Fig. 2: Gate set calibration (GSC).

a Numerical pulse optimization based on a realistic but inaccurate qubit model provides initial optimal control pulses (blue) for 36 ns long π/2_x and π/2_y gates. According to the model, the pulses shown in red are actually seen by the qubit. b Next, these pulses are optimized on the experiment using closed-loop feedback. 8 error syndromes \(\tilde{{S}_{i}}\) are extracted in each iteration by applying the gate sequences from Table 1. In order to remove gate errors, the syndromes \(\tilde{{S}_{i}}\) are minimized by adjusting the pulse segments' amplitudes \({\epsilon }_{j}^{g}\). c Typically, GSC converges within 15 iterations and can recover from charge rearrangements in the quantum dot (indicated by a red dot, see Supplementary Note 16). Before iteration 1, the gate fidelity is typically so low that randomized benchmarking³³ (RB) can not be used to reliably extract the gate fidelity. This is remedied by scaling the pulses before the first iteration, leading to an average Clifford gate fidelity between 63 and 70%. In this specific calibration run, the feedback loop improved the fidelity of the gate set first to 99.0 ± 0.1%, then to 99.3 ± 0.1% (after disabling the decoherence syndromes \(\tilde{{S}_{7}}\) and \(\tilde{{S}_{8}}\)) and eventually to 99.50 ± 0.04% (after adding a small correction of 0.05 to S_M). All of these fidelities are extracted using RB. d, e For a different gate set consisting of two 24 ns long pulses, we performed self-consistent state tomography³². After a few GSC iterations, the simulated Bloch sphere trajectories (right) can be reproduced in the experiment (left). A major portion of the remaining deviation can be attributed to concatenation errors with the measurement pulses, specifically when states following large J pulses are determined.