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.
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