omega _{L1}$. Considering the time evolution under different initial states, the bus is set to be in the $|g_1\rangle$ state initially and the two single-mode resonators in the Fock state subspace {$|0\rangle _{R1}|0\rangle _{R2}$, $|0\rangle _{R1}|1\rangle _{R2}$, $|1\rangle _{R1}|0\rangle _{R2}$, $|1\rangle _{R1}|1\rangle _{R2}$}. For $|0\rangle _{R1}|0\rangle _{R2}$ and $|0\rangle _{R1}|1\rangle _{R2}$ being the initial states of two single-mode resonators, the whole system has no evolution and the transition between the bus and resonator $1$ is prohibited owing to no photon in the resonator $1$ being absorbed initially by the bus for the excitation $|g_1\rangle \rightarrow |e_1\rangle$.

 

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.

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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

$$H_{\Pi,2}=\frac{g_1}{\sqrt{2}}\left(e^{i\Omega_1t}|\Phi_+\rangle+e^{{-}i\Omega_1t}|\Phi_-\rangle\right)\langle\phi_1|+\textrm{H.c.},$$
in which $|\Phi _{\pm }\rangle =(|\phi _2\rangle \pm |\phi _3\rangle )/\sqrt 2$ are the dressed states of $H_{\Omega }$ corresponding to the eigenvalues $\pm \Omega _1$. Based on the Hamiltonian $H_{\Pi ,2}$, the large energy splitting between dressed states $|\Phi _{\pm }\rangle$ will lead to the very slow energy exchanges between $|\phi _1\rangle$ and $|\Phi _{\pm }\rangle$ if we set the parameter condition $|\Omega _{1}|\gg |g_1|,|g_2|$. The Stark shifts of $|\phi _1\rangle$ induced by the two strongly-dispersive interactions between $|\phi _1\rangle$ and $|\Phi _{\pm }\rangle$ will offset each other, and thus the Hamiltonian $H_{\Pi ,2}$ is invalid for the evolution of the system with $|1\rangle _{R1}|0\rangle _{R2}$ being the initial state.

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

$$\begin{aligned} H^{'}_{\Pi,2}&=\frac{g_1}{\sqrt2}(e^{i\Omega_1 t}|\Psi_+\rangle +e^{{-}i\Omega_1t}|\Psi_-\rangle ) \langle\varphi_1|\\ &+\frac{g_2}{\sqrt2}(e^{i\Omega_1t}|\Psi_+\rangle -e^{{-}i\Omega_1t}|\Psi_-\rangle )\langle\varphi_4|+\textrm{H.c.}, \end{aligned}$$
in which $|\Psi _\pm \rangle =(|\varphi _2\rangle \pm |\varphi _3\rangle )/\sqrt 2$ are the dressed states of $H^{'}_{\Omega }$ corresponding to the eigenvalues $\pm \Omega _1$. For the further simplification of the $H^{'}_{\Pi ,2}$, we consider the condition $|\Omega _1|\gg |g_1|,|g_2|$, so the Hamiltonian of the system can be developed as [55]
$$H_{{\textrm eff}}=g_{{\textrm eff}}|\varphi_4\rangle\langle\varphi_1|+\textrm{H.c.}$$
with the effective coupling constant $g_{\textrm {eff}}=g_1g_2/\Omega _1$. Therefore, with the choice of the operation time duration $t_I=\pi /g_{\textrm {eff}}$, we can easily obtain
$$\begin{aligned} |g_1\rangle|0\rangle_{R1}|0\rangle_{R2}&\rightarrow|g_1\rangle|0\rangle_{R1}|0\rangle_{R2},\\ |g_1\rangle|0\rangle_{R1}|1\rangle_{R2}&\rightarrow|g_1\rangle|0\rangle_{R1}|1\rangle_{R2},\\ |g_1\rangle|1\rangle_{R1}|0\rangle_{R2}&\rightarrow|g_1\rangle|1\rangle_{R1}|0\rangle_{R2},\\ |g_1\rangle|1\rangle_{R1}|1\rangle_{R2}&\rightarrow-|g_1\rangle|1\rangle_{R1}|1\rangle_{R2}. \end{aligned}$$
It is noted that the state $|g_1\rangle |1\rangle _{R1}|1\rangle _{R2}$ has a $\pi$-phase flip, while the other three states remain invariable. So we finally obtain a $\pi$-phase gate between two single-mode resonators.

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

$$H_\textrm{I,3}=\sum^{3}_{m=1}g_ma_m|e_m\rangle\langle g_m|+\sum^{2}_{n=1}\Omega_n|e_n\rangle\langle g_{n+1}|+\textrm{H.c.}$$
in which $g_m$ is coupling strength between resonator $m$ and the bus transition $|e_m\rangle \leftrightarrow |g_m\rangle$, and $\Omega _n$ is the Rabi frequency of the $n$th classical field driving resonantly the bus transition $|e_n\rangle \leftrightarrow |g_{n+1}\rangle$.

