# Rotation gates

**Rotation gates** ($R_X$, $R_Y$, $R_Z$) are parametric single-qubit gates that apply rotations around the x, y, z axes of the Bloch sphere by angle $\theta$. They are fundamental building blocks for state preparation and variational quantum algorithms, since any single-qubit unitary can be decomposed as a product of rotations around two perpendicular axes.

## The three rotation gates

- **[[quantum-gate-rx|RX gate]]** ($R_X(\theta)$): rotation around x-axis by $\theta$; has complex entries
- **[[quantum-gate-ry|RY gate]]** ($R_Y(\theta)$): rotation around y-axis by $\theta$; real-valued matrix
- **[[quantum-gate-rz|RZ gate]]** ($R_Z(\theta)$): rotation around z-axis by $\theta$; diagonal (phase shifts only)

## Matrix representations

$$R_X(\theta) = \begin{pmatrix} \cos(\theta/2) & -i\sin(\theta/2) \\ -i\sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$

$$R_Y(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$

$$R_Z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$$

## Common properties

All three rotation gates share:

- **Periodicity**: $R(\theta + 2\pi) = R(\theta)$
- **Inverse**: $R(\theta)^\dagger = R(-\theta)$
- **Composition**: $R_i(\alpha) R_i(\beta) = R_i(\alpha + \beta)$ (same axis)
- **Exponential form**: $R_i(\theta) = e^{-i\theta \sigma_i / 2}$ where $\sigma_i$ is the corresponding Pauli matrix

## Special cases and relations to Pauli gates

At $\theta = \pi$, rotations relate to [[quantum-gate-pauli|Pauli gates]] (up to global phase):

- $R_X(\pi) = -i X$
- $R_Y(\pi) = -i Y$
- $R_Z(\pi) = -i Z$

$R_Z(\pi/2)$ equals the [[quantum-gate-s|S gate]]; $R_Z(\pi/4)$ equals the [[quantum-gate-t|T gate]].

## Euler angle decomposition

Any single-qubit unitary can be decomposed as:

$$U(\phi, \theta, \lambda) = R_Z(\phi) R_Y(\theta) R_Z(\lambda)$$

This is the standard Euler angle parametrization. The [[quantum-gate-u|U gate]] implements this directly. Alternatively, $R_X R_Y R_Z$ or other axis orderings work depending on convention.

## Uses

- **State preparation**: Rotate computational basis states to arbitrary points on the Bloch sphere
- **Variational algorithms**: [[cuda-q-vqe|VQE]] and [[cuda-q-qaoa|QAOA]] use parameterized rotation layers as ansatze
- **Circuit synthesis**: Arbitrary single-qubit unitaries decompose into two rotations around perpendicular axes
- **Measurement basis change**: Rotate before measurement to extract different observables
- **Native gates**: Most quantum platforms implement rotations efficiently as single pulses

## Implementation notes

- **Superconducting qubits**: RX and RY typically ~20–40 ns; RZ is often virtual (reference frame adjustment)
- **Trapped ions**: rotations via laser pulses at various frequencies; gate times ~1–10 μs
- **Photonic**: rotations via wave plates or optical phase shifters
- **Fidelity**: typically 99–99.9% depending on platform

## Relations

- [[quantum-gate-rx|RX gate]], [[quantum-gate-ry|RY gate]], [[quantum-gate-rz|RZ gate]]: individual gate articles
- [[quantum-gate-u|U gate]]: universal single-qubit gate (Euler angles)
- [[quantum-gate-single-qubit|Single-qubit gates]]: rotations generate all of $\mathrm{SU}(2)$
- [[quantum-gate-pauli|Pauli gates]]: special cases of rotations at $\theta = \pi$
- [[quantum-gate-s|S gate]] and [[quantum-gate-t|T gate]]: special cases of RZ at specific angles