Controlled-Y gate

Definition

$$\mathrm{CY} = |0\rangle\langle 0|\otimes I + |1\rangle\langle 1|\otimes Y = \begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&0&-i\\0&0&i&0 \end{pmatrix}$$

with the control as the first (leftmost) qubit. Like $\mathrm{CX}$, this matrix is a bit-flip on the target; unlike $\mathrm{CX}$, it also attaches a $\pm i$ phase, since $Y|0\rangle = i|1\rangle$ and $Y|1\rangle = -i|0\rangle$.

Action on basis states

$$\mathrm{CY}|0,t\rangle = |0,t\rangle, \qquad \mathrm{CY}|1,0\rangle = i|1,1\rangle, \qquad \mathrm{CY}|1,1\rangle = -i|1,0\rangle$$

The control is left untouched in every case; the target flips exactly when the control is $|1\rangle$, exactly as with $\mathrm{CX}$, but the flipped amplitude also picks up a phase of $\pm i$ depending on direction.

Decomposition

Using the single-qubit identity $Y = S\,X\,S^\dagger$ (verified directly: $S X S^\dagger = \begin{pmatrix}0&-i\\i&0\end{pmatrix} = Y$), conjugating a plain $\mathrm{CX}$'s target by $S^\dagger$ before and $S$ after reproduces $\mathrm{CY}$ exactly:

$$\mathrm{CY} = (I\otimes S)\,\mathrm{CX}\,(I\otimes S^\dagger)$$

so the circuit applies $S^\dagger$ to the target, then $\mathrm{CX}$, then $S$ to the target — a direct three-gate analogue of the $\mathrm{CZ} = (I\otimes H)\,\mathrm{CX}\,(I\otimes H)$ identity elsewhere in this catalog.

Key identities

• $\mathrm{CY}^2 = I\otimes I$: $\mathrm{CY}$ is its own inverse, since $Y^2=I$.
• $\mathrm{CY}$ is Clifford: it maps Pauli operators to Pauli operators under conjugation, just like $\mathrm{CX}$ and $\mathrm{CZ}$.
• Together, $\mathrm{CX}$, $\mathrm{CY}$, and $\mathrm{CZ}$ are the three controlled-Pauli gates; any one can be built from either of the others plus single-qubit Clifford conjugation ($H$ for $\mathrm{CZ}\leftrightarrow\mathrm{CX}$, $S$ for $\mathrm{CY}\leftrightarrow\mathrm{CX}$).