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

$$\begin{aligned} H_{\Pi,3}&=\frac{g_1}{\sqrt2}e^{i\Omega_1t}\left(|\psi_+\rangle\langle\varphi^{'}_1|+|\varphi^{'}_1\rangle\langle\psi_-|\right)+\frac{g_2}{2}e^{i(\Omega_1-\Omega_2)t}\left(|\psi_+\rangle\langle \psi^{'}_+|-|\psi^{'}_-\rangle\langle\psi_-|\right)\\ &+\frac{g_2}{2}e^{i(\Omega_1+\Omega_2)t}\left(|\psi_+\rangle\langle \psi^{'}_-|-|\psi^{'}_+\rangle\langle\psi_-|\right) +\frac{g_3}{\sqrt2}e^{i\Omega_2t}\left(|\psi^{'}_+\rangle\langle\varphi^{'}_6|-|\varphi^{'}_6\rangle\langle\psi^{'}_-|\right)\\ &+\textrm{H.c}. \end{aligned}$$
From $H_{\Pi ,3}$, the resonant transition $|\varphi ^{'}_1\rangle \leftrightarrow |\varphi ^{'}_6\rangle$ is mediated by $|\psi _{\pm }\rangle$ and $|\psi ^{'}_{\pm }\rangle$, and an effective coupling can be obtained by the following three-order perturbation with the effective coupling strength [32], labeled as $g_\textrm {eff,3}$
$$\begin{aligned} &\frac{\langle \varphi^{'}_6|H_{\Pi,3}|\psi^{'}_+\rangle\langle \psi^{'}_+| H_{\Pi,3}|\psi_+\rangle\langle\psi_+|H_{\Pi,3}|\phi^{'}_1\rangle}{\Omega_1\Omega_2}-\frac{\langle \varphi^{'}_6|H_{\Pi,3}|\psi^{'}_-\rangle\langle \psi^{'}_-| H_{\Pi,3}|\psi_+\rangle\langle\psi_+|H_{\Pi,3}|\phi^{'}_1\rangle}{\Omega_1\Omega_2}\\ &-\frac{\langle \varphi^{'}_6|H_{\Pi,3}|\psi^{'}_+\rangle\langle \psi^{'}_+| H_{\Pi,3}|\psi_-\rangle\langle\psi_-|H_{\Pi,3}|\varphi^{'}_1\rangle}{\Omega_1\Omega_2}+\frac{\langle \varphi^{'}_6|H_{\Pi,3}|\psi^{'}_-\rangle\langle \psi^{'}_-| H_{\Pi,3}|\psi_-\rangle\langle\psi_-|H_{\Pi,3}|\phi^{'}_1\rangle}{\Omega_1\Omega_2}\\ &=\frac{g_1g_2g_3}{\Omega_1\Omega_2}. \end{aligned}$$
However, Stark shifts of $|\varphi ^{'}_1\rangle$ and $|\varphi ^{'}_6\rangle$ are zero because the stark shift from the $|\psi _+\rangle$ ($|\psi ^{'}_+\rangle$) balances out the Stark shift from the $|\psi _-\rangle$ ($|\psi ^{'}_-\rangle$) which can be proved by,
$$\frac{\langle \varphi^{'}_1|H_{\Pi,3}|\psi_+\rangle\langle \psi_+| H_{\Pi,3}|\varphi^{'}_1\rangle}{-\Omega_1}+\frac{\langle \varphi^{'}_1|H_{\Pi,3}|\psi_-\rangle\langle \psi_-| H_{\Pi,3}|\varphi^{'}_1\rangle}{\Omega_1}=0,$$
$$\frac{\langle \varphi^{'}_6|H_{\Pi,3}|\psi^{'}_+\rangle\langle \psi^{'}_+| H_{\Pi,3}|\varphi^{'}_6\rangle}{-\Omega_2}+\frac{\langle \varphi^{'}_6|H_{\Pi,3}|\psi^{'}_-\rangle\langle \psi^{'}_-| H_{\Pi,3}|\varphi^{'}_6\rangle}{\Omega_2}=0.$$
Thus, we obtain the effective Hamiltonian of the whole system when the system is in $|\varphi ^{'}_1\rangle$ initially,
$$H^{'}_\textrm{eff,3}=g_\textrm{eff,3}|\varphi^{'}_6\rangle\langle\varphi^{'}_1|+\textrm{H.c}.$$
If the state is in $|g_1\rangle |1\rangle _{R1}|1\rangle _{R2}|1\rangle _{R3}$, the system undergoes a Rabi oscillation with an effective Rabi frequency $g_\textrm {eff,3}$ while other states remain unchanged. By selecting the operation time duration $t^{'}_{II}=\pi /g_\textrm {eff,3}$, therefore we can obtain
$$|g_1\rangle|x\rangle_{R1}|y\rangle_{R2}|z\rangle_{R3}\rightarrow e^{ixyz\pi}|g_1\rangle|x\rangle_{R1}|y\rangle_{R2}|z\rangle_{R3},$$
in which $x, y, z=0,1$. This is a $\pi$-phase gate between three single-mode resonators.

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

$$H_\textrm{I,n}=\sum^{n}_{i=1}g_ia_i|e_i\rangle\langle g_i|+\sum^{n-1}_{j=1}\Omega_{j}|e_{j}\rangle\langle g_{j+1}|$$
in which $g_i$ is coupling strength between resonator $i$ and the $|e_i\rangle \rightarrow |g_i\rangle$ transition of the bus, and $\Omega _j$ is the Rabi frequency of the $j$th classical field for the $|e_j\rangle \leftrightarrow |g_{j+1}\rangle$ transition of the bus.

