P phase gate

Definition

$$P(\lambda) = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\lambda} \end{pmatrix}$$

$P$ fixes the $|0\rangle$ amplitude at exactly $1$ and rotates only the $|1\rangle$ amplitude by $\lambda$ — a convention choice, since any diagonal unitary is determined by its two phases up to an overall factor.

Action on basis states

$$P(\lambda)|0\rangle = |0\rangle, \qquad P(\lambda)|1\rangle = e^{i\lambda}|1\rangle$$

Applied to a superposition $\alpha|0\rangle+\beta|1\rangle$, only the relative phase between the two amplitudes changes, by exactly $\lambda$.

Key identities and special cases

$$P(\pi/2) = S, \qquad P(\pi/4) = T, \qquad P(\pi) = Z, \qquad P(2\pi) = I$$

so $P$ is the continuous parent of the entire discrete phase-gate ladder $\{I, T, S, Z\}$ built from repeated $\pi/4$ steps.

• $P(\lambda) = e^{i\lambda/2}R_Z(\lambda)$: $P$ and $R_Z$ implement the same rotation up to a global phase; $P$ is the more convenient convention when composing controlled-phase gates, since a controlled-$P$ needs no extra correction for the global phase (a controlled-$R_Z$ does).
• $P(\lambda_1)P(\lambda_2) = P(\lambda_1+\lambda_2)$: phases add, so repeated $P$ gates telescope into one gate at synthesis time.
• Controlled-$P(\lambda)$ is the entangling phase primitive used at every stage of the quantum Fourier transform, where the angles are $\pi/2, \pi/4, \pi/8, \dots$.