Pauli-X gate
| Identifier | Operator | Example statement |
|---|---|---|
| X | \(X\) | X q[0] |
Description
The Pauli-X, or X, gate is an anti-clockwise rotation of \(\pi\) [rad] about the \(\hat{\mathbf{x}}\)-axis and a global phase of \(\pi/2\) [rad].
Aliases
Also known as the bit-flip gate.
Properties
Representation
\[X = \sigma_x = \sigma_1 = \left(\begin{matrix}
0 & 1 \\
1 & 0
\end{matrix}\right)\]
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-X gate is given by:
\[\begin{align}
X &= R_\hat{\mathbf{n}}\left([1, 0, 0]^T, \pi, \frac{\pi}{2}\right) = e^{i\frac{\pi}{2}} \cdot e^{-i\frac{\pi}{2} \sigma_x}, \\
\\
X &= \left(\begin{matrix}
0 & 1 \\
1 & 0
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the Pauli-X gate \(X_H\) is given by:
\[X_H = HXH = \left(\begin{matrix}
1 & 0 \\
0 & -1
\end{matrix}\right)=Z.\]
Operation examples
Standard basis
\[\begin{align}
X\,|0\rangle &= |1\rangle \\
\\
X\,|1\rangle &= |0\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
X\,|+\rangle &= |+\rangle \\
\\
X\,|-\rangle &= -|-\rangle
\end{align}\]