By using the similar conditions mentioned above from Eq. (10) to Eq. (15), an effective Hamiltonian can be expressed as

$$H_\textrm{eff,n}=g_\textrm{eff,n}|\psi_n\rangle\langle \psi_1|+\textrm{H.c}$$
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.

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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 $g_1=g_2=g_3=g(t)$]

$$g(t)=\frac{g_m}{2}\left[1-\cos\left(\frac{2\pi t}{t^{'}_{II}}+\frac{\pi}{3}\right)\right]$$
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.

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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 $|\Psi (0)\rangle$.

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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

$$\begin{aligned} \frac{d\rho}{dt}&=-i\Big[H_{II},~\rho\Big]+\sum^{3}_{i=1}\kappa_i D[a_i]+\gamma_{g_1,e_1}D[\sigma_1]+\gamma_{g_2,e_1}D[\sigma_2]\\ &+\gamma_{g_2,e_2}D[\sigma_3]+\gamma_{g_3,e_2}D[\sigma_4]+\gamma_{g_3,e_3}D[\sigma_5], \end{aligned}$$
where $\rho$ is the density operator of the system, $\sigma _1=|g_1\rangle \langle e_1|$, $\sigma _2=|g_2\rangle \langle e_1|$, $\sigma _3=|g_2\rangle \langle e_2|$, $\sigma _4=|g_3\rangle \langle e_2|$, $\sigma _5=|g_3\rangle \langle e_3|$, $D[A]=A\rho A^{+}-A^{+}A\rho /2-\rho A^{+}A/2$ with $A=a_i$ or $\sigma _j$. Parameters of $\gamma _{g_1,e_1}, \gamma _{g_2,e_1}, \gamma _{g_2,e_2}, \gamma _{g_3,e_2},$ and $\gamma _{g_3,e_3}$ are energy relaxation rates of excited states $|e_1\rangle$, $|e_2\rangle$ and $|e_3\rangle$ for the bus. For convenience, we assume that $\kappa _1=\kappa _2=\kappa _3=\kappa$ and $\gamma _{g_1,e_1}=\gamma _{g_2,e_1}=\gamma _{g_2,e_2}=\gamma _{g_3,e_2}=\gamma _{g_3,e_3}=\gamma _b$.

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 $\gamma =0.40\pi$.

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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

$$\begin{aligned} H_\textrm{un}&=\Big(\Omega_1e^{i\delta t}+g_2e^{2i\delta t}a^{{\dagger}}_2\Big)|g_1\rangle\langle e_1|+\Big(g_1e^{i\delta t}a_1+g_2e^{{-}i\delta t}a_2\Big)|e_1\rangle\langle g_2|\\ &+\Big(\Omega_1e^{{-}i\delta t}+g_1e^{{-}2i\delta t}a^{{\dagger}}_1\Big)|g_2\rangle\langle e_2|+\textrm{H.c}, \end{aligned}$$
According to the Schrödinger equation of the Hamiltonian $\tilde {H}=H_\textrm {I,2}+H_\textrm {un}$, Fig. 7(a) shows the final fidelity of the two-qubit gate with varying $\delta$, for which we choose the initial state as $|\psi (0)\rangle =|g_1\rangle _{bus}\otimes (\cos \theta |0\rangle _{R1}+\sin \theta |1\rangle _{R1})\otimes (\cos \eta |0\rangle _{R2}+\sin \eta |1\rangle _{R2})$ with $\theta =0.24\pi$ and $\eta =0.49\pi$. In Fig. 7(a), we also simulate the separate effect of the unwanted transitions induced by the quantum fields in of resonators or the classical field. We learn from Fig. 7(a) that the unwanted transitions induced by the quantum fields have a puny influence on the fidelity and when $\delta /2\pi >1$ GHZ their effect can be neglected. The unwanted transitions induced by the classical fields play a major role in the destruction of the gate fidelity because in Fig. 7(a) the dotted green line and the solid red line basically coincide. Overall, the gate fidelity is very close to unity under the condition of $\delta /2\pi >20$ GHz. To this end, in Fig. 7(b) we pick up $\delta /2\pi =25$ GHz and plot the fidelity evolution of the gate fidelity. The two-qubit $\pi$-phase gate can be achieved finally with the fidelity near unity (over $0.9975$). Therefore, we can choose suitable differences among the transition frequencies to suppress the unwanted transitions.

 

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.

