Experimentally realizable controlled NOT gate in a flux qubit/resonator system

We present an experimentally realizable microwave pulse sequence that effects a controlled-NOT (C-NOT) gate operation on a Josephson-junction-based flux qubit/resonator system with high-process fidelity. We obtained a C-NOT gate process fidelity of 0.988 (0.980) for a two-(three-)qubit/resonator system under ideal conditions and a fidelity of 0.903 for a two-qubit/resonator system under the best, currently achieved, experimental conditions. Our simulations show that this gate is a feasible first step toward multiqubit quantum-information processing with flux qubit/resonator systems.

Figure 1

(Color online) Multiple flux qubits (blue rectangles with three crosses), each addressed by its own microwave line (green line), coupled to a resonator via a mutual inductance $M$. The resonator is schematically represented by the inductor $L$ and the capacitor $C$.

Figure 2

(Color online) (a) Energy levels of the two-qubit/resonator system and transitions used in our pulse sequence. Solid (dashed) lines represent the energy levels without (with) the coupling between the qubits and the resonator. Additional shifts $\delta$ occur during the irradiation of the BSB pulse. Energy difference between the dashed arrows is utilized to perform the controlled-$Z$ gate between the resonator and the target qubit. (b) Basic operational sequence of target, control, and BSB pulses on our two qubits, using values defined in Sec. 4, that enact a C-NOT gate in our system. Each pulse is represented by the microwave frequency (carrier frequency or BSB frequency), the rotation angle, and the phase of the microwave pulse. Note that the parameters $T$, the free evolution time, and $\varphi$, the microwave pulse phase, are variable and are defined by the experimental system to those values that optimize the operation of the gate. Unitary matrix representations of each pulse are shown by $Y_{\text{C}}$, $Y_{\text{T}}$, and $U_{\text{CZ}}$, which are also used in Eq. (1).

Figure 3

(Color online) Process fidelity as a function of qubit decay rate. The upper (lower) curve represents the fidelity when the quality factor of the resonator is ${10}^6\left(3\times {10}^5\right)$.

Figure 4

(Color) From top: initial (left) and final (middle and right) density matrices (real components only) showing the $|00⟩\mapsto |00⟩$, $|01⟩\mapsto |01⟩$, $|10⟩\mapsto |11⟩$, and $|11⟩\mapsto |10⟩$ operations, and the corresponding qubit leakage into higher resonator modes $\left(n>0\right)$ due to the pulse sequence values defined in Sec. 4 without (middle) and with (right) decoherence ($Q={10}^6$ and ${\Gamma }_1={\Gamma }_2=0.25\text{MHz}$). The color scheme used is a nonlinear gradient: zero values are white with negative values greenish and positive values reddish.