X90 gate
| Identifier | Operator | Example statement |
|---|---|---|
| X90 | \(X^{1/2}\) | X90 q[0] |
Description
The X90 gate is an anti-clockwise rotation of \(\pi/2\) [rad] about the \(\hat{\mathbf{x}}\)-axis and a global phase of \(\pi/4\) [rad].
It is equal to half the \(X\) rotation: \(X^{1/2}\)
Aliases
Also known as the V gate.
Properties
Representation
\[\begin{align}
X^{1/2} &= V = \frac{1}{2}\left(\begin{matrix}
1 + i & 1 - i \\
1 - i & 1 + i
\end{matrix}\right).
\end{align}\]
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 X90 gate is given by:
\[\begin{align}
X^{1/2} &= R_\hat{\mathbf{n}}\left([1, 0, 0]^T, \frac{\pi}{2}, \frac{\pi}{4}\right) = e^{i\frac{\pi}{4}} \cdot e^{-i\frac{\pi}{4}\sigma_x}, \\
\\
X^{1/2} &= \frac{1}{2}\left(\begin{matrix}
1 + i & 1 - i \\
1 - i & 1 + i
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the X90 gate \(X^{1/2}_H\) is given by:
\[X^{1/2}_H = HX^{1/2}H = \left(\begin{matrix}
1 & 0 \\
0 & i
\end{matrix}\right)=S.\]
Operation examples
Standard basis
\[\begin{align}
X^{1/2}\,|0\rangle &= \frac{1 + i}{2}|0\rangle + \frac{1 - i}{2}|1\rangle \\
\\
X^{1/2}\,|1\rangle &= \frac{1 - i}{2}|0\rangle + \frac{1 + i}{2}|1\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
X^{1/2}\,|+\rangle &= |+\rangle \\
\\
X^{1/2}\,|-\rangle &= i|-\rangle
\end{align}\]