Optimal two-qubit quantum circuits using exchange interactions

We give the optimal decomposition of a universal two-qubit circuit using Heisenberg exchange interactions and single qubit rotations. Tuning the strength and duration of the Heisenberg exchange allows one to implement (SWAP){sup {alpha}} gates. Our optimal circuit is constructed from only three (SWAP){sup {alpha}} gates and six single qubit gates. We show that three (SWAP){sup {alpha}} gates are not only sufficient, but necessary. Since six single-qubit gates are already known to be necessary, our implementation is optimal in gate count.