Extended Data Fig. 7: Analysis of fidelities and rates when entangling two remote ¹⁷¹Yb spin qubits.

a, Entanglement fidelity analysis. We characterize the entanglement fidelity by performing simulations based on independently measured experimental parameters. This leads to a simulated fidelity of \({\mathcal{F}}=0.729\,\pm \,0.004\) (dashed vertical line with grey region for error bar), consistent with experimental results. We characterize the individual fidelity contributions in two ways: first by simulating the fidelity when a given error source is removed (blue bars), and secondly, by simulating the fidelity when the error source is exclusively present (orange bars). We find three dominant sources of error: undetected spontaneous emission during the dynamic rephasing protocol; noise counts i.e. photons that don’t originate from our two ¹⁷¹Yb ions; and errors in the optical gates applied to the ions. Error bars are obtained by performing 50 simulation repetitions, each with input parameters sampled from Gaussian distributions with mean and standard deviation given by the experimentally determined values and errors presented in Supplementary Information section V. More detail on the fidelity estimation can be found in Supplementary Information sections VII and VIII. b, Entanglement rate analysis. The predicted entanglement rate of \({\mathcal{R}}=3.48\,\pm \,0.03\) Hz is obtained from the experiment repetition rate, 12.3 kHz, multiplied by the entanglement success probability, (2.83 ± 0.02) × 10⁻⁴. A slight deviation from the measured result is due to slow drift of experimental parameters on the timescale of several days between measurement and calibration. The dominant limitation to the experiment repetition rate arises from the qubit initialization time, requiring 33 μs per attempt. We also list contributions to the success rate consisting of: the photon detection efficiency, the fraction of photons within the 500 ns acceptance window, and the weak superposition states used in the single photon heralding protocol. More detail on this estimation can be found in Supplementary Information section IX.