CY applies the Y gate to the target qubit if the control qubit is $|1\rangle$. Like CX, it is an asymmetric Clifford two-qubit gate used for entanglement, though less common than CX or CZ on most quantum platforms.

Matrix (control on qubit 0, target on qubit 1):

$$\text{CY} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \end{pmatrix}$$

Action on basis states:

$$|00\rangle \to |00\rangle, \quad |01\rangle \to |01\rangle, \quad |10\rangle \to i|11\rangle, \quad |11\rangle \to -i|10\rangle$$

CY can be constructed from CX using phase gates on the target:

$$\text{CY}_{01} = (I \otimes S^\dagger) \text{CX}_{01} (I \otimes S)$$

where $S$ and $S^\dagger$ are phase gates that rotate the Y axis to align with X. Alternatively:

$$\text{CY} = (I \otimes S) \text{CZ} (I \otimes S^\dagger)$$

This shows CY has equivalent power to CX and CZ (all Clifford entangling gates).

• Self-inverse: $\text{CY}^2 = I$
• Asymmetric: control and target are distinct; $\text{CY}_{01} \neq \text{CY}_{10}$
• Clifford gate: preserves Pauli group under conjugation
• Relates to Paulis: applies Y (which involves $\pm i$ phase) to target

• Alternative entangling gate: less commonly used than CX, but equivalent in power
• Clifford circuits: part of the Clifford group; can be simulated classically
• Algorithm equivalence: CY, CX, CZ are interconvertible via single-qubit gates

CY can be decomposed using single-qubit gates and CX (or CZ):

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

Gate count: two single-qubit gates (S and S†) and one CX. Useful when CX is more efficient than native CY on a given platform.

• Superconducting qubits: less commonly native; typically decomposed into CX or CZ plus single-qubit gates; gate time similar to CX (~20–100 ns if native)
• Trapped ions: can be implemented directly via entangling laser pulses, or decomposed into CX/CZ
• Photonic: challenging; typically decomposed

• CX gate: controlled X; interconvertible via single-qubit rotations
• CZ gate: controlled Z; related by phase gate conjugation
• Pauli Y: the gate applied to target when control is |1⟩
• Phase gates: used in CY decomposition
• Clifford gates: CY is a Clifford gate (preserves Pauli closure)
• Two-qubit gates: general category