a transmon qubit with long coherence times and demonstrate single-qubit gates with the average gate error below 10−4, i.e. (7.42±0.04)×10−5 by randomized benchmarking (RB). To understand the error sources, we experimentally obtain an error budget, consisting of the decoherence errors lower bounded by (4.62±0.04)×10−5 and the leakage rate per gate of (1.16±0.04)×10−5. Moreover, we reconstruct the process matrices for the single-qubit gates by the gate set tomography (GST), with which we simulate RB sequences and obtain single-qubit fedlities consistent with experimental results. We also observe non-Markovian behavior in the experiment of long-sequence GST, which may provide guidance for further calibration. The demonstration extends the upper limit that the average fidelity of single-qubit gates can reach in a transmon-qubit system, and thus can be an essential step towards practical and reliable quantum computation in the near future.