An asymmetric and fast Rydberg gate protocol for long range entanglement

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.

💡 Research Summary

The paper introduces a modified Rydberg gate protocol that builds on the classic π‑2π‑π scheme originally proposed by Jaksch et al. The key innovation is the addition of a detuning Δ to the 2π pulse applied to the target qubit, creating an asymmetric configuration where the control and target atoms can be driven with different Rabi frequencies. By choosing Δ = V/2 (with V the two‑atom interaction strength) and setting the Rabi frequency to Ω = √3 V/2, the authors ensure that both the |00⟩ and |10⟩ computational states undergo exact 2π rotations, eliminating coherent rotation errors even when the interaction strength is only comparable to the Rabi rate.

A detailed error analysis shows that the average population in the Rydberg state during the gate is P_r = π(33+8√3)/(24 V), leading to a scattering‑limited error ϵ ≈ 2.39 · ϵ_DDP, where ϵ_DDP = (1+π/2)/(V τ) is the fundamental limit set by the Rydberg lifetime τ. By allowing the control atom to be driven more strongly (Ω_c = p Ω with p>1), the error can be reduced further to ϵ ≈ (11π/8)/(V τ), approaching 1.68 · ϵ_DDP as p → ∞. This asymmetric design thus achieves high fidelity without requiring extremely strong interactions, enabling long‑range entanglement.

The protocol is generalized to arbitrary controlled phases. By enforcing integer multiples of 2π rotations for both the blocked and unblocked pathways (parameters n₀ and n_V), the authors derive analytic expressions for the required detuning, Rabi frequency, and pulse duration. This yields a continuous tunability of the controlled phase θ from –π to +π, as illustrated in Fig. 3.

Time‑optimal pulse shaping is explored using GRAPE to modulate the laser phase ξ(t) on the target atom while keeping Ω fixed. The optimization reveals two limiting regimes: (i) fixed Ω with V → ∞ gives a gate time τ → 2π/Ω (the original π‑2π‑π limit); (ii) fixed V with Ω → ∞ yields τ → π/V (the interaction‑gate limit). The analytic solution with V = 2Ω/√3 is found to be globally optimal for a given Ω, achieving τ ≈ 5.44/Ω.

Robustness against static parameter fluctuations is addressed through robust optimal control. Assuming a realistic ratio Ω/Γ = 2π × 150 (Γ being the Rydberg decay rate), the authors design phase‑modulated pulses that tolerate ±5 % variations in Ω or V while keeping the gate error low. The robust solutions require longer durations (≈1.5–2 t_opt, where t_opt = 2π/V) but reduce the average error by a factor of three under the specified noise levels.

In the discussion, the new asymmetric gate is compared with several state‑of‑the‑art Rydberg gates, including dark‑state, time‑optimal symmetric, and dressing schemes. While the symmetric time‑optimal gate can reach ϵ ≈ 1.33 · ϵ_DDP, the asymmetric protocol achieves comparable or better performance without demanding the strongest possible interactions. This makes it especially attractive for applications requiring long‑range entanglement, such as non‑local error‑correcting codes.

Overall, the paper provides a comprehensive theoretical framework for an asymmetric, fast, and robust Rydberg CZ gate, offering practical pathways to high‑fidelity two‑qubit operations in neutral‑atom quantum processors where interaction strengths are moderate and laser power is limited.