  Implementing arbitrary single-qubit gates with near perfect fidelity is among
the most fundamental requirements in gate-based quantum information processing.
In this work, we fabric a transmon qubit with long coherence times and
demonstrate single-qubit gates with the average gate error below $10^{-4}$,
i.e. $(7.42\pm0.04)\times10^{-5}$ by randomized benchmarking (RB). To
understand the error sources, we experimentally obtain an error budget,
consisting of the decoherence errors lower bounded by
$(4.62\pm0.04)\times10^{-5}$ and the leakage rate per gate of
$(1.16\pm0.04)\times10^{-5}$. 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.
