  State of the art qubit systems are reaching the gate fidelities required for
scalable quantum computation architectures. Further improvements in the
fidelity of quantum gates demands characterization and benchmarking protocols
that are efficient, reliable and extremely accurate. Ideally, a benchmarking
protocol should also provide information on how to rectify residual errors.
Gate Set Tomography (GST) is one such protocol designed to give detailed
characterization of as-built qubits. We implemented GST on a high-fidelity
electron-spin qubit confined by a single $^{31}$P atom in $^{28}$Si. The
results reveal systematic errors that a randomized benchmarking analysis could
measure but not identify, whereas GST indicated the need for improved
calibration of the length of the control pulses. After introducing this
modification, we measured a new benchmark average gate fidelity of
$99.942(8)\%$, an improvement on the previous value of $99.90(2)\%$.
Furthermore, GST revealed high levels of non-Markovian noise in the system,
which will need to be understood and addressed when the qubit is used within a
fault-tolerant quantum computation scheme.
