Exact & formalGatesRotation gate2026-07-16

RY rotation gate

A parametrized single-qubit rotation about the Bloch-sphere Y axis whose matrix entries are always real, making it the standard tool for encoding real-valued amplitudes.

rotation gatesingle qubitreal amplitude

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RY(θ) is the gate of choice whenever a circuit needs to move probability amplitude between |0⟩ and |1⟩ without introducing a complex relative phase.

Circuit & simulation

q[0]

RY(π/2)

P(|0⟩) at θ=π/250%

P(|1⟩) at θ=π/250%

What this takes and returns

Takes1 qubitWhat joins here

Takes a 1-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 1 qubits can feed it.

15 entries meet this end. Named below.

Returns1 qubitWhat joins here

Returns a 1-qubit register a next stage can take. This is the end that joins.

13 entries meet this end. Named below.

A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →

Fits before this 15

Fits after this 13

How it works

Definition

R_Y(\theta) = e^{-i\theta Y/2} = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}

Unlike $R_X$ or $R_Z$ , every entry of $R_Y(\theta)$ is a real number for real $\theta$ : the $-i$ prefactor from the exponential exactly cancels the $\pm i$ entries of $Y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}$ , leaving a real $2\times 2$ rotation matrix — the same form as a classical rotation matrix in the $(|0\rangle, |1\rangle)$ plane.

Action on basis states

R_Y(\theta)|0\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle, \qquad R_Y(\theta)|1\rangle = -\sin(\theta/2)|0\rangle + \cos(\theta/2)|1\rangle

At $\theta = \pi/2$ , $R_Y(\pi/2)|0\rangle = \tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ — the same state Hadamard produces on $|0\rangle$ , but $R_Y(\pi/2)|1\rangle = \tfrac{1}{\sqrt2}(-|0\rangle + |1\rangle)$ differs from $H|1\rangle$ by an overall sign.

Key identities

$R_Y(\pi) = -iY$ : a full $\pi$ rotation reproduces the Pauli-Y flip up to global phase $-i$ .
$X R_Y(\theta) X = R_Y(-\theta)$ : since $XYX = -Y$ , conjugating by $X$ reverses the rotation direction.
Because its output amplitudes are real, $R_Y(\theta)$ is the standard primitive for amplitude encoding — loading a classical probability vector $(\cos(\theta/2), \sin(\theta/2))$ into a qubit — and for the mixer layers of variational circuits such as hardware-efficient ansätze.
$R_Y(\theta) = R_Z(-\pi/2)\,R_X(\theta)\,R_Z(\pi/2)$ : an axis change from $X$ to $Y$ via conjugation by a $Z$ rotation, mirroring the $H$ -conjugation identity that relates $R_X$ and $R_Z$ .

Implementation

FrameworkNative

ry-rotation.qasm

OPENQASM 3.0;
include "stdgates.inc";
qubit q;
ry(pi/2) q;

Quantum vs classical

Classical baseline

Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.

Quantum claim

The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.

How to compare

Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.

Declared gaps

Nobody has reviewed this record for gaps yet.

Literature & references

Quantum Computation and Quantum Information: 10th Anniversary Edition2010 · Michael A. Nielsen and Isaac L. Chuang

Standard reference for single- and multi-qubit gate matrices, Euler-angle decomposition, and universal gate sets.

doi.org/10.1017/cbo9780511976667 ↗