  We show that the exchange interaction of a singlet-triplet spin qubit
confined in double quantum dots, when being controlled by the barrier method,
is insensitive to a charged impurity lying along certain directions away from
the center of the double-dot system. These directions differ from the polar
axis of the double dots by the magic angle, equaling
$\arccos\left(1/\sqrt{3}\right)\approx 54.7^\circ$, a value previously found in
atomic physics and nuclear magnetic resonance. This phenomenon can be
understood from an expansion of the additional Coulomb interaction created by
the impurity, but also relies on the fact that the exchange interaction solely
depends on the tunnel coupling in the barrier-control scheme. Our results
suggest that for a scaled-up qubit array, when all pairs of double dots rotate
their respective polar axes from the same reference line by the magic angle,
cross-talks between qubits can be eliminated, allowing clean single-qubit
operations. While our model is a rather simplified version of actual
experiments, our results suggest that it is possible to minimize unwanted
couplings by judiciously designing the layout of the qubits.
