  A most intuitive realization of a qubit is a single electron charge sitting
at two well-defined positions, such as the left and right sides of a double
quantum dot. This qubit is not just simple but also has the potential for
high-speed operation, because of the strong coupling of electric fields to the
electron. However, charge noise also couples strongly to this qubit, resulting
in rapid dephasing at nearly all operating points, with the exception of one
special 'sweet spot'. Fast dc voltage pulses have been used to manipulate
semiconductor charge qubits, but these previous experiments did not achieve
high-fidelity control, because dc gating requires excursions away from the
sweet spot. Here, by using resonant ac microwave driving, we achieve coherent
manipulation of a semiconductor charge qubit, demonstrating a Rabi frequency of
up to 2GHz, a value approaching the intrinsic qubit frequency of 4.5GHz. Z-axis
rotations of the qubit are well-protected at the sweet spot, and by using ac
gating, we demonstrate the same protection for rotations about arbitrary axes
in the X-Y plane of the qubit Bloch sphere. We characterize operations on the
qubit using two independent tomographic approaches: standard process tomography
and a newly developed method known as gate set tomography. Both approaches show
that this qubit can be operated with process fidelities greater than 86% with
respect to a universal set of unitary single-qubit operations.
