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.

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

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

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)

• 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)

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

• 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