Phase gates are single-qubit quantum gates that change the relative phase of the $\lvert 1\rangle$ component of a qubit while leaving the $\lvert 0\rangle$ component unchanged. The general phase gate is usually written as $P(\phi)$:
$$P(\phi) = \begin{pmatrix}1 & 0\\0 & e^{i\phi}\end{pmatrix}$$
Applied to a qubit in the state $\lvert\psi\rangle = \alpha\lvert0\rangle + \beta\lvert1\rangle$, it produces
$$P(\phi)\lvert\psi\rangle = \alpha\lvert0\rangle + e^{i\phi}\beta\lvert1\rangle$$
The phase is not directly visible when the qubit is measured in the computational basis. It becomes observable when the amplitudes interfere, for example after applying a Hadamard gate.
The most commonly used phase gates are:
| Gate | Matrix | Phase $\phi$ | Relationship |
| — | — | — | — |
| P gate | $\begin{pmatrix}1 & 0\\0 & e^{i\phi}\end{pmatrix}$ | arbitrary | general phase gate |
| S gate | $\begin{pmatrix}1 & 0\\0 & i\end{pmatrix}$ | $\pi/2$ | $S = P(\pi/2)$ |
| T gate | $\begin{pmatrix}1 & 0\\0 & e^{i\pi/4}\end{pmatrix}$ | $\pi/4$ | $T = P(\pi/4)$ |
| Z gate | $\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}$ | $\pi$ | $Z = P(\pi)$ |
The $Z$ gate is also one of the Pauli gates. The $S$ gate is a square root of $Z$ ($S^2 = Z$), while $T$ is a fourth root of $Z$ ($T^4 = Z$). Naming conventions can vary between software libraries.
Because phase gates add their angles when multiplied,
$$P(\phi_1)P(\phi_2) = P(\phi_1 + \phi_2)$$
Important identities include
$$S^2 = Z \qquad S^4 = I$$
$$T^2 = S \qquad T^4 = Z \qquad T^8 = I$$
Their inverse gates subtract the phase:
$$S^\dagger = P(-\pi/2) \qquad T^\dagger = P(-\pi/4) \qquad Z^\dagger = Z$$
On the Bloch sphere, $P(\phi)$ rotates the state around the $z$-axis by angle $\phi$, up to a physically irrelevant global phase. Therefore, $S$, $T$, and $Z$ correspond to rotations by $\pi/2$, $\pi/4$, and $\pi$, respectively.