 Solution

The Hadamard basis changes the Pauli matrices by $Z_S={\tilde{X}}_S$ and $X_D={\tilde{Z}}_D$. Thus, keeping tensor factors ordered as $S,D$, the Hamiltonian operator isIf $D$ has new-basis label plus, the second factor vanishes. If $D$ has label minus, the integrated generator on $S$ is $\pi (I-{\tilde{X}}_S)/2$, whose exponential is ${\tilde{X}}_S$ by part (a). The generator therefore has the same controlled form with $D$ as control in the complementary bases.