Rz gate
| Identifier |
Operator |
Example statement |
| Rz |
\(R_z(\theta)\) |
Rz(pi) q[0] |
Description
The Rz gate is one of the three standard rotation operators:
an anti-clockwise rotation with an angle of \(\theta\) [rad] about the \(\hat{\mathbf{z}}\)-axis.
The Rz gate is given by:
\[
R_z(\theta) = e^{-i\frac{\theta}{2}\sigma_z } = \left(\begin{matrix}
\cos\left(\theta / 2\right) - i \sin\left(\theta / 2\right) & 0 \\
0 & \cos\left(\theta / 2\right) + i \sin\left(\theta / 2\right)
\end{matrix}\right)
\]
The rotation operators are generated by exponentiation of the Pauli matrices according to
\[e^{-i \frac{\theta}{2} \sigma_j } = \cos\left ( \theta / 2 \right )I-i\sin\left (\theta / 2 \right )\sigma_j \]
where \(\sigma_j\) is one of the three Pauli matrices.
The three rotation matrices \(R_x\), \(R_y\) and \(R_z\) are special cases of the canonical \(R_\hat{n}\) operation.
Properties
- The rotation gates Rx, Ry, and Rz (at rotation angles of \(\pi\)) differ from the respective Pauli gates only by a
global phase:
\[R_x\left( \pi \right) = -iX,~~~~ R_y\left( \pi \right) = -iY,~~~~ R_z\left( \pi \right) = -iZ.\]
Representation
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 Rz gate is given by:
\[\begin{align}
R_z(\theta) &= R_\hat{\mathbf{n}}\left([0, 0, 1]^T, \theta, 0\right) = e^{-i\frac{\theta}{2}\sigma_z}, \\
\\
R_z(\theta) &= \left(\begin{matrix}
\cos\left(\theta / 2\right) - i \sin\left(\theta / 2\right) & 0 \\
0 & \cos\left(\theta / 2\right) + i \sin\left(\theta / 2\right)
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the Rz gate \(R_{z,H}\) is given by:
\[R_{z,H}(\theta) = HR_z(\theta) H = \left(\begin{matrix}
\cos\left(\theta / 2\right) & - i \sin\left(\theta / 2\right) \\
- i \sin\left(\theta / 2\right) & \cos\left(\theta / 2\right)
\end{matrix}\right) = R_x(\theta).\]
Operation examples
Standard basis
\[\begin{align}
R_z(\theta)\,|0\rangle &= \left[\cos\left(\theta / 2\right) - i \sin\left(\theta / 2\right)\right]|0\rangle \\
\\
R_z(\theta)\,|1\rangle &= \left[\cos\left(\theta / 2\right) + i \sin\left(\theta / 2\right)\right]|1\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
R_z(\theta)\,|+\rangle &= \cos\left(\theta / 2\right)|+\rangle - i \sin\left(\theta / 2\right)|-\rangle \\
\\
R_z(\theta)\,|-\rangle &= - i \sin\left(\theta / 2\right)|+\rangle + \cos\left(\theta / 2\right)|-\rangle \\
\end{align}\]