An asymmetric and fast Rydberg gate protocol for long range entanglement

Abstract: We analyze a new Rydberg gate design based on the original $π-2π-π$ protocol [Jaksch, et. al. Phys. Rev. Lett. {\bf 85}, 2208 (2000)] that is modified to enable high fidelity operation without requiring a strong Rydberg interaction. The gate retains the $π-2π-π$ structure with an additional detuning added to the $2π$ pulse on the target qubit. The protocol reaches within a factor of 2.39 (1.68) of the fundamental fidelity limit set by Rydberg lifetime for equal (asymmetric) Rabi frequencies on the control and target qubits. We generalize the gate protocol to arbitrary controlled phases. We design optimal target-qubit phase waveforms to generalize the gate across a range of interaction strengths and we find that, within this family of gates, the constant-phase protocol is time-optimal for a fixed laser Rabi frequency and tunable interaction strength. Robust control methods are used to design gates that are robust against variations in Rydberg Rabi frequency or interaction strength.

• The paper presents an asymmetric π-2π-π Rydberg gate that mitigates coherent rotation errors even at moderate interaction strengths.
• It employs a tunable detuning strategy on the target qubit to optimize phase accumulation and suppress residual errors during entanglement operations.
• Robust control via GRAPE and phase modulation extends operational range and enhances fidelity, offering practical advances for scalable quantum computing.

Asymmetric Fast Rydberg Gate Protocols for Long-Range Entanglement

Introduction

The investigated work presents a thorough analysis of a novel Rydberg gate based on a modified $\pi$-$2\pi$-$\pi$ protocol, tailored for high-fidelity two-qubit entanglement over extended atomic separations. Distinct from symmetric pulse protocols, the asymmetric gate incorporates a tunable detuning on the target qubit’s $2\pi$ pulse, offering notable enhancements regarding operational range and gate error scaling—especially in regimes where interaction strength $V$ is comparable to the Rabi frequency $\Omega$. The protocol allows for deterministic, error-free coherent evolution in the absence of dissipation and experimental imperfections, even outside the strong blockade regime, a regime where previous symmetric approaches incur significant coherent rotation errors.

Protocol Design and Error Scaling

The asymmetric gate operates via a three-pulse sequence: a $\pi$ pulse on the control qubit, a detuned $2\pi$ pulse on the target, and a final $\pi$ pulse on the control. The detuning is chosen such that for the detuned $2\pi$ pulse, the population returns perfectly to the ground state both with and without the Rydberg interaction, obviating residual rotation errors even in the partial blockade limit.

Figure 1: Schematic of the Rydberg CZ protocol and the asymmetric detuned target pulse.

By detuning the target pulse appropriately ($\Delta = V/2$ for the optimal case), the protocol ensures phase accumulation necessary for the CZ operation while minimizing depopulation. The error in this protocol is dominated by Rydberg state decay and scales as $\epsilon \propto (V\tau)^{-1}$, achieving a minimum error within a factor of 2.39 of the dissipation-dominated lower bound ($\epsilon_{\rm DDP}$), and down to 1.68 in the limit of a highly asymmetric Rabi frequency ($p \rightarrow \infty$).

Figure 2: Scattering-limited gate errors as a function of control and target Rabi frequency ratio, showcasing proximity to the dissipation-limited bound.

This situates the asymmetric protocol among the highest-performing Rydberg two-qubit gates with respect to fidelities attainable in current neutral-atom architectures. Unlike protocols requiring extreme $V/\Omega$ ratios for error minimization, similar gate error bounds are achieved here for moderate interaction strengths, enabling longer-range operations.

Generalization to Controlled-Phase Gates

A salient feature is the analytic and numerical extension to general controlled-phase operations ($U = \mathrm{diag}[1,1,1,e^{i\theta}]$), achieved by tuning the Rabi frequency and target pulse duration to fulfill the evolution constraints for arbitrary angles $\theta$. Multi-loop generalizations and non-canonical detuning cases are derived, demonstrating full coverage over the phase interval $[-\pi, \pi]$.

Figure 3: Controlled-phase gate mapping as a function of Rabi rate $\Omega$, showing tunable and continuous phase acquisition.

Time-Optimal and Robust Control

Optimization of phase modulation using GRAPE demonstrates that, under fixed-$\Omega$ and tunable-$V$, the constant-phase protocol is time-optimal within the family considered. As $\Omega/V$ increases, the minimum gate time transitions from the analytic $\pi$-$2\pi$-$\pi$ duration to the interaction gate limit, reflecting the tunable hardware landscape in neutral-atom platforms.

Figure 4: (a) GRAPE-optimized gate durations vs. $\Omega/V$; (b) corresponding phase waveform, revealing detuning modulation in the fast-gate regime.

Robustified versions of the asymmetric protocol were engineered using phase modulation to suppress gate infidelity caused by static errors in $\Omega$ and $V$. Robust gates are demonstrably less sensitive to parameter drifts but necessary robustness comes at the cost of increased gate duration and therefore higher integrated Rydberg population and decay errors. Quantitative analysis establishes optimal compromise points for various noise scales, relevant for thermal motion and technical noise-limited systems.

Figure 5: Robust (blue) vs. analytic (orange) phase modulation for Rabi error suppression, with enhanced fidelity across parameter perturbations.

For interaction-strength robust gates, modulation likewise mitigates infidelity induced by inter-atomic separation fluctuations, showing order-of-magnitude reductions in average error at moderate overhead in pulse duration.

Figure 6: Interaction-strength robust phase waveform and corresponding fidelity improvement across variation bands.

Comparative Analysis and Implementation Trade-Offs

When benchmarked against previously reported protocols—such as dark state, symmetric adiabatic, and time-optimal schemes—the asymmetric gate achieves competitive error scaling near the fundamental limit and extends entanglement-capable range without requiring strong blockade. However, it is more susceptible to error from atomic position and coupling variations, a practical consideration given trap anharmonicities and finite temperature effects in experimental settings. Furthermore, fast gates with large double-excited state populations can incur additional heating and technical errors; the trade-off between these errors and the benefits of long-range, fast gates is implementation-dependent.

A notable implementation reported in the manuscript is with $^{133}$Cs atoms, where an experimental fidelity of 0.964 was achieved at $R \sim 6\,\mu$m interatomic spacing, approaching the theoretical expectations set by the asymmetric protocol and limited by technical rather than intrinsic quantum noise.

Conclusion

This work establishes an asymmetric, detuned $\pi$-$2\pi$-$\pi$ Rydberg gate protocol as a practical and theoretically compelling approach for long-range, high-fidelity quantum entanglement in neutral-atom qubit platforms. Its suppression of coherent rotation errors at moderate interaction strengths, tunability for arbitrary controlled-phase logic, and extensibility via robust optimal control techniques position it as a significant advance for scalable quantum logic and distributed quantum error correction. Future developments will involve hardware-optimized gate design, integration with noise-adaptive control, and further minimization of decay and position-based technical limitations, facilitating robust large-scale neutral atom quantum computing.