One-step multiqubit Greenberger-Horne-Zeilinger state generation in a circuit QED system

We propose a one-step scheme to generate Greenberger-Horne-Zeilinger (GHZ) states for superconducting flux qubits or charge qubits in a circuit QED setup. The GHZ state can be produced within the coherence time of the multiqubit system. Our scheme is independent of the initial state of the transmission line resonator and works in the presence of higher harmonic modes. Our analysis also shows that the scheme is robust to various operation errors and environmental noise.

Figure 1

(Color online) Schematic diagram of our setup. (a) The qubits are coupled through a superconducting stripline resonator (the blue stripe). Each “crossed box” denotes one qubit which can be either a charge qubit or a flux qubit; the red dashed line shows the magnitude of the magnetic field. (b) Detailed schematic of a charge qubit. The crosses denote Josephson junctions. (c) Detailed schematic of a gradiometer-type flux qubit. The crosses denote Josephson junctions.

Figure 2

(Color online) Time dependence of the fidelity of the prepared GHZ state for two different initial resonator states: the ground state (blue solid line) and the thermal state (red dashed line) in the case of (a) two qubits and (b) four qubits. The black dots indicate the time when the resonator and qubits are effectively decoupled. The green dotted lines limit the regime in which the fidelity is larger than 90%. The following parameter values were used: qubit frequency $\Omega =10\text{GHz}$, resonator frequency $\omega =1\text{GHz}$, and coupling strength $g=144\text{MHz}$. The time is given in units of $T_{\text{min}}$.

Figure 3

(Color online) Dependence of the fidelity $F$ on the error of the coupling coefficient $\delta$. The curves correspond to $N=2$ (top), 4, 6, and 8 (bottom).

Figure 4

(Color online) Time dependence of the error due to decoherence. $\Delta F$ is the difference of the fidelity of the prepared GHZ state with/without environment decoherence for (a) two qubits (b) four qubits. The red dots mark the times at which the GHZ state is prepared. The following parameter values were used: qubit frequency $\Omega =10\text{GHz}$, resonator frequency $\omega =1\text{GHz}$, coupling strength $g=144\text{MHz}$, qubit decay rate $\gamma =10\text{MHz}$, and resonator decay rate $\kappa =0.5\text{MHz}$. The time is given in units of $T_{\text{min}}$.