  We propose an optimal set of quantum gates for a singlet-triplet qubit in a
double quantum dot with two electrons utilizing the $S$-$T_-$ subspace. Qubit
rotations are driven by the applied magnetic field and an orthogonal field
gradient provided by a micromagnet. We optimize the fidelity of this qubit as a
function of magnetic fields, taking advantage of "sweet spots" where the
rotation frequencies are independent of the energy level detuning, providing
protection against charge noise. We simulate gate operations and qubit
rotations in the presence of quasistatic noise from charge and nuclear spins as
well as leakage to nonqubit states, and predict that in silicon quantum dots
gate fidelities greater than $99\%$ can be achieved for two nearly-orthogonal
rotation axes.
