A two-qubit Rydberg CZ gate must achieve raw fidelity F > 99% to meet common fault-tolerance thresholds for practical quantum error correction (surface-code thresholds are order 1%, depending on noise model and decoder). But which error source dominates? This tool builds a complete error budget across 8 contributions, spontaneous emission from the Rydberg state, Doppler dephasing, laser phase noise, blockade leakage, SPAM, atom loss, B-field dephasing, and Rabi inhomogeneity, so you can immediately see which term to attack first. Compare against published Rydberg-gate benchmarks from Rb and Yb platforms.
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
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90%95%99%99.5%99.9%100%
Rydberg lifetime τ
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Total error Σεᵢ
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= 1 − F
Dominant error source
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Per-source error budget
| Error source | εᵢ | Fraction | Contribution |
|---|---|---|---|
| Computing… | |||
Error breakdown, relative contributions
Error waterfall — cumulative build-up
Each bar adds its error to the running total. The dashed line marks the fault-tolerance threshold (0.1% error = 99.9% fidelity).
03 State-of-the-Art Comparison
Evered et al. 2023 (Harvard)
99.5%
Rb-87, n=70 CZ gate with 60 pairs in parallel. Nature 622, 2023.
Muniz et al. 2025 (Atom/JILA)
99.72%
¹⁷¹Yb nuclear-spin CZ gate with post-selection; 99.40% without post-selection. PRX Quantum 2025.
Ma et al. 2023 (Princeton)
98.0%
Metastable ¹⁷¹Yb qubit with mid-circuit erasure conversion. Nature 622, 2023.
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: $$\varepsilon_{se} = \Gamma_{total} \times t_{gate} = t_{gate}/\tau_{Ryd}$$
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, $\varepsilon \approx 2.4\times10^{-2}$
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
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:
$$\varepsilon_{\rm blockade} \approx \left(\frac{\Omega}{U}\right)^2$$
$U/\Omega \geq 10$: $\varepsilon < 1\%$; $U/\Omega \geq 50$: $\varepsilon < 0.04\%$; $U/\Omega \geq 100$: $\varepsilon < 0.01\%$
$C_6/h$ for alkali $nS$ states near $n\sim60$--$70$ is typically order $10^2$--$10^3$ GHz$\cdot\mu$m$^6$ and is strongly state- and Förster-defect-dependent. At $r=5\,\mu$m this gives interaction shifts from tens to hundreds of MHz, so $U/\Omega \gg 1$ is achievable for $\Omega/2\pi\sim1$ MHz.
$C_6/h$ for alkali $nS$ states near $n\sim60$--$70$ is typically order $10^2$--$10^3$ GHz$\cdot\mu$m$^6$ and is strongly state- and Förster-defect-dependent. At $r=5\,\mu$m this gives interaction shifts from tens to hundreds of MHz, so $U/\Omega \gg 1$ is achievable for $\Omega/2\pi\sim1$ 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)
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)
Muniz et al. (2025), High-fidelity universal gates in the ¹⁷¹Yb nuclear-spin qubit (PRX Quantum)
Ma et al. (2023), High-fidelity gates and mid-circuit erasure conversion in ¹⁷¹Yb (Nature)
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)