# YY Gate (Parametric)

**YY gate** is a parametric two-qubit gate implementing correlated Y rotations on both qubits.

Matrix:

$$\text{YY}(\theta) = e^{-i\theta Y_1 Y_2 / 2} = \begin{pmatrix} \cos(\theta/2) & 0 & 0 & i\sin(\theta/2) \\ 0 & \cos(\theta/2) & -i\sin(\theta/2) & 0 \\ 0 & -i\sin(\theta/2) & \cos(\theta/2) & 0 \\ i\sin(\theta/2) & 0 & 0 & \cos(\theta/2) \end{pmatrix}$$

Action: entangling two-qubit rotation around YY axis.

## Special Cases

- $\text{YY}(0) = I$
- $\text{YY}(\pi/2)$ creates entanglement
- $\text{YY}(\pi)$ full rotation

## Properties

- **Parametric**: tune interaction strength
- **Symmetric**: both qubits interact equally
- **Entangling gate**
- **Hamiltonian form**: $\text{YY}(\theta) = e^{-i\theta Y \otimes Y / 2}$

## Physical Origin

Arises from:

- **Superconducting qubits**: some coupling geometries
- **Trapped ions**: laser interactions
- **Spin systems**: YY coupling (less common than XX or ZZ in nature, but engineerable)

## Relation to XX and ZZ

- **Gate family**: Pauli axis determines interaction character
- **XX**: $e^{-i\theta X \otimes X / 2}$ (correlate X rotations)
- **YY**: $e^{-i\theta Y \otimes Y / 2}$ (correlate Y rotations)
- **ZZ**: $e^{-i\theta Z \otimes Z / 2}$ (correlate Z rotations)

## Uses

- **Ansatze**: YY gates in variational circuits for certain problems
- **Quantum simulation**: simulate YY-coupled systems
- **Native gates**: some platforms implement YY directly

## Decomposition

- **Typical composition**: X rotations, CNOT gates, and single-qubit gates if not native
- **Availability**: less common native gate than XX or ZZ, but available on some platforms