Extended Data Fig. 3: Atomic physics error-level diagram and numerical comparison of benchmarking methods.

a, Level diagram shows the eight states assumed in the simulation. We assume a 88-μs Rydberg-state lifetime (based on measured T₁ with 1,013-nm scattering lifetime subtracted) and a 110-ns lifetime for the intermediate states. We assume the following branching ratios for the intermediate states⁹⁹: η_e→L = 0.6142, η_e→1 = 0.2504, η_e→0 = 0.1354, and the following ones for the Rydberg states¹⁰⁰: η_r→L = 0.894, η_r→1 = 0.053, η_r→0 = 0.053. We use the branching ratios between different channels of intermediate-state scattering as reported in ref. ⁹⁹ and we also assume a simplified model in which all indirect paths (through 4D and 6S) populate the ground-state manifold uniformly. The Rydberg lifetime has both radiative decay (170 μs) and black-body decay (128 μs) components, which we obtain by rescaling the values in ref. ¹⁰⁰ to n = 53. The microwave component results purely in atom loss and we assume that the radiative decay populates the ground-state manifold uniformly. We note that a more accurate treatment of the decay channels³⁶ could increase error modelling precision in future work. b, Benchmarking of the CZ-π-CZ sequence with global random rotations, which is insensitive to the single-particle phase. c, Benchmarking a standalone CZ gate with global random rotations, which enables separate calibration of the single-particle phase. d, The usual interleaved randomized benchmarking method using random two-qubit Clifford gates (not performed in this work). e, Numerical simulation of the presented benchmarking methods and the Bell-state-preparation method, using the realistic error model developed in this work. All approaches give consistent results, with the Bell-state fidelity measurement lower-bounding the other curves.