  We propose a scheme for implementing the CNOT gate over qubits encoded in a
pair of electron spins in a double quantum dot. The scheme is based on exchange
and spin orbit interactions and on local gradients in Zeeman fields. We find
that the optimal device geometry for this implementation involves effective
magnetic fields that are parallel to the symmetry axis of the spin orbit
interaction. We show that the switching times for the CNOT gate can be as fast
as a few nanoseconds for realistic parameter values in GaAs semiconductors.
Guided by recent advances in surface codes, we also consider the perpendicular
geometry. In this case, leakage errors due to spin orbit interaction occur but
can be suppressed in strong magnetic fields.