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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

$$\delta_1=\frac{\Omega_1^2}{\delta},\quad \delta_2={-}\frac{\Omega_1^2}{\delta},\quad \delta_3={-}\frac{\Omega_1^2}{\delta},\quad \delta_4=\frac{\Omega_1^2}{\delta}.$$
Then these Stark shifts will change the three desired resonant transitions, $|g_1\rangle \leftrightarrow |e_1\rangle$, $|e_1\rangle \leftrightarrow |g_2\rangle$, and $|g_2\rangle \leftrightarrow |e_2\rangle$ described by Eq. (5), into off-resonant transitions with detunings $|\delta _1-\delta _2|$, $|\delta _2-\delta _3|$, and $|\delta _3-\delta _4|$, respectively. Therefore, in order to compensate for the Stark shifts, we can reversely introduce detunings to modify Eq. (5) into
$$H'_{\textrm{I},2}=g_1e^{i\delta_{12}t}a^{{\dagger}}_1|g_1\rangle\langle e_1|+\Omega_1e^{i\delta_{23} t}|e_1\rangle\langle g_2|+g_2e^{i\delta_{34}t}a^{{\dagger}}_2|g_2\rangle\langle e_2|+{\textrm H.c}.$$
In order to eliminate the second-order Stark shifts in Eq. (22), one should choose
$$\delta_{12}=\delta_2-\delta_1,\quad\delta_{23}=\delta_3-\delta_2,\quad\delta_{34}=\delta_4-\delta_3.$$
However, since the effective coupling strength $g_1g_2/\Omega _1$ is small, higher-order Stark shifts may also influence the gate fidelity. Therefore, the relations in Eq. (24) may be not correct sufficiently. To this end, it is necessary to seek for the optimized detuning compensation for suppressing unwanted transitions. Here we use the numerical search algorithm to optimize detunings in Eq. (24). We first change the parameters in Eq. (22) into
$$\delta_1=\frac{\Omega_1^2}{\delta},\quad \delta_2= D_2,\quad \delta_3= D_3,\quad \delta_4=\frac{\Omega_1^2}{\delta}.$$
Then we search numerically for the suitable values of $D_2$ and $D_3$ by scanning the range around $-{\Omega _1^2}/{\delta }$ so that the detunings defined in Eq. (24) can ensure the high-fidelity gate. For instance, with $\delta /2\pi =10$ GHz that is not large enough to ensure the high-fidelity gate [see Fig. 7(a)] and the Hamiltonian $\tilde {H}^{'}=H'_\textrm {I,2}+H_\textrm {un}$, in Fig. 8(a) we simulate the final ($t=\pi \Omega _1/g_1g_2$) fidelity of the two-qubit phase gate with varying $D_2$ and $D_3$, for which we have made the preliminary work to know the rough ranges of $D_2$ and $D_3$ that ensure the high fidelity through the numerical search algorithm. There exists a range that guarantees the high-fidelity (over 0.9975) two-qubit gate. Further, in Fig. 8(b) we compare the cases of the detuning compensation and the optimized detuning compensation by plotting the fidelity evolutions of the two-qubit phase gate. In Fig. 8(b), the gate fidelity for the case of the optimized detuning compensation reaches over 0.9975 finally, which indicates that the unwanted transitions have been efficiently suppressed. For the multi-qubit phase gates, the similar processing can also be conducted.

 

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.

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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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46. X. Li, Y. Ma, J. Han, T. Chen, Y. X W. Cai, H. Wang, Y. P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Y. Sun, “Perfect quantum state transfer in a superconducting qubit chain with parametrically tunable couplings,” Phys. Rev. Appl. 10(5), 054009 (2018). [CrossRef]  

47. T. Chen and Z. Y. Xue, “Nonadiabatic geometric quantum computation with parametrically tunable coupling,” Phys. Rev. Appl. 10(5), 054051 (2018). [CrossRef]  

48. Y. l. Wu, L.-P. Yang, M. Gong, Y. r. Zheng, H. Deng, Z. g. Yan, Y. j. Zhao, K. q. Huang, A. D. Castellano, W. J. Munro, K. Nemoto, D.-N. Zheng, C. P. Sun, Y. -x. Liu, X. b. Zhu, and L. Lu, “An efficient and compact switch for quantum circuits,” NPJ Quantum Inf. 4(1), 50 (2018). [CrossRef]  

49. S. A. Caldwell, N. Didier, C. A. Ryan, E. A. Sete, A. Hudson, P. Karalekas, R. Manenti, M. P. da Silva, R. Sinclair, E. Acala, N. Alidoust, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, K. Kuang, M. Lenihan, T. Manning, A. Marchenkov, J. Marshall, R. Maydra, Y. Mohan, W. O’Brien, C. Osborn, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, G. Prawiroatmodjo, V. Rawat, M. Reagor, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Scheer, M. Selvanayagam, R. Smith, A. Staley, M. Suska, N. Tezak, D. C. Thompson, T.-W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. Rigetti, “Parametrically activated entangling gates using transmon qubits,” Phys. Rev. Appl. 10(3), 034050 (2018). [CrossRef]  

50. M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroatmodjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. d. Silva, E. Acala, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, S. Caldwell, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. E. Bouayadi, D. Girshovich, S. Hong, A. Hudson, P. Karalekas, K. Kuang, M. Lenihan, R. Manenti, T. Manning, J. Marshall, Y. Mohan, W. Brien, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, V. Rawat, C. A. Ryan, R. Renzas, N. Rubin, D. Russel, M. Rust, D. Scarabelli, M. Selvanayagam, R. Sinclair, R. Smith, M. Suska, T. -W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. T. Rigetti, “Demonstration of universal parametric entangling gates on a multi-qubit lattice,” Sci. Adv. 4(2), eaao3603 (2018). [CrossRef]  

51. O. Kyriienko and A. S. Sørensen, “Floquet quantum simulation with superconducting qubits,” Phys. Rev. Appl. 9(6), 064029 (2018). [CrossRef]  

