  We use quantum optimal control to identify fast collision-based two-qubit
$\sqrt{\text{SWAP}}$ gates in ultracold atoms. We show that a significant speed
up can be achieved by optimizing the full gate instead of separately optimizing
the merge-wait-separate sequence of the trapping potentials. Our optimal
strategy does not rely on the atoms populating the lowest eigenstates of the
merged potential, and it crucially includes accumulation of quantum phases
before the potentials are fully merged. Our analyses transcend the particular
trapping geometry, but to compare with previous works, we present systematic
results for an optical lattice and find greatly improved gate durations and
fidelities.
