Single-pulse two-qubit gates for Rydberg atoms with noncyclic geometric control

Arrays of neutral atoms have emerged as promising platforms for quantum computing. The realization of high-fidelity two-qubit gates with robustness is currently a significantly important task for large-scale operations. In this paper, we present a convenient approach for implementing a two-qubit controlled-phase gate with Rydberg blockade. We achieve noncyclic and geometric control within a single modulated pulse. Compared with the control scheme by cyclic evolution determined by dynamical parameters, the robustness of our proposal against systematic errors will be remarkably improved due to its geometric characteristics. Importantly, the noncyclic geometric control reduces the gate time for small rotation angles and is more insensitive to decoherence effects. Furthermore, we accelerate the adiabatic control with the aid of shortcuts to adiabaticity to further shorten the operation time. We apply our protocol to the quantum Fourier transformation algorithm to demonstrate the actual acceleration. Therefore, the proposed scheme provides analytical wave forms for arbitrary two-qubit gates and may play an important role in experiments involving atomic arrays.

Figure 1

Scheme of $C_Z$ gate based on noncyclic geometric quantum control. (a) Coupling configuration of two interacting atoms under the two-qubit basis. Since the Rydberg-excitation lasers coupling $|1\rangle$ and $|r\rangle$ of each qubit, $|00\rangle$ is decoupled from the system. $|10\rangle (|01\rangle )$ are coupled to $|r0\rangle (|0r\rangle )$ with Rabi frequency $\mathrm{\Omega }$ while $|11\rangle$ is coupled to $|R\rangle =\left(\right|1r\rangle +|r1\rangle )/\sqrt{2}$ with $\sqrt{2}\mathrm{\Omega }$ due to the collective excitation of Rydberg states. All the couplings are shifted by a detuning $\mathrm{\Delta }$. $|rr\rangle$ will not be involved under the Rydberg blockade condition. (b) Control wave forms of NCGC. Black-dashed line: $\mathrm{\Omega }$, red solid line: $\phi$. The adiabatic control can be accelerated by adding an auxiliary detuning ${\mathrm{\Delta }}_a$ (blue dashed-dotted line) which induces $\mathrm{\Omega }T=4\pi , T$ is the evolution period. (c), (d) Dynamics of different basis under the driving. (c) symbols the dynamics of $|11\rangle$ while (d) symbols the ones of $|10\rangle$ or $|01\rangle$. Blue dashed lines: population $P^{\left(\eta \right)}$. Red solid lines: acquired phases ${\theta }^{\left(\eta \right)}$. As can be seen that all basis return to the initial population with a $\pi$ phase acquired after the driving and thus the $C_Z$ gate is achieved.

Figure 2

Robustness of $C_z$ gate realized by NCGC scheme. (a) Fidelity $F$ of $C_z$ gates realized by different protocols against the variation of Rabi frequencies. (a) Fidelity $F$ of $C_z$ gates realized by different protocols against the variation of detuning. Red solid lines: NCGC, blue dotted lines: PM, green dashed dotted lines: RM, where NCGC adopts the wave forms in Eq. (10), RM adopts the ones in Eq. (11) and PM using Eq. (12). (c) The numerical simulation of gate fidelity $F$ against random noise of Rabi frequency and detuning, in which ${\mathrm{\Omega }}_r\left(t\right)={\kappa }_1^{\prime }\left(t\right)\mathrm{\Omega }+\mathrm{\Omega }$ and ${\mathrm{\Delta }}_r\left(t\right)={\kappa }_2^{\prime }\left(t\right)\mathrm{\Omega }+\mathrm{\Delta }\left(t\right)$. ${\kappa }_1^{\prime }, {\kappa }_2^{\prime }$ are time-varying random number and ${\kappa }_1^{\prime }\left({\kappa }_2^{\prime }\right)\in [0,0.1]$. The fidelity of each point is obtained by averaging 50 times.

Figure 3

Fidelity $F$ of $C_Z$ gate against Rydberg interacting strength $V$. Red lines: NCGC, blue lines: PM. Solid lines: $T=500\mathrm{ns}$, dashed lines: $T=250\mathrm{ns}$. The robustness of NCGC scheme against $V$ will be enhanced when the operation time shortened while the robustness of PM scheme will be weakened.

Figure 4

Numerical simulation of gate fidelity $F$ against the evolution time $T$ under the effect of decay and dephasing. The simulation includes the decay from upper energy level $|{\lambda }_{+}^{\left(\eta \right)}\rangle$ to lower energy level $|{\lambda }_{-}^{\left(\eta \right)}\rangle$ and upper energy level $|{\lambda }_{+}^{\left(\eta \right)}\rangle$ to the energy level outside the system $|W\rangle$, as well as a dephasing between the upper energy level $|{\lambda }_{+}^{\left(\eta \right)}\rangle$ and lower energy level $|{\lambda }_{-}^{\left(\eta \right)}\rangle$.

Figure 5

Numerical simulation of total evolution time of quantum Fourier transformation versus the number of qubits $N$. Blue dotted line: cyclic scheme, red solid line: noncyclic scheme. The NCGC scheme rises slower than cyclic scheme as the amount of two-qubit gates with small rotating angles increases polynomially.