52. F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, Fernando G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrá, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574(7779), 505–510 (2019). [CrossRef]  

53. L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Demonstration of two-qubit algorithms with a superconducting quantum processor,” Nature 460(7252), 240–244 (2009). [CrossRef]  

54. C.-P. Yang, S. Chu, and S. y. Han, “Possible realization of entanglement, logical gates, and quantum-information transfer with superconducting-quantum-interference-device qubits in cavity QED,” Phys. Rev. A 67(4), 042311 (2003). [CrossRef]  

55. D. F. V. James and J. Jerke, “Effective Hamiltonian theory and its applications in quantum information,” Can. J. Phys. 85(6), 625–632 (2007). [CrossRef]  

56. P. Mundada, G. Zhang, T. Hazard, and A. Houck, “Suppression of qubit crosstalk in a tunable coupling superconducting circuit,” Phys. Rev. Appl. 12(5), 054023 (2019). [CrossRef]  

57. I. M. Pop, K. Geerlings, G. Catelani, R. J. Schoelkopf, L. I. Glazman, and M. H. Devoret, “Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles,” Nature 508(7496), 369–372 (2014). [CrossRef]  

58. X. Gu, A. F. Kockum, A. Miranowicz, Y. X. Liu, and F. Nori, “Microwave photonics with superconducting quantum circuits,” Phys. Rep. 718-719, 1–102 (2017). [CrossRef]  

59. M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O’Connell, H. Wang, A. N. Cleland, and J. M. Martinis, “Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state,” Nat. Phys. 4(7), 523–526 (2008). [CrossRef]  

60. J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42–47 (2005). [CrossRef]  

61. C.-P. Yang, S.-B. Zheng, and F. Nori, “Multiqubit tunable phase gate of one qubit simultaneously controlling n qubits in a cavity,” Phys. Rev. A 82(6), 062326 (2010). [CrossRef]  

62. M. Sandberg, C. M. Wilson, F. Persson, T. Bauch, G. Johansson, V. Shumeiko, T. Duty, and P. Delsing, “Tuning the field in a microwave resonator faster than the photon life time,” Appl. Phys. Lett. 92(20), 203501 (2008). [CrossRef]  

63. Z. L. Wang, Y. P. Zhong, L. J. He, H. Wang, J. M. Martinis, A. N. Cleland, and Q. W. Xie, “Quantum state characterization of a fast tunable superconducting resonator,” Appl. Phys. Lett. 102(16), 163503 (2013). [CrossRef]  

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2017 (2)

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H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Observation of high coherence in josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. 107(24), 240501 (2011).
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S. A. Caldwell, N. Didier, C. A. Ryan, E. A. Sete, A. Hudson, P. Karalekas, R. Manenti, M. P. da Silva, R. Sinclair, E. Acala, N. Alidoust, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, K. Kuang, M. Lenihan, T. Manning, A. Marchenkov, J. Marshall, R. Maydra, Y. Mohan, W. O’Brien, C. Osborn, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, G. Prawiroatmodjo, V. Rawat, M. Reagor, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Scheer, M. Selvanayagam, R. Smith, A. Staley, M. Suska, N. Tezak, D. C. Thompson, T.-W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. Rigetti, “Parametrically activated entangling gates using transmon qubits,” Phys. Rev. Appl. 10(3), 034050 (2018).
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F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, Fernando G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrá, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574(7779), 505–510 (2019).
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Rigetti, C.

S. A. Caldwell, N. Didier, C. A. Ryan, E. A. Sete, A. Hudson, P. Karalekas, R. Manenti, M. P. da Silva, R. Sinclair, E. Acala, N. Alidoust, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, K. Kuang, M. Lenihan, T. Manning, A. Marchenkov, J. Marshall, R. Maydra, Y. Mohan, W. O’Brien, C. Osborn, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, G. Prawiroatmodjo, V. Rawat, M. Reagor, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Scheer, M. Selvanayagam, R. Smith, A. Staley, M. Suska, N. Tezak, D. C. Thompson, T.-W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. Rigetti, “Parametrically activated entangling gates using transmon qubits,” Phys. Rev. Appl. 10(3), 034050 (2018).
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Rigetti, C. T.

M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroatmodjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. d. Silva, E. Acala, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, S. Caldwell, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. E. Bouayadi, D. Girshovich, S. Hong, A. Hudson, P. Karalekas, K. Kuang, M. Lenihan, R. Manenti, T. Manning, J. Marshall, Y. Mohan, W. Brien, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, V. Rawat, C. A. Ryan, R. Renzas, N. Rubin, D. Russel, M. Rust, D. Scarabelli, M. Selvanayagam, R. Sinclair, R. Smith, M. Suska, T. -W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. T. Rigetti, “Demonstration of universal parametric entangling gates on a multi-qubit lattice,” Sci. Adv. 4(2), eaao3603 (2018).
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Roth, M.

M. Roth, M. Ganzhorn, N. Moll, S. Filipp, G. Salis, and S. Schmidt, “Analysis of a parametrically driven exchange-type gate and a two-photon excitation gate between superconducting qubits,” Phys. Rev. A 96(6), 062323 (2017).
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Roushan, P.

