One-step construction of the multiple-qubit Rydberg controlled-phase gate

Although the three-body Rydberg antiblockade regime (RABR) can produce Rabi oscillation between the Rydberg collective excited state and the collective ground state, it is still hard to use the RABR to construct the three-qubit quantum logic gate in one step since the effective Hamiltonian is always accompanied by undesired Stark shifts. In order to overcome this difficulty, an additional laser is introduced to eliminate the Stark shifts in the ground-state subspace. And the initial RABR condition is modified to eliminate the remaining undesired Stark shifts in the collective-excitation subspace. The modified RABR is then generalized to the $n (n>3)$-qubit case. Based on the proposed regime, one-step schemes to construct three- and $n$-qubit quantum controlled-PHASE gates are proposed without the requirement of atomic addressability. The asymmetric Rydberg-Rydberg interaction, which is more practical for Rydberg atoms, is also discussed and proven to be feasible for the modified RABR and quantum controlled-PHASE gate in theory. A full-Hamiltonian-based master equation is used to evaluate the performance and some experimental parameters are also considered.

Figure 1

Schematic of the many-body RABR and the multiple-qubit controlled-PHASE gate. (a) Three-qubit case. (b) Four-qubit case. (c) Energy level and the corresponding laser couplings for each Rydberg atom. $|0\rangle$ and $|1\rangle$ are two ground states that were used to encode the quantum information. $|R\rangle$ denotes the high-lying Rydberg state, and $|a\rangle$ denotes the ordinary auxiliary state. $\mathrm{\Omega }\left(\omega \right)$ is the Rabi frequency of the transition $|1\rangle \leftrightarrow |R\left(a\right)\rangle$ with the detuning $-\mathrm{\Delta }\left(\delta \right)$.

Figure 2

Energy level diagram and dynamic processes of the Hamiltonian relevant to $\mathrm{\Omega }$ as well as RRIs (a) and $\omega$ (b) in the three-atom basis. The states cannot be excited to the high-excitation subspace except for $|111\rangle$, due to the antiblockade regime. Nevertheless, Stark shifts would be obtained for these states due to the second-order perturbation theory. E denotes the energy of the Rydberg state in a single atom;V denotes the RRI strength. $\mathrm{\Delta }$ and $\delta$ are laser detunings.

Figure 3

Fidelities of the quantum controlled-PHASE gate plotted with the full [Eqs. (1) plus (2)] and effective [Eq. (12)] Hamiltonians. (a) Three-qubit case. (b) Four-qubit case. In (a), the dot-dashed green line denotes the effective-Hamiltonian-based result, and the solid saffron yellow line denotes the full-Hamiltonian-based result. In (b), the dashed red line denotes the effective-Hamiltonian-based result, and the solid blue line denotes the full-Hamiltonian-based result. The initial state is set as ${\sum }_{i=1}^n\otimes \left(\right|0\rangle +{|1\rangle )}_i/\sqrt{2}$. The final states given by the ideal and practical controlled-PHASE gate, respectively, are used to plot the fidelity. The parameters are chosen as $n=3$, $\mathrm{\Delta }/\mathrm{\Omega }=12$, $\omega /\mathrm{\Omega }=3.25$, and $\gamma =0$ in (a) and $n=4$, $\mathrm{\Delta }/\mathrm{\Omega }=10.6$, $\omega /\mathrm{\Omega }=3.8$, and $\gamma =0$ in (b).

Figure 4

Average fidelities of the three-qubit controlled-PHASE gate versus $\gamma$. (a) Random-initial-state method. (b) Integral method. For simplicity, we set ${|\psi \rangle }_1=sin\eta \left(\right|000\rangle +|001\rangle +|010\rangle +|011\rangle +|100\rangle +|101\rangle +|110\rangle )/\sqrt{7}+cos\eta |111\rangle$, ${|\psi \rangle }_2=sin\eta \left(\right|000\rangle +|001\rangle +|010\rangle +|011\rangle +|100\rangle +|101\rangle )/\sqrt{6}+cos\eta \left(\right|110\rangle +|111\rangle )/\sqrt{2}$, ${|\psi \rangle }_3=sin\eta \left(\right|000\rangle +|001\rangle +|010\rangle +|011\rangle +|100\rangle )/\sqrt{5}+cos\eta \left(\right|101\rangle +|110\rangle +|111\rangle )/\sqrt{3}$, and ${|\psi \rangle }_4=sin\eta \left(\right|000\rangle +|001\rangle +|010\rangle +|011\rangle )/2+cos\eta \left(\right|100\rangle +|101\rangle +|110\rangle +|111\rangle )/2$. The parameters are chosen as $\mathrm{\Delta }/\mathrm{\Omega }=12$, $\omega /\mathrm{\Omega }=3.25$; $\gamma /\mathrm{\Omega }$ ranges from ${10}^{-4}$ to ${10}^{-2}$. $\delta$ is chosen based on Eq. (7).

Figure 5

Fidelities of the three-qubit controlled-PHASE gate in one evolution period with two groups of asymmetric Rydberg parameters. The initial and final states are the same as those in Fig. 3 and the full Hamiltonian is used for numerical simulation. The parameters are chosen as $V_{12}/\mathrm{\Omega }=10$, $\omega /\mathrm{\Omega }=3.25$. ${\mathrm{\Delta }}_i$ and ${\delta }_i$ are chosen based on Eqs. (24) and (22), respectively.

Figure 6

Fidelity of the quantum controlled-PHASE gate with variation of parameters for symmetric RRI. All coordinate values are percentages. For (b) and (c), 100 groups of parameters that satisfy the Gauss distribution are considered for one deviation point of the parameters. Here we use one specific initial state, ${\sum }_{i=1}^3\otimes [\left(\right|0\rangle +|1\rangle {)}_i/\sqrt{2}]$, and the corresponding final state to simulate the fidelity. The parameters are chosen as $\mathrm{\Delta }/\mathrm{\Omega }=12$, $\omega /\mathrm{\Omega }=3.25$, and $\gamma /\mathrm{\Omega }={10}^{-4}$. $\delta$ is chosen based on Eq. (7).