Two-qubit entangling gates between distant atomic qubits in a lattice

Arrays of qubits encoded in the ground-state manifold of neutral atoms trapped in optical (or magnetic) lattices appear to be a promising platform for the realization of a scalable quantum computer. Two-qubit conditional gates between nearest-neighbor qubits in the array can be implemented by exploiting the Rydberg blockade mechanism, as was shown by D. Jaksch et al. [Phys. Rev. Lett. 85, 2208 (2000)]. However, the energy shift due to dipole-dipole interactions causing the blockade falls off rapidly with the interatomic distance, and protocols based on direct Rydberg blockade typically fail to operate between atoms separated by more than one lattice site. In this work, we propose an extension of the protocol of Jaksch et al. for controlled-$Z$ and controlled-not gates which works in the general case where the qubits are not nearest neighbors in the array. Our proposal relies on the Rydberg excitation hopping along a chain of ancilla noncoding atoms connecting the qubits on which the gate is to be applied. The dependence of the gate fidelity on the number of ancilla atoms, the blockade strength, and the decay rates of the Rydberg states is investigated. A comparison between our implementation of a distant controlled-not gate and one based on a sequence of nearest-neighbor two-qubit gates is also provided.

Figure 1

(a) Qubits are encoded in a 1D chain of atoms (blue). A parallel chain of ancilla noncoding atoms (green) is used to connect the control and target qubits through the nearest-neighbor Rydberg blockade (red arrows). (b) Internal structure of the qubit and ancilla atoms. For clarity, the subscript denoting the atom to which the different states belong is omitted.

Figure 2

Pulse sequence implementing the cz and cnot gates. Note that the pulse sequence on ancilla atoms is the same regardless of the parity of $n_{\mathrm{A}}$.

Figure 3

Process fidelity $F_{\mathrm{pro}}$ of the distant qubits protocol implementing a modified cz gate as a function of $n_{\mathrm{A}}$ for different values of the decay rates: from top to bottom $\gamma /\mathrm{\Omega }=4\times {10}^{-5}$ (black dots), $32\times {10}^{-5}$ (blue triangles), $128\times {10}^{-5}$ (green squares), and $512\times {10}^{-5}$ (red diamonds). (a) ${\gamma }_0={\gamma }_1=\gamma /2$ and ${\gamma }_{\mathrm{A}}=0$, (b) ${\gamma }_0={\gamma }_1=0$ and ${\gamma }_{\mathrm{A}}=\gamma$, (c) ${\gamma }_0={\gamma }_1=\gamma /2, {\gamma }_{\mathrm{A}}=\gamma$. The lower and upper bounds on the process fidelity (12) delimit the shaded area. The symbols are the values of $F_{\mathrm{pro}}$ obtained through numerical simulations and the lines correspond to the fits of the data using (15) with $t_{\mathrm{q}}$ and $t_{\mathrm{A}}$ given by (16) and (17) with $t_{\mathrm{eff}}^{\pi }$ as only fitting parameter (see text).

Figure 4

Same as in Fig. 3, but for the cnot gate. The data for $F_{\mathrm{pro}}$ have been fitted using Eq. (15) with $t_{\mathrm{q}}$ and $t_{\mathrm{A}}$ given by (18) and (19) with $t_{\mathrm{eff}}^{\pi }$ as only a fitting parameter (see text).

Figure 5

Possible implementation of a cnot quantum gate between nonadjacent qubits using only nearest-neighbor cnot gates [53].

Figure 6

Comparison of the process fidelity obtained using only the nearest-neighbor cnot gate [53] and the distant-qubit protocol using noncoding ancilla atoms with ${\gamma }_{\mathrm{A}}={\gamma }_{\mathrm{q}}=\gamma$ and ${\gamma }_0={\gamma }_1$. The black dots correspond to the case of two ancilla atoms, while the black open circles represent the corresponding fidelity for the nearest-neighbor-qubit implementation with a single qubit between the control and target atoms. The red solid triangles represent the process fidelity in the case of three ancilla atoms, and the red open triangles correspond to the case where the control and target are separated by two other qubits.

Figure 7

Gain in the fidelity (20) resulting from the use of non-coding ancilla atoms in a distant qubits cnot gate as a function of the decay rate (${\gamma }_{\mathrm{A}}={\gamma }_{\mathrm{q}}=\gamma$ and ${\gamma }_0={\gamma }_1$) for (from bottom to top) $n_{\mathrm{A}}=2$ (black dots), 3 (blue triangles), 4 (green dashed curve) and 5 (red dotted curve). The symbols correspond to the gain computed from our numerical simulations.