Controlled-Hadamard gate

Definition

$$\mathrm{CH} = |0\rangle\langle 0|\otimes I + |1\rangle\langle 1|\otimes H = \begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&\tfrac{1}{\sqrt2}&\tfrac{1}{\sqrt2}\\0&0&\tfrac{1}{\sqrt2}&-\tfrac{1}{\sqrt2} \end{pmatrix}$$

with the control as the first (leftmost) qubit. Unlike $\mathrm{CZ}$ or $\mathrm{CP}$, this matrix is not diagonal, and unlike $\mathrm{CX}$, the applied operation $H$ is not a Pauli matrix.

Action on basis states

$$\mathrm{CH}|0,t\rangle = |0,t\rangle, \qquad \mathrm{CH}|1,0\rangle = |1\rangle\otimes\tfrac{|0\rangle+|1\rangle}{\sqrt2}, \qquad \mathrm{CH}|1,1\rangle = |1\rangle\otimes\tfrac{|0\rangle-|1\rangle}{\sqrt2}$$

When the control is $|0\rangle$ the target is left completely untouched; when the control is $|1\rangle$ the target is put into superposition exactly as a bare $H$ would.

Decomposition

Because $H$ is neither diagonal nor Pauli, $\mathrm{CH}$ is not directly native on most hardware and is built with the general controlled-$U$ recipe of Barenco et al. (1995): find single-qubit gates $A,B,C$ with

$$ABC = I, \qquad AXBXC = H \text{ (up to global phase)}$$

and implement controlled-$H$ as $C$–CNOT–$B$–CNOT–$A$ on the target, conditioned on the control via two CNOTs. A common concrete choice uses $A = R_Y(\pi/4)$, $B = R_Y(-\pi/4)$, and a compensating phase gate for $C$, giving a 2-CNOT circuit.

Key identities

• $\mathrm{CH}^2 = I$: applying the same controlled-Hadamard twice restores the input, since $H^2 = I$.
• $\mathrm{CH}$ conditioned on $|+\rangle$ on the control produces genuine entanglement between control and target, unlike $\mathrm{CZ}$/$\mathrm{CP}$ acting on product states, because $H$ mixes the target's basis rather than only phasing it.