Square root of the SWAP gate ($\sqrt{\mathrm{SWAP}}=P_T+i{P}_S$)

For two spin-one-half qubits, the SWAP gate acts as one on the triplet sector and minus one on the singlet sector. The displayed unitary squares to SWAP. An isotropic exchange interaction with integrated angle $\pi /2$ implements it up to a common phase. There are other square-root branches; a pulse decomposition must use the declared branch consistently.

Two square roots of the SWAP gate, separated and followed by local $z$ rotations, implement a controlled phase gate. Local $y$ rotations of its target convert it into a CNOT gate. The construction assumes isotropic exchange and independent one-qubit control; pulse errors and extra Hamiltonian terms change the implemented operation.