  We present a method to construct high-fidelity quantum phase gates, which are
insensitive to errors in various experimental parameters. The phase gates
consist of a pair of two sequential broadband composite pulses, with a phase
difference $\pi+\alpha/2$ between them, where $\alpha$ is the desired gate
phase. By using composite pulses which compensate systematic errors in the
pulse area, the frequency detuning, or both the area and the detuning, we
thereby construct composite phase gates which compensate errors in the same
parameters. Particularly interesting are phase gates which use the recently
discovered universal composite pulses, which compensate systematic errors in
any parameter of the driving field, which keep the evolution Hermitian (e.g.,
pulse amplitude and duration, pulse shape, frequency detuning, Stark shifts,
residual frequency chirps, etc.
