quantum gate directory

The controlled-Hadamard gate performs a conditional change of basis, moving the target between the computational basis and the Hadamard basis only when the control is $|1\rangle$.

$$ \mathrm{C}H = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \tfrac{1}{\sqrt{2}} & \tfrac{1}{\sqrt{2}} \\ 0 & 0 & \tfrac{1}{\sqrt{2}} & -\tfrac{1}{\sqrt{2}} \end{bmatrix} $$

Properties

• Hermitian and self-inverse, like the Hadamard itself.
• Real, so it lies in the orthogonal group.
• Although $H$ is Clifford, $\mathrm{C}H$ is not. Adding a control to a Clifford gate does not preserve Clifford-ness in general (compare CNOT, which stays Clifford, versus $\mathrm{C}H$ and $\mathrm{C}S$, which do not).

Decompositions

• A single entangling gate suffices: $\mathrm{C}H = (I \otimes R_y(\pi/4)) \, \mathrm{C}Z \, (I \otimes R_y(-\pi/4))$, which works because $R_y(\pi/4) \, Z \, R_y(-\pi/4) = H$.