Exchange-pulse CNOT synthesis

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.

With $S=P_{+}+i{P}_{-}$ the conventional square root of the SWAP gate, $P=\mathrm{diag}(1,i)$ the S gate, and operators acting rightmost first, $S{Z}_{1}S=\mathrm{diag}(1,i,-i,-1)$. Therefore $H_{2}S{Z}_{1}S{P}_2{H}_2=|0⟩⟨0|\otimes X-i|1⟩⟨1|\otimes I$. Appending $X_2$ changes the zero-controlled flip into a one-controlled flip, and $P_1$ removes the control-dependent phase. This proves the displayed exact CNOT gate identity. Both the exchange branch and the phase gate square-root convention matter; changing either requires recalculating the correction.