Resonant quantum gates in circuit quantum electrodynamics

We propose the implementation of fast resonant gates in circuit quantum electrodynamics for quantum information processing. We show how a suitable utilization of three-level superconducting qubits inside a resonator constitutes a key tool to perform diverse two-qubit resonant gates, improving the operation speed when compared to slower dispersive techniques. To illustrate the benefit of resonant two-qubit gates in circuit quantum electrodynamics, we consider the implementation of a two-dimensional cluster state in an array of $N\times N$ superconducting qubits by using resonant controlled-phase and one-qubit gates, where the generation time grows linearly with $N$. For $N=3$, and taking into account decoherence mechanisms, a fidelity over 60% for the generation of this cluster state is obtained.

Figure 1

(Color online) Sketch and protocol for the resonant implementation of a CPHASE gate between qubits 1 and 2. The coupling values given in parentheses are the ones used for the simulation presented in Fig. 2.

Figure 2

(Color online) Density operator of the initial state ${\Psi }_i$ (with $\alpha =\delta =1$, $\beta =\gamma =0$) and of the final state after the implementation of the CPHASE gate, in a simulation involving qubit relaxation and dephasing, and cavity decay; expressed in the computational basis of the two qubits.

Figure 3

(Color online) Sketch and protocol for the resonant implementation of an iSWAP gate between qubits 1 and 2.

Figure 4

Protocol for the resonant implementation of a Bogoliubov gate between qubits 1 and 2. ${\Omega }_R$ is the frequency of an external classical field resonant to the transition frequency between qubit states $|{\text{g}}_1⟩$ and $|{\text{e}}_1⟩$.

Figure 5

(Color online) Generation of a cluster state in a cavity grid by implementing simultaneous CPHASE gates following steps (a)–(d). The grid is composed of $2N$ cavities and contains $N^2$ qubits, one at each crossing point. The performed CPHASE gates for the lines are displayed in red and for the columns in blue. The row is shifted to the top and the column to the right. The missing gates at the crossings are indicated as dashed black lines.