T gate
| Identifier | Operator | Example statement |
|---|---|---|
| T | \(T\) | T q[0] |
Description
The T gate is an anti-clockwise rotation of \(\pi / 4\) [rad] about the \(\hat{\mathbf{z}}\)-axis and a global phase of \(\pi / 8\) [rad].
It is equal to half the \(S\) rotation and a quarter of the \(Z\) rotation: \(T = S^{1/2} = Z^{1/4}\).
Representation
\[\begin{align}
T &= \left(\begin{matrix}
1 & 0 \\
0 & \frac{1 + i}{\sqrt{2}}
\end{matrix}\right)
\end{align}\]
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 T gate is given by:
\[\begin{align}
T &= R_\hat{\mathbf{n}}\left([0, 0, 1]^T, \frac{\pi}{4}, \frac{\pi}{8}\right) = e^{i\frac{\pi}{8}} \cdot e^{-i\frac{\pi}{8}\sigma_z}, \\
\\
T &= \left(\begin{matrix}
1 & 0 \\
0 & \frac{1 + i}{\sqrt{2}}
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the T gate \(T_H\) is given by:
\[T_H = HT H = \frac{1}{2\sqrt{2}}\left(\begin{matrix}
\sqrt{2} + 1 + i & \sqrt{2} - 1 - i \\
\sqrt{2} - 1 - i & \sqrt{2} + 1 + i
\end{matrix}\right).\]
Operation examples
Standard basis
\[\begin{align}
T\,|0\rangle &= |0\rangle \\
\\
T\,|1\rangle &= \frac{1 + i}{\sqrt{2}} |1\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
T\,|+\rangle &= \frac{\sqrt{2} + 1 + i}{2\sqrt{2}}|+\rangle + \frac{\sqrt{2} - 1 - i}{2\sqrt{2}}|-\rangle \\
\\
T\,|-\rangle &= \frac{\sqrt{2} - 1 - i}{2\sqrt{2}}|+\rangle + \frac{\sqrt{2} + 1 + i}{2\sqrt{2}}|-\rangle \\
\end{align}\]