  A nuclear spin can act as a quantum switch that turns on or off ultracold
collisions between atoms even when there is neither interaction between nuclear
spins nor between the nuclear and electron spins. This "exchange blockade" is a
new mechanism for implementing quantum logic gates that arises from the
symmetry of composite identical particles, rather than direct coupling between
qubits. We study the implementation of the entangling $\sqrt{\text{SWAP}}$ gate
based on this mechanism for a model system of two atoms with ground electron
configuration $^1S_0$, spin 1/2 nuclei, trapped in optical tweezers. We
evaluate a proof-of-principle protocol based on adiabatic evolution of a one
dimensional double Gaussian well, calculating fidelities of operation as a
function of interaction strength, gate time, and temperature.
