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Implementing arbitrary single-qubit gates with near perfect fidelity is amongthe most fundamental requirements in gate-based quantum information processing.In this work, we fabric a transmon qubit with long coherence times anddemonstrate single-qubit gates with the average gate error below $10^{-4}$,i.e. $(7.42\pm0.04)\times10^{-5}$ by randomized benchmarking (RB). Tounderstand the error sources, we experimentally obtain an error budget,consisting of the decoherence errors lower bounded by$(4.62\pm0.04)\times10^{-5}$ and the leakage rate per gate of$(1.16\pm0.04)\times10^{-5}$. Moreover, we reconstruct the process matrices forthe single-qubit gates by the gate set tomography (GST), with which we simulateRB sequences and obtain single-qubit fedlities consistent with experimentalresults. We also observe non-Markovian behavior in the experiment oflong-sequence GST, which may provide guidance for further calibration. Thedemonstration extends the upper limit that the average fidelity of single-qubitgates can reach in a transmon-qubit system, and thus can be an essential steptowards practical and reliable quantum computation in the near future.

Error per single-qubit gate below $10^{-4}$ in a superconducting qubit