  Singlet-triplet spin qubits in six-electron double quantum dots, in moderate
magnetic fields, can show superior immunity to charge noise. This immunity
results from the symmetry of orbitals in the second energy shell of circular
quantum dots: singlet and triplet states in this shell have identical charge
distributions. Our phase-gate simulations, which include $1/f$ charge noise
from fluctuating traps, show that this symmetry is most effectively exploited
if the gate operation switches rapidly between sweet spots deep in the (3,3)
and (4,2) charge stability regions; fidelities very close to one are predicted
if subnanosecond switching can be performed.
