Identity gate
| Identifier | Operator | Example statement |
|---|---|---|
| I | \(I\) | I q[0] |
Description
The Identity gate is a gate that leaves the qubit state unchanged (NOP).
Note
Internally, an identity gate may be defined as a rotation of \(0\) radians about the z-axis and a global phase of \(0\).
Properties
- Clifford gate;
- Involutory operation (its own inverse);
- Special Unitary \(SU\).
Representation
\[I = \left(\begin{matrix}
1 & 0 \\
0 & 1
\end{matrix}\right)\]
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 anti-clockwise angle of rotation [rad], and \(\phi\in[0,2\pi)\) the global phase angle [rad].
The Identity gate can be represented in canonical form as:
\[\begin{align}
I &= R_\hat{\mathbf{n}}\left(\left[0, 0, 1\right]^T, 0, 0\right) \\
\\
&= \left(\begin{matrix}
1 & 0 \\
0 & 1
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the Identity gate \(I_H\) is given by:
\[I_H = HIH = \left(\begin{matrix}
1 & 0 \\
0 & 1
\end{matrix}\right)=I.\]
Operation examples
Standard basis
\[\begin{align}
I\,|0\rangle &= |0\rangle \\
\\
I\,|1\rangle &= |1\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
I\,|+\rangle &= |+\rangle \\
\\
I\,|-\rangle &= |-\rangle
\end{align}\]