Skip to content

RY

RY gate tile from the Operations catalog

RY rotates a qubit by an angle \( \theta \) about the Y-axis of the Bloch sphere. Unlike RX and RZ, its matrix entries are all real numbers, which makes it a common choice for building superpositions with adjustable probabilities.
Qubits 1
Parameters \( \theta \) - rotation angle, in radians
Inverse RY(−θ)
Also called Y-rotation

What it does

RY(θ) turns \( |0\rangle \) into a superposition weighted by \( \cos(\theta/2) \) and \( \sin(\theta/2) \), with no complex phase at all - useful whenever you want to dial in a specific measurement probability directly.

Matrix

\[ RY(\theta) = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix} \]

Action on basis states

  • \( |0\rangle \rightarrow \cos(\theta/2)\,|0\rangle + \sin(\theta/2)\,|1\rangle \)
  • \( |1\rangle \rightarrow -\sin(\theta/2)\,|0\rangle + \cos(\theta/2)\,|1\rangle \)

Example

RY with \( \theta = \pi/3 \) on q[0] (starting at \( |0\rangle \)) gives \( \cos(\pi/6)|0\rangle + \sin(\pi/6)|1\rangle \), which the Probabilities panel shows as a 75%/25% split.

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 ry(theta) q[0];
Qiskit circuit.ry[theta, q[0]]
Cirq cirq.ry(theta)(q[0])
Q# Ry(theta, q[0]);
  • RX
  • RZ
  • U - RY(θ) = U(θ, 0, 0)