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# Pauli Y Gate

**Pauli Y** combines a bit flip (X) and a phase flip (Z), with a relative phase.

Matrix:

$$Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$$

Action: $Y|0\rangle = i|1\rangle$, $Y|1\rangle = -i|0\rangle$.

## Properties

- **Self-inverse**: $Y^2 = I$
- **Eigenvalues**: $+1, -1$ with eigenvectors $(|0\rangle + i|1\rangle)/\sqrt{2}$ and $(|0\rangle - i|1\rangle)/\sqrt{2}$
- **Commutation**: anticommutes with X and Z
- **Relation to X and Z**: $Y = iXZ = iZX$

## Uses

- **Basis rotation**: measure in Y eigenbasis (Y-basis)
- **Algorithm**: less common than X or Z; appears in specific variational ansatze
- **Bloch sphere**: rotation by $\pi$ around y-axis

## Implementation

- **Superconducting qubits**: RY($\pi$) via microwave pulse with appropriate phase; same duration as X gate
- **Trapped ions**: two resonant laser pulses (or single pulse at angle) to achieve Y rotation
- **Decomposition**: often as $Y = RX(\pi/2) Z RX(-\pi/2)$ if not native
- **Fidelity**: typically 99.5–99.9%