Error budgeting for a controlled-phase gate with strontium-88 Rydberg atoms

We study the implementation of a high fidelity controlled-phase gate in a Rydberg quantum computer. The protocol is based on a symmetric gate with respect to the two qubits as experimentally realized by Levine et al. [Phys. Rev. Lett. 123, 170503 (2019)], but allows for arbitrary pulse shapes with time-dependent detuning. Optimizing the pulse shapes, we introduce laser pulses which shorten the time spent in the Rydberg state by 10% and reduce the leading contribution to the gate infidelity, i.e., the decay from the Rydberg state. Remarkably, this reduction can be achieved for smooth pulses in detuning and smooth turning on of the Rabi frequency as required in any experimental realization. We carefully analyze the influence of fundamental error sources such as the photon recoil and the microscopic interaction potential, as well as the harmonic trapping of the atoms for an experimentally realistic setup based on strontium-88 atoms. We find that an average gate fidelity above 99.9% is possible for a very conservative estimation of experimental parameters.

Figure 1

Setup for the realization of a controlled-phase gate with neutral atoms. The two atoms are trapped in optical tweezers (in blue) at a fixed distance $R$ along the $x$ direction. In the idealized description, we consider a three-level structure with states $|0\rangle$, $|1\rangle$ and the Rydberg state $|r\rangle$ for each atom. The Rydberg state is coupled with the state $|1\rangle$ by homogeneously driving a global laser, i.e., a plane wave (in red), with Rabi frequency $\mathrm{\Omega }\left(t\right)$ and detuning $\mathrm{\Delta }\left(t\right)$ along the $z$ direction. The Rydberg states interact via a van der Waals interaction $V\left(R\right)$ (green arrow), which strongly depends on the distance between the atoms.

Figure 2

Analysis for the realization of a controlled-phase gate for two qubits via the interaction of their Rydberg states. (a) The controlled-phase gate can be realized by using different time-dependent detuning shapes $\mathrm{\Delta }\left(t\right)$ at interaction strength $V/{\mathrm{\Omega }}_0\hslash =21.1$. The shape (I) is a $\delta$ function as recently applied by Levine et al. [28]. The shape (II) is an isosceles triangle. The shape (III.A) is a Gaussian with width $w{\mathrm{\Omega }}_0=1.7$. A solution with a Gaussian detuning can be found also introducing a finite rise time $\kappa {\mathrm{\Omega }}_0=0.31$ on the Rabi frequency $\mathrm{\Omega }\left(t\right)$ as one can see in (III.B). Then, there is the solution (IV) obtained by using the optimal control algorithm dCRAB. (b), (c) Dynamics on the Bloch sphere for a controlled-phase gate realized by assuming the shape (III.A). (d) Comparing the total time spent in the Rydberg state $T_r^{\alpha }$ for the different shapes by considering separately $\alpha =|11\rangle$ and $\alpha =|01\rangle$ as initial state for the evolution. The average time in the Rydberg state is ${\overline{T}}_r=(T_r^{|01\rangle }+T_r^{|10\rangle }+T_r^{|11\rangle })/3$. In order to show that we can drop the assumption of a perfect blockade, we also illustrate the total time spent in the Rydberg state as a function of the Rydberg interaction.

Figure 3

Time spent in the Rydberg state $T_r^{\alpha }$ as a function of the Gaussian width $w{\mathrm{\Omega }}_0$ by assuming a Gaussian detuning protocol. Each data point optimizes the peak's center and height after fixing the width of the Gaussian and is the solution with minimal infidelity, which is zero up to numerical precision. The point at $w=0$ is obtained with the $\delta$-function protocol. The gray line is the Gaussian width $w{\mathrm{\Omega }}_0=1.7$ of the best pulse (III.A).

Figure 4

Time spent in the Rydberg state $T_r^{\alpha }$ as a function of the characteristic time scale $\kappa {\mathrm{\Omega }}_0$ by assuming a Gaussian detuning protocol with a realistic turning on of the Rabi frequency. The inset shows the pulse shape for the detuning (blue) and the Rabi frequency (red) for the slowest turning on with $\kappa {\mathrm{\Omega }}_0=1.2$.

Figure 5

Intrinsic infidelity. For pulse (III.B), we analyze the infidelity caused by various effects that are intrinsic to realistic setups. The gray lines depict the parameters that we consider for our strontium-88 setup. (a) We simulate the Rydberg decay in a system where the atoms are at a fixed position. The dashed line depicts the analytic result of Eq. (13). The infidelity $1-F_d$ increases proportionally to the decay rate $\gamma$ and the average time spent in the Rydberg state ${\overline{T}}_r$. (b) The infidelity $1-F_r$ caused by photon recoil is studied in a system where the atoms are placed in one-dimensional harmonic traps along the direction of the laser $z$. The infidelity increases with the trap frequency ${\omega }_z$. We simulate the system for different temperatures $T$ of the atoms in the harmonic traps. The dashed curves show the analytic result of Eq. (15). (c) The infidelity $1-F_i$, which is caused by the force due to van der Waals interaction, is simulated for a system where the atoms are placed in one-dimensional harmonic traps along $x$, i.e., the direction of the interatomic axis. The effect is only significant for trap frequencies ${\omega }_x\ll 100\text{kHz}$. The dashed curves are the analytic estimates from Eq. (17).