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 $|111\rangle \to -i|110\rangle$.

• 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

• 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

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.

• 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

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