F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, Fernando G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrá, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574(7779), 505–510 (2019).
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Rubin, N.

M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroatmodjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. d. Silva, E. Acala, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, S. Caldwell, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. E. Bouayadi, D. Girshovich, S. Hong, A. Hudson, P. Karalekas, K. Kuang, M. Lenihan, R. Manenti, T. Manning, J. Marshall, Y. Mohan, W. Brien, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, V. Rawat, C. A. Ryan, R. Renzas, N. Rubin, D. Russel, M. Rust, D. Scarabelli, M. Selvanayagam, R. Sinclair, R. Smith, M. Suska, T. -W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. T. Rigetti, “Demonstration of universal parametric entangling gates on a multi-qubit lattice,” Sci. Adv. 4(2), eaao3603 (2018).
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S. A. Caldwell, N. Didier, C. A. Ryan, E. A. Sete, A. Hudson, P. Karalekas, R. Manenti, M. P. da Silva, R. Sinclair, E. Acala, N. Alidoust, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, K. Kuang, M. Lenihan, T. Manning, A. Marchenkov, J. Marshall, R. Maydra, Y. Mohan, W. O’Brien, C. Osborn, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, G. Prawiroatmodjo, V. Rawat, M. Reagor, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Scheer, M. Selvanayagam, R. Smith, A. Staley, M. Suska, N. Tezak, D. C. Thompson, T.-W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. Rigetti, “Parametrically activated entangling gates using transmon qubits,” Phys. Rev. Appl. 10(3), 034050 (2018).
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Rubin, N. C.

F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, Fernando G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrá, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574(7779), 505–510 (2019).
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Russel, D.

M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroatmodjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. d. Silva, E. Acala, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, S. Caldwell, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. E. Bouayadi, D. Girshovich, S. Hong, A. Hudson, P. Karalekas, K. Kuang, M. Lenihan, R. Manenti, T. Manning, J. Marshall, Y. Mohan, W. Brien, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, V. Rawat, C. A. Ryan, R. Renzas, N. Rubin, D. Russel, M. Rust, D. Scarabelli, M. Selvanayagam, R. Sinclair, R. Smith, M. Suska, T. -W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. T. Rigetti, “Demonstration of universal parametric entangling gates on a multi-qubit lattice,” Sci. Adv. 4(2), eaao3603 (2018).
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Russell, D.

S. A. Caldwell, N. Didier, C. A. Ryan, E. A. Sete, A. Hudson, P. Karalekas, R. Manenti, M. P. da Silva, R. Sinclair, E. Acala, N. Alidoust, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, K. Kuang, M. Lenihan, T. Manning, A. Marchenkov, J. Marshall, R. Maydra, Y. Mohan, W. O’Brien, C. Osborn, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, G. Prawiroatmodjo, V. Rawat, M. Reagor, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Scheer, M. Selvanayagam, R. Smith, A. Staley, M. Suska, N. Tezak, D. C. Thompson, T.-W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. Rigetti, “Parametrically activated entangling gates using transmon qubits,” Phys. Rev. Appl. 10(3), 034050 (2018).
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Rust, M.

S. A. Caldwell, N. Didier, C. A. Ryan, E. A. Sete, A. Hudson, P. Karalekas, R. Manenti, M. P. da Silva, R. Sinclair, E. Acala, N. Alidoust, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, K. Kuang, M. Lenihan, T. Manning, A. Marchenkov, J. Marshall, R. Maydra, Y. Mohan, W. O’Brien, C. Osborn, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, G. Prawiroatmodjo, V. Rawat, M. Reagor, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Scheer, M. Selvanayagam, R. Smith, A. Staley, M. Suska, N. Tezak, D. C. Thompson, T.-W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. Rigetti, “Parametrically activated entangling gates using transmon qubits,” Phys. Rev. Appl. 10(3), 034050 (2018).
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M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroatmodjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. d. Silva, E. Acala, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, S. Caldwell, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. E. Bouayadi, D. Girshovich, S. Hong, A. Hudson, P. Karalekas, K. Kuang, M. Lenihan, R. Manenti, T. Manning, J. Marshall, Y. Mohan, W. Brien, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, V. Rawat, C. A. Ryan, R. Renzas, N. Rubin, D. Russel, M. Rust, D. Scarabelli, M. Selvanayagam, R. Sinclair, R. Smith, M. Suska, T. -W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. T. Rigetti, “Demonstration of universal parametric entangling gates on a multi-qubit lattice,” Sci. Adv. 4(2), eaao3603 (2018).
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Ryan, C. A.

