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

• 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: $Y = iXZ = iZX$

• 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

• 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%