Phase-corrected exchange network for a CNOT gate (${\mathrm{CNOT}}_{1\to 2}=P_1{X}_2{H}_{2}S{Z}_{1}S{P}_2{H}_2$)

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.