U (general single-qubit gate)¶
U is the most general possible single-qubit gate - with the right three angles, it can reproduce any single-qubit unitary, including every other gate on this page.
| Qubits | 1 |
| Parameters | \( \theta, \varphi, \lambda \) - three angles, in radians |
| Inverse | U(−θ, −λ, −φ) |
| Also called | U3 gate |
What it does¶
U(θ, φ, λ) covers every possible way to rotate and phase-shift a single qubit. Every gate on this reference page is a special case of U with particular angles plugged in.
Matrix¶
\[
U(\theta, \varphi, \lambda) = \begin{pmatrix}
\cos(\theta/2) & -e^{i\lambda}\sin(\theta/2) \\
e^{i\varphi}\sin(\theta/2) & e^{i(\varphi+\lambda)}\cos(\theta/2)
\end{pmatrix}
\]
Example¶
U with \( (\theta,\varphi,\lambda) = (\pi/2, 0, \pi) \) on q[0] produces exactly the same state as an H gate - a good way to see how U generalizes the gates you already know.
Other notations¶
Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.
| Language | Typical form |
|---|---|
| OpenQASM 2.0 | u(theta, phi, lambda) q[0]; |
| Qiskit | circuit.u[theta, phi, lam, q[0]] |
| Cirq | - |
| Q# | - |
Notes¶
Some useful special cases (all exact, no leftover global phase):
- \( U(0, 0, \lambda) = P(\lambda) \) - see P
- \( U(\theta, -\pi/2, \pi/2) = RX(\theta) \) - see RX
- \( U(\theta, 0, 0) = RY(\theta) \) - see RY
- \( U(\pi, 0, \pi) = X \) - see Pauli-X
- \( U(\pi/2, 0, \pi) = H \) - see Hadamard