Rn gate

Identifier	Operator	Example statement
Rn	\(R_\hat{\mathbf{n}}(\hat{\mathbf{n}}, \theta, \phi)\)	Rn(1,0,0,pi,pi/2) q[0]

Description

The canonical Rn gate is an anti-clockwise rotation with an angle of \(\theta\) [rad] about the specified axis \(\hat{\mathbf{n}}=[n_x, n_y, n_z]^T\) and a global phase of \(\phi\).

Note

The Rn instruction expects the separate components of the axis \(\hat{\mathbf{n}}\), i.e. \(n_x\), \(n_y\), and \(n_z\), as input arguments:

Rn(\(n_x\), \(n_y\), \(n_z\), \(\theta\), \(\phi\)) qubit-argument

as shown in the Example statement in the table above.

Properties

The canonical representation of unitary matrices \(U(2)\) yields special unitary matrices \(SU(2)\), with determinant \(\text{det} = +1\), when \(\phi\) equals zero. \(SU(2)\) is the special unitary group of degree \(2\), and has a double cover of \(SO(3)\), the group of rotations in 3-dimensional space. This means that each element in \(SO(3)\) corresponds to two elements in \(SU(2)\).

Operations on Spin-\(\tfrac{1}{2}\) qubits, such as electron spin qubits, are represented by \(SU(2)\) matrices. As an example, when a spin-qubit is operated on by an Rx gate it does not return to the same state after a \(2\pi\) rotation, but only after a \(4\pi\) rotation.

The difference is a global phase of \(\pi\), which is insignificant by itself, but will be relevant when using it in combination with the control modifier ctrl, which changes the gate \(U\) into the controlled- \(U\) gate.

• Special Unitary \(SU\)

Representation

Any single-qubit operation in \(U(2)\) (including global phase) can be expressed by 5 parameters in the canonical representation:

\[R_\hat{\mathbf{n}}\left([n_x, n_y, n_z]^T, \theta, \phi\right) = e^{i\phi} \cdot e^{-i\frac{\theta}{2}\left(n_x\cdot\sigma_x + n_y\cdot\sigma_y + n_z\cdot\sigma_z\right)},\]

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].

Which expands to the matrix representation:

\[ R_\hat{\mathbf{n}}\left(\hat{\mathbf{n}}, \theta, \phi\right) = e^{i\phi} \cdot \left(\begin{matrix} \cos\left(\theta / 2\right) - i n_z \sin\left(\theta / 2\right) & -n_y \sin\left(\theta / 2\right) - i n_x \sin\left(\theta / 2\right) \\ n_y \sin\left(\theta / 2\right) - i n_x \sin\left(\theta / 2\right) & \cos\left(\theta / 2\right) + i n_z \sin\left(\theta / 2\right) \end{matrix}\right). \]

In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the Rn gate \(R_{\hat{\mathbf{n}},H}(...)\) is given by:

\[R_{\hat{\mathbf{n}},H}\left(\hat{\mathbf{n}}, \theta, \phi\right) = H R_\hat{\mathbf{n}}(\theta) H = e^{i\phi} \cdot \left(\begin{matrix} \cos\left(\theta / 2\right) - i n_x \sin\left(\theta / 2\right) & n_y \sin\left(\theta / 2\right) - i n_z \sin\left(\theta / 2\right) \\ -n_y \sin\left(\theta / 2\right) - i n_z \sin\left(\theta / 2\right) & \cos\left(\theta / 2\right) + i n_x \sin\left(\theta / 2\right) \end{matrix}\right)\]

Operation examples

Standard basis

\[\begin{align} R_\hat{\mathbf{n}}\left(\hat{\mathbf{n}}, \theta, \phi\right)\,|0\rangle &= \left[\cos\left(\theta / 2\right) - i n_z \sin\left(\theta / 2\right)\right]|0\rangle + \left[n_y \sin\left(\theta / 2\right) - i n_x \sin\left(\theta / 2\right)\right]|1\rangle \\ \\ R_\hat{\mathbf{n}}\left(\hat{\mathbf{n}}, \theta, \phi\right)\,|1\rangle &= \left[-n_y \sin\left(\theta / 2\right) - i n_x \sin\left(\theta / 2\right)\right]|0\rangle + \left[\cos\left(\theta / 2\right) + i n_z \sin\left(\theta / 2\right)\right]|1\rangle \\ \end{align}\]

Hadamard basis

\[\begin{align} R_\hat{\mathbf{n}}\left(\hat{\mathbf{n}}, \theta, \phi\right)\,|+\rangle &= \left[\cos\left(\theta / 2\right) - i n_x \sin\left(\theta / 2\right)\right]|+\rangle + \left[-n_y \sin\left(\theta / 2\right) - i n_z \sin\left(\theta / 2\right)\right]|-\rangle \\ \\ R_\hat{\mathbf{n}}\left(\hat{\mathbf{n}}, \theta, \phi\right)\,|-\rangle &= \left[n_y \sin\left(\theta / 2\right) - i n_z \sin\left(\theta / 2\right)\right]|+\rangle + \left[\cos\left(\theta / 2\right) + i n_x \sin\left(\theta / 2\right)\right]|-\rangle \\ \end{align}\]