# CY gate (Controlled-Y)

**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:

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

## Relation to CX and CZ

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).

## Properties

- **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

## Uses

- **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

## Decomposition

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.

## Implementation

- **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

## Relations

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