Pauli-Z gate
| Identifier | Operator | Example statement |
|---|---|---|
| Z | \(Z\) | Z q[0] |
Description
The Pauli-Z, or Z, gate is an anti-clockwise rotation of \(\pi\) [rad] about the \(\hat{\mathbf{z}}\)-axis and a global phase of \(\pi/2\) [rad].
Aliases
Also known as the phase-flip gate.
Properties
Representation
\[\begin{align}
Z = \sigma_z = \sigma_3 = \left(\begin{matrix}
1 & 0 \\
0 & -1
\end{matrix}\right)
\end{align}\]
Any single-qubit operation in \(U(2)\) (including global phase) can be expressed by 5 parameters in the canonical representation \(R_\hat{\mathbf{n}}\)
\[R_\hat{\mathbf{n}}\left([n_x, n_y, n_z]^T, \theta, \phi\right) = e^{i\phi} \cdot e^{-i\frac{\theta}{2}\left(n_x\cdot\sigma_x + n_y\cdot\sigma_y + n_z\cdot\sigma_z\right)},\]
where \(\hat{\mathbf{n}}=[n_x, n_y, n_z]^T\) denotes the axis of rotation, \(\theta\in(-\pi, \pi]\) the angle of rotation [rad], and \(\phi\in[0,2\pi)\) the global phase angle [rad].
The Pauli-Z gate is given by:
\[\begin{align}
Z &= R_\hat{\mathbf{n}}\left([0, 0, 1]^T, \pi, \frac{\pi}{2}\right) = e^{i\frac{\pi}{2}} \cdot e^{-i\frac{\pi}{2}\sigma_z}, \\
\\
Z &= \left(\begin{matrix}
1 & 0 \\
0 & -1
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the Pauli-Z gate \(Z_H\) is given by:
\[Z_H = HZH = \left(\begin{matrix}
0 & 1 \\
1 & 0
\end{matrix}\right)=X.\]
Operation examples
Standard basis
\[\begin{align}
Z\,|0\rangle &= |0\rangle \\
\\
Z\,|1\rangle &= -|1\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
Z\,|+\rangle &= |-\rangle \\
\\
Z\,|-\rangle &= |+\rangle
\end{align}\]