# CCY gate (Controlled-Controlled-Y)

**CCY** (Controlled-Controlled-Y) is a three-qubit gate: applies Y to the target qubit if both control qubits are $|1\rangle$. Like CCX, it is a doubly-controlled gate but less commonly native on quantum platforms.

Action: $|c_1 c_2 t\rangle \to |c_1 c_2 (Y|t\rangle)\rangle$ where $c_1, c_2$ are controls and $t$ is target. Applies Y only if both controls are 1.

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

Basis state mapping: acts as identity on all states except $|110\rangle$ and $|111\rangle$. Maps $|110\rangle \to i|111\rangle$ and $|111\rangle \to -i|110\rangle$.

## Properties

- **Self-inverse**: $\text{CCY}^2 = I$
- **Asymmetric**: like CCX, control and target are distinct
- **Related to CCX**: CCY can be constructed from CCX using single-qubit phase gate rotations on the target

## Uses

- **Alternative multi-control gate**: less common than CCX but equivalent in power for some algorithms
- **Quantum arithmetic**: phase-based operations in quantum circuits
- **Gate equivalence**: CCY, CCX, CCZ are interconvertible via single-qubit gates

## Decomposition

CCY can be decomposed using CCX and single-qubit gates:

$$\text{CCY} = (I \otimes I \otimes S^\dagger) \text{CCX} (I \otimes I \otimes S)$$

where $S$ and $S^\dagger$ are phase gates. Gate count: two single-qubit gates and one CCX.

## Implementation

- **Superconducting qubits**: typically decomposed into CCX or CX chains with phase gates; native implementation uncommon
- **Trapped ions**: can implement via laser interactions or decompose into native gates
- **Photonic**: challenging; usually decomposed

## Relations

- [[quantum-gate-ccx|CCX]]: doubly-controlled X; interconvertible via single-qubit rotations
- [[quantum-gate-ccz|CCZ]]: doubly-controlled Z; related by phase gate conjugation
- [[quantum-gate-y|Pauli Y]]: the gate applied to target when both controls are $|1\rangle$
- [[quantum-gate-phase|Phase gates]]: used in CCY decomposition
- [[quantum-gate-three-qubit|Three-qubit gates]]: general category