M. Reagor, C. B. Osborn, N. Tezak, A. Staley, G. Prawiroatmodjo, M. Scheer, N. Alidoust, E. A. Sete, N. Didier, M. P. d. Silva, E. Acala, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, S. Caldwell, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. E. Bouayadi, D. Girshovich, S. Hong, A. Hudson, P. Karalekas, K. Kuang, M. Lenihan, R. Manenti, T. Manning, J. Marshall, Y. Mohan, W. Brien, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, V. Rawat, C. A. Ryan, R. Renzas, N. Rubin, D. Russel, M. Rust, D. Scarabelli, M. Selvanayagam, R. Sinclair, R. Smith, M. Suska, T. -W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. T. Rigetti, “Demonstration of universal parametric entangling gates on a multi-qubit lattice,” Sci. Adv. 4(2), eaao3603 (2018).
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S. A. Caldwell, N. Didier, C. A. Ryan, E. A. Sete, A. Hudson, P. Karalekas, R. Manenti, M. P. da Silva, R. Sinclair, E. Acala, N. Alidoust, J. Angeles, A. Bestwick, M. Block, B. Bloom, A. Bradley, C. Bui, L. Capelluto, R. Chilcott, J. Cordova, G. Crossman, M. Curtis, S. Deshpande, T. El Bouayadi, D. Girshovich, S. Hong, K. Kuang, M. Lenihan, T. Manning, A. Marchenkov, J. Marshall, R. Maydra, Y. Mohan, W. O’Brien, C. Osborn, J. Otterbach, A. Papageorge, J.-P. Paquette, M. Pelstring, A. Polloreno, G. Prawiroatmodjo, V. Rawat, M. Reagor, R. Renzas, N. Rubin, D. Russell, M. Rust, D. Scarabelli, M. Scheer, M. Selvanayagam, R. Smith, A. Staley, M. Suska, N. Tezak, D. C. Thompson, T.-W. To, M. Vahidpour, N. Vodrahalli, T. Whyland, K. Yadav, W. Zeng, and C. Rigetti, “Parametrically activated entangling gates using transmon qubits,” Phys. Rev. Appl. 10(3), 034050 (2018).
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M. Roth, M. Ganzhorn, N. Moll, S. Filipp, G. Salis, and S. Schmidt, “Analysis of a parametrically driven exchange-type gate and a two-photon excitation gate between superconducting qubits,” Phys. Rev. A 96(6), 062323 (2017).
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F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, Fernando G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrá, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574(7779), 505–510 (2019).
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H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, “Improving the coherence time of superconducting coplanar resonators,” Appl. Phys. Lett. 95(23), 233508 (2009).
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M. Sasura and V. Buzek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A 64(1), 012305 (2001).
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Satzinger, K. J.

F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, Fernando G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrá, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, “Quantum supremacy using a programmable superconducting processor,” Nature 574(7779), 505–510 (2019).
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Scarabelli, D.

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Figures (8)

Fig. 1.
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.

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Fig. 2.
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.

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Fig. 3.
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.

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Fig. 4.
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.

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Fig. 5.
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 $|\Psi (0)\rangle$.

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Fig. 6.
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 $\gamma =0.40\pi$.

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Fig. 7.
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.

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Fig. 8.
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.

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Equations (25)

