RYDBERG QUANTUM GATES

Gate Fidelity Budget

Quantify every error source in a Rydberg two-qubit gate — from spontaneous emission and Doppler dephasing to laser noise, finite blockade, and SPAM. Identify your dominant bottleneck and compare to state-of-the-art experimental fidelities.

8error sources
FTotal gate fidelity
ε_iPer-source budget
99.5%SOTA benchmark

01 Gate & Atom Setup

Rydberg Gate Parameters
n=30n=60n=100+
0.1 MHz1 MHz10 MHz
0.1 μs2 μs100 μs
Noise Sources
0.1 Hz1 kHz1 MHz
1× (weak)50×200× (strong)

02 Gate Fidelity & Error Budget

Total Gate Fidelity F = 1 − Σεᵢ —
90%95%99%99.5%99.9%100%
Rydberg lifetime τ
—
—
Total error Σεᵢ
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= 1 − F
Dominant error source
—
—
Per-source error budget
Error source εᵢ Fraction Contribution
Computing…
Error breakdown — relative contributions

03 State-of-the-Art Comparison

Evered et al. 2023 (Harvard)
99.5%
Rb-87, n=70 CZ gate. Best published two-qubit Rydberg fidelity. Nature 604, 2023.
Ma et al. 2023 (Princeton)
99.3%
¹⁷¹Yb clock qubit, 302 nm Rydberg excitation. Phys. Rev. X 2023.
Levine et al. 2022 (QuEra)
99.0%
Rb-87, n=70, parallel gates in 256-atom array. Nature 2022.
FTQC threshold (surface code)
99.0%
Typical threshold for fault-tolerant QC with surface codes (~1% physical error rate).
Typical lab (optimised)
97–99%
Well-optimised Rydberg gate without extensive noise reduction (Δν~1 kHz, T~5 μK).
Your simulation
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Set parameters above to compute.

04 Error Source Formulas

Radiative decay during gate

During a Rydberg gate, atoms spend time in the Rydberg state (lifetime τ_Ryd ∝ n*³). Any spontaneous emission event destroys the coherence and constitutes an error. At room temperature, blackbody radiation (BBR) drives transitions to neighbouring Rydberg states, effectively halving the lifetime for n ~ 50–70.

$$\tau_{\rm Ryd} \approx \tau_0 \times n^{*3} \qquad (\tau_0 \sim 1\text{ ns for alkali }nS)$$ With BBR (300 K): $\Gamma_{\rm total} = \Gamma_{\rm rad} + \Gamma_{\rm BBR} \approx 2\Gamma_{\rm rad}$ ($n \sim 60$)
Cryogenic ($<10$ K): $\Gamma_{\rm total} \approx \Gamma_{\rm rad}$ (BBR suppressed)
Spontaneous emission error: $$\varepsilon_{se} = \Gamma_{total} \times t_{gate} = t_{gate}/\tau_{Ryd}$$

Random Doppler phase accumulated during the gate

A thermal atom moving at speed v accumulates a random phase δφ = k_eff × v × t_gate from the Rydberg excitation laser. For a Gaussian velocity distribution, the gate fidelity is reduced by the dephasing factor.

$$v_{\rm rms} = \sqrt{\frac{k_{\rm B} T}{m}} \quad \text{[1D thermal velocity]}$$ Phase uncertainty: $\delta\phi = k_{\rm eff} \cdot v_{\rm rms} \cdot t_{\rm gate}$
Dephasing error: $$\varepsilon_D = \frac{1}{2}(k_{\rm eff} \cdot v_{\rm rms} \cdot t_{\rm gate})^2$$ Single-photon excitation ($\lambda \sim 300$–$480$ nm): $k_{eff} = 2\pi/\lambda_{Ryd}$
Example: Rb, $T=5\,\mu$K, $\lambda=480$ nm, $t_{gate}=2\,\mu$s $\Rightarrow$ $v_{rms} = 8.5$ mm/s, $\delta\phi = 0.22$ rad, $\varepsilon \approx 2.4\times10^{-2}$

Laser coherence time vs gate duration

A laser with Lorentzian linewidth Δν has coherence time T_coh = 1/(π Δν). If the gate time t_gate ≳ T_coh, phase noise significantly degrades the fidelity. Narrow-linewidth lasers (<1 Hz) are needed for high-fidelity Rydberg gates.

$$T_{\rm coh} = \frac{1}{\pi\,\Delta\nu}$$ Phase noise error (white frequency noise): $$\varepsilon_\phi \approx \pi\,\Delta\nu\cdot t_{\rm gate}$$ For $t_{\rm gate} \ll T_{\rm coh}$: $\varepsilon_\phi \ll 1$ ✓; for $t_{\rm gate} = T_{\rm coh}$: $\varepsilon_\phi \approx 1$ ✗
State-of-the-art: $\Delta\nu < 1$ Hz (ULE cavity) $\Rightarrow T_{coh} > 300$ ms $\Rightarrow \varepsilon_\phi < 10^{-5}$ for $t_{gate} = 2\,\mu$s

Leakage into doubly-excited |rr⟩ state

The Rydberg blockade assumes the interaction U ≫ Ω. When this is not satisfied, the doubly-excited |rr⟩ state acquires a small amplitude ~ Ω/U, causing gate errors. For the standard CZ protocol:

$$\varepsilon_{\rm blockade} \approx \left(\frac{\Omega}{U}\right)^2$$ $U/\Omega \geq 10$: $\varepsilon < 1\%$; $U/\Omega \geq 50$: $\varepsilon < 0.04\%$; $U/\Omega \geq 100$: $\varepsilon < 0.01\%$
$C_6$ at $n\sim60$ (Rb): $\sim10^5$ GHz$\cdot\mu$m$^6$; $r=5\,\mu$m $\Rightarrow U/2\pi = C_6/r^6 \sim 130$ MHz $\Rightarrow U/\Omega \approx 130$ for $\Omega/2\pi = 1$ MHz ✓
Magnetic field noise ($g_F m_F \sim 1$): $$\delta\phi_B = \frac{\mu_B \cdot \Delta B \cdot t_{\rm gate}}{\hbar} \qquad \varepsilon_B = \frac{1}{2}\left(\frac{\mu_B \cdot \Delta B \cdot t_{\rm gate}}{\hbar}\right)^2$$ Use clock states ($m_F=0$) to suppress by $\sim10^3$
Rabi inhomogeneity: $$\varepsilon_\Omega = \frac{1}{2}\left(\frac{\pi\,\delta\Omega/\Omega}{2}\right)^2 \quad \text{[suppress with composite pulses/DRAG]}$$ SPAM: $\varepsilon_{\rm SPAM} = p_{\rm prep} + p_{\rm meas}$
Atom loss: $\varepsilon_{\rm loss} = t_{\rm gate}/\tau_{\rm trap}$ ($\tau_{\rm trap} \sim 10$–$100$ s in UHV tweezers)

05 Key References

📄 Evered et al. (2023) — High-fidelity parallel entangling gates on a neutral-atom QC (Nature) 📄 Ma et al. (2023) — High-fidelity gates with ¹⁷¹Yb Rydberg qubits (Phys. Rev. X) 📄 Jaksch et al. (2000) — Fast quantum gates for neutral atoms (original Rydberg gate proposal) 📄 Saffman, Walker & Mølmer (2010) — Quantum information with Rydberg atoms (Rev. Mod. Phys.) 📄 Levine et al. (2022) — Dispersive optical systems for scalable Rydberg gates (Nature)