Source-verifiedLevel 2
Ry Gate
The Ry gate performs a rotation by angle theta about the y-axis of the Bloch sphere, uniquely producing real-valued superpositions from basis states.
What it means
The Ry(theta) gate rotates the qubit state by angle theta about the y-axis of the Bloch sphere.Its matrix is [[cos(theta/2), -sin(theta/2)], [sin(theta/2), cos(theta/2)]].Notably, Ry is the only standard rotation gate with a purely real matrix, meaning it maps real states to real states.Ry(pi/2) applied to |0⟩ produces the |+⟩ state, similar to the Hadamard gate.Ry gates are extensively used in state preparation circuits and variational quantum algorithms, where they provide a natural parameterization for creating arbitrary superpositions.Everyday analogy
The Ry gate is like tilting a globe along the north-south direction by a precise angle.
Think of it as adjusting the balance between |0⟩ and |1⟩ smoothly, like a volume dial between silence and full sound.
Common misconceptions
- Ry is NOT the same as the Hadamard gate, even though both can create equal superpositions -- Ry(pi/2)|0⟩ = |+⟩ but with different phase conventions.
- The Ry gate does NOT introduce complex phases -- its matrix entries are purely real.
Key takeaways
- Ry(theta) rotates about the y-axis with a purely real matrix.
- Ry is ideal for state preparation since it creates real-valued superpositions.
- Together with Rx and Rz, Ry completes the set of rotation gates for arbitrary single-qubit unitaries.
Check your understanding
What is unique about the Ry gate's matrix compared to Rx and Rz?
- A.It is diagonal
- B.It has purely real entries
- C.It is anti-Hermitian
- D.It has eigenvalues 0 and 1
Show the answer
Answer: B. It has purely real entries
Why: The Ry gate matrix has purely real entries (cos and sin), unlike Rx (which has -i factors) and Rz (which has complex exponentials).
Builds on
Primary source: Barenco et al., Phys. Rev. A 52, 3457 (1995), doi:10.1103/PhysRevA.52.3457
Learn it hands-on
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