# Phase gates
**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.

## Common phase gates

The most commonly used phase gates are:

| Gate | Matrix | Phase $\phi$ | Relationship |
| --- | --- | --- | --- |
| [[p-gate|P gate]] | $\begin{pmatrix}1 & 0\\0 & e^{i\phi}\end{pmatrix}$ | arbitrary | general phase gate |
| [[s-gate|S gate]] | $\begin{pmatrix}1 & 0\\0 & i\end{pmatrix}$ | $\pi/2$ | $S = P(\pi/2)$ |
| [[t-gate|T gate]] | $\begin{pmatrix}1 & 0\\0 & e^{i\pi/4}\end{pmatrix}$ | $\pi/4$ | $T = P(\pi/4)$ |
| [[z-gate|Z gate]] | $\begin{pmatrix}1 & 0\\0 & -1\end{pmatrix}$ | $\pi$ | $Z = P(\pi)$ |

The $Z$ gate is also one of the [[pauli-gates|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.

## Powers and inverses

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:

$$S^\dagger = P(-\pi/2) \qquad T^\dagger = P(-\pi/4) \qquad Z^\dagger = Z$$

## Bloch sphere interpretation

On the [[bloch-sphere|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.

## List of code implementations

- [[p-gate-qiskit|P gate (Qiskit)]]
- [[p-gate-custatevec|P gate (cuStateVec)]]
- [[p-gate-cudaq|P gate (CUDA-Q)]]
- [[z-gate-qiskit|Z gate (Qiskit)]]
- [[z-gate-custatevec|Z gate (cuStateVec)]]
- [[z-gate-cudaq|Z gate (CUDA-Q)]]