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H b u s = Q 2 2 C + ( Φ − Φ x ) 2 2 L − E J cos ⁡ 2 π Φ Φ 0 ,
V = V 0 { 1 2 [ 2 π ( Φ − Φ x ) Φ 0 ] − E J V 0 cos ⁡ ( 2 π Φ Φ 0 ) } ,
Ω j = 1 2 L ℏ ⟨ g j + 1 | Φ | e j ⟩ ∫ S B μ ω j ( r → , t ) ⋅ d S , g j = 1 L ω j 2 μ 0 ℏ ⟨ g j | Φ | e j ⟩ ∫ S B r j ( r → , t ) ⋅ d S ,
H = H 0 + H i , H 0 = ω 1 a 1 † a 1 + ω 2 a 2 † a 2 + ∑ l = g , e ∑ j = 1 , 2 ω l j | l j ⟩ ⟨ l j | , H i = ∑ n = 1 2 g n a n | e n ⟩ ⟨ g n | + Ω 1 | e 1 ⟩ ⟨ g 2 | e − i ω L 1 t + H.c. ,
H I , 2 = ∑ n = 1 2 g n a n | e n ⟩ ⟨ g n | + Ω 1 | e 1 ⟩ ⟨ g 2 | + H . c . ,
H Π , 2 = g 1 2 ( e i Ω 1 t | Φ + ⟩ + e − i Ω 1 t | Φ − ⟩ ) ⟨ ϕ 1 | + H.c. ,
H Π , 2 ′ = g 1 2 ( e i Ω 1 t | Ψ + ⟩ + e − i Ω 1 t | Ψ − ⟩ ) ⟨ φ 1 | + g 2 2 ( e i Ω 1 t | Ψ + ⟩ − e − i Ω 1 t | Ψ − ⟩ ) ⟨ φ 4 | + H.c. ,
H e f f = g e f f | φ 4 ⟩ ⟨ φ 1 | + H.c.
| g 1 ⟩ | 0 ⟩ R 1 | 0 ⟩ R 2 → | g 1 ⟩ | 0 ⟩ R 1 | 0 ⟩ R 2 , | g 1 ⟩ | 0 ⟩ R 1 | 1 ⟩ R 2 → | g 1 ⟩ | 0 ⟩ R 1 | 1 ⟩ R 2 , | g 1 ⟩ | 1 ⟩ R 1 | 0 ⟩ R 2 → | g 1 ⟩ | 1 ⟩ R 1 | 0 ⟩ R 2 , | g 1 ⟩ | 1 ⟩ R 1 | 1 ⟩ R 2 → − | g 1 ⟩ | 1 ⟩ R 1 | 1 ⟩ R 2 .
H I,3 = ∑ m = 1 3 g m a m | e m ⟩ ⟨ g m | + ∑ n = 1 2 Ω n | e n ⟩ ⟨ g n + 1 | + H.c.
H Π , 3 = g 1 2 e i Ω 1 t ( | ψ + ⟩ ⟨ φ 1 ′ | + | φ 1 ′ ⟩ ⟨ ψ − | ) + g 2 2 e i ( Ω 1 − Ω 2 ) t ( | ψ + ⟩ ⟨ ψ + ′ | − | ψ − ′ ⟩ ⟨ ψ − | ) + g 2 2 e i ( Ω 1 + Ω 2 ) t ( | ψ + ⟩ ⟨ ψ − ′ | − | ψ + ′ ⟩ ⟨ ψ − | ) + g 3 2 e i Ω 2 t ( | ψ + ′ ⟩ ⟨ φ 6 ′ | − | φ 6 ′ ⟩ ⟨ ψ − ′ | ) + H.c .
⟨ φ 6 ′ | H Π , 3 | ψ + ′ ⟩ ⟨ ψ + ′ | H Π , 3 | ψ + ⟩ ⟨ ψ + | H Π , 3 | ϕ 1 ′ ⟩ Ω 1 Ω 2 − ⟨ φ 6 ′ | H Π , 3 | ψ − ′ ⟩ ⟨ ψ − ′ | H Π , 3 | ψ + ⟩ ⟨ ψ + | H Π , 3 | ϕ 1 ′ ⟩ Ω 1 Ω 2 − ⟨ φ 6 ′ | H Π , 3 | ψ + ′ ⟩ ⟨ ψ + ′ | H Π , 3 | ψ − ⟩ ⟨ ψ − | H Π , 3 | φ 1 ′ ⟩ Ω 1 Ω 2 + ⟨ φ 6 ′ | H Π , 3 | ψ − ′ ⟩ ⟨ ψ − ′ | H Π , 3 | ψ − ⟩ ⟨ ψ − | H Π , 3 | ϕ 1 ′ ⟩ Ω 1 Ω 2 = g 1 g 2 g 3 Ω 1 Ω 2 .
⟨ φ 1 ′ | H Π , 3 | ψ + ⟩ ⟨ ψ + | H Π , 3 | φ 1 ′ ⟩ − Ω 1 + ⟨ φ 1 ′ | H Π , 3 | ψ − ⟩ ⟨ ψ − | H Π , 3 | φ 1 ′ ⟩ Ω 1 = 0 ,
⟨ φ 6 ′ | H Π , 3 | ψ + ′ ⟩ ⟨ ψ + ′ | H Π , 3 | φ 6 ′ ⟩ − Ω 2 + ⟨ φ 6 ′ | H Π , 3 | ψ − ′ ⟩ ⟨ ψ − ′ | H Π , 3 | φ 6 ′ ⟩ Ω 2 = 0.
H eff,3 ′ = g eff,3 | φ 6 ′ ⟩ ⟨ φ 1 ′ | + H.c .
| g 1 ⟩ | x ⟩ R 1 | y ⟩ R 2 | z ⟩ R 3 → e i x y z π | g 1 ⟩ | x ⟩ R 1 | y ⟩ R 2 | z ⟩ R 3 ,
H I,n = ∑ i = 1 n g i a i | e i ⟩ ⟨ g i | + ∑ j = 1 n − 1 Ω j | e j ⟩ ⟨ g j + 1 |
H eff,n = g eff,n | ψ n ⟩ ⟨ ψ 1 | + H.c
g ( t ) = g m 2 [ 1 − cos ⁡ ( 2 π t t I I ′ + π 3 ) ]
d ρ d t = − i [ H I I ,   ρ ] + ∑ i = 1 3 κ i D [ a i ] + γ g 1 , e 1 D [ σ 1 ] + γ g 2 , e 1 D [ σ 2 ] + γ g 2 , e 2 D [ σ 3 ] + γ g 3 , e 2 D [ σ 4 ] + γ g 3 , e 3 D [ σ 5 ] ,
H un = ( Ω 1 e i δ t + g 2 e 2 i δ t a 2 † ) | g 1 ⟩ ⟨ e 1 | + ( g 1 e i δ t a 1 + g 2 e − i δ t a 2 ) | e 1 ⟩ ⟨ g 2 | + ( Ω 1 e − i δ t + g 1 e − 2 i δ t a 1 † ) | g 2 ⟩ ⟨ e 2 | + H.c ,
δ 1 = Ω 1 2 δ , δ 2 = − Ω 1 2 δ , δ 3 = − Ω 1 2 δ , δ 4 = Ω 1 2 δ .
H I , 2 ′ = g 1 e i δ 12 t a 1 † | g 1 ⟩ ⟨ e 1 | + Ω 1 e i δ 23 t | e 1 ⟩ ⟨ g 2 | + g 2 e i δ 34 t a 2 † | g 2 ⟩ ⟨ e 2 | + H . c .
δ 12 = δ 2 − δ 1 , δ 23 = δ 3 − δ 2 , δ 34 = δ 4 − δ 3 .
δ 1 = Ω 1 2 δ , δ 2 = D 2 , δ 3 = D 3 , δ 4 = Ω 1 2 δ .

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