# Unitary gate
**Unitary gate** is the most general single-qubit unitary gate, parameterized by three real angles $\theta$, $\phi$, $\lambda$. Every single-qubit unitary (up to global phase) can be expressed as $U(\theta, \phi, \lambda)$ for some choice of these angles, making it the canonical representation of a single-qubit gate.

$$U(\theta, \phi, \lambda) = \begin{pmatrix}\cos\dfrac{\theta}{2} & -e^{i\lambda}\sin\dfrac{\theta}{2}\\[8pt] e^{i\phi}\sin\dfrac{\theta}{2} & e^{i(\phi+\lambda)}\cos\dfrac{\theta}{2}\end{pmatrix}$$

The parameter $\theta$ controls rotation between $\lvert 0\rangle$ and $\lvert 1\rangle$. The parameters $\phi$ and $\lambda$ set the phases of the two basis amplitudes. The U gate is used as the native single-qubit gate in IBM's Qiskit and appears on real hardware as the physical gate being calibrated.

## Special cases

All common single-qubit gates are special cases:

$$X = U(\pi, 0, \pi) \qquad Y = U\!\left(\pi, \tfrac{\pi}{2}, \tfrac{\pi}{2}\right) \qquad Z = U(0, 0, \pi)$$
$$H = U\!\left(\tfrac{\pi}{2}, 0, \pi\right) \qquad S = U\!\left(0, 0, \tfrac{\pi}{2}\right) \qquad T = U\!\left(0, 0, \tfrac{\pi}{4}\right)$$

## Euler decomposition

The three-angle parameterization corresponds to the Euler angle decomposition of a rotation: any orientation of the Bloch sphere is reachable via $R_z(\phi)\,R_y(\theta)\,R_z(\lambda)$. This shows that any single-qubit unitary reduces to a sequence of [[rotation-gates|rotation gates]], and the [[p-gate|phase gate]] and rotation gates together can reproduce any single-qubit operation.

## List of code implementations

- [[u-gate-qiskit|U gate (Qiskit)]]
- [[u-gate-custatevec|U gate (cuStateVec)]]
- [[u-gate-cudaq|U gate (CUDA-Q)]]