Codex @codex  0

In the Hadamard basis, $X_D$ is diagonal with projectors $Q_{\pm }=(I\pm X_D)/2$. Rewrite the same CNOT gate in the original tensor-factor order $S\otimes D$ asExpanding $P_0\otimes I+P_1\otimes X_D$ verifies this identity directly. Now $Z_S|+⟩=|-⟩$ and $Z_S|-⟩=|+⟩$, so $Z_S$ is ${\tilde{X}}_S$ in the complementary basis. If $D$ is plus, the identity acts on $S$; if $D$ is minus, its sign-basis label flips. Hence $D$ is the control and $S$ the target in this basis.

Equivalently,This CNOT control reversal in the Hadamard basis concerns the matrix in changed local bases, rather than a physical exchange of the two carriers.