Fast multiqubit phase gate in circuit QED beyond the rotating-wave approximation

We propose a two-step approach to realize a multiqubit phase gate in circuit QED beyond the rotating-wave approximation. The multiqubit gate can be implemented in a few nanoseconds, which is much faster than that under the rotating-wave approximation. Also the operation time of the gate is independent of the number of qubits. Due to the absence of the resonator degree of freedom in the effective Hamiltonian, our scheme is insensitive to the initial state of the resonator and immune to the leakage of photons. Numerical simulation shows that the scheme is robust against the main decoherence sources.

Figure 1

Schematic diagram of the setup consisting of gradiometer-type flux qubits coupled to a transmission line resonator. The diagram in the red box shows the detailed schematics of a flux qubit which is fabricated in the middle of the resonator. The red crosses denote the Josephson junctions. The external flux is applied though the bias lines $I_1,I_2$, and $I_{\alpha }$.

Figure 2

Time dependence of fidelity of four-qubits case for ${\omega }_r=10\times 2\pi \mathrm{GHz}$ and $g=0.1021{\omega }_r=1.021\times 2\pi \mathrm{GHz}$. The blue line and red line are simulated based on the full Hamiltonian in Eq. (25) and the effective Hamiltonian in Eq. (10), respectively. The green lines indicate the ending times of step one and step two. The unity fidelity is reached at the time $t_2$.

Figure 3

Maximal fidelity versus different ratio of coupling strength to resonator frequency $g/{\omega }_r$ of four qubits case. The fidelity gets close to unity in a wavy pattern, which indicates the approximation in the dispersive regime is better as $g/{\omega }_r$ goes smaller. The red dots represent the decoupling points.

Figure 4

Evolution of $\Delta F$ as a function of time. The multiqubit phase gate is realized at time $t_2$. The used parameters are $g=1.021\times 2\pi \mathrm{GHz},{\omega }_r=10\times 2\pi \mathrm{GHz},\kappa =5\times 2\pi \mathrm{MHz},\gamma =0.1\times 2\pi \mathrm{MHz}$.

Figure 5

Effects of inhomogeneous parameters with (a) $\epsilon$, (b) $\Omega$, and (c) $g$. The mean maximal fidelities versus scaled standard deviations of parameters are shown. The error bars show the standard deviation of $\left\{F_{\max }^j\right\}$. The parameters are $g=0.1021\times 2\pi \mathrm{GHz},{\omega }_r=10\times 2\pi \mathrm{GHz}$, and $n_r=200$.