HomeQuantum PhysicsLessonsRY Gate (Y-Rotation)

RY Gate (Y-Rotation)

RotationSingle-qubitY-rot

Rotates around the Y axis by angle θ.

Intuition

RY rotates around the Y axis of the Bloch sphere. What makes it special compared to RX and RZ is that it creates real-valued superpositions — no imaginary phases appear. Starting from |0⟩, an RY rotation tilts toward the equator through the XZ plane, producing states like cos(θ/2)|0⟩ + sin(θ/2)|1⟩ with purely real amplitudes. This makes RY particularly clean for state preparation.

Matrix representation

(cos \frac{θ}{2} sin \frac{θ}{2} - sin \frac{θ}{2} cos \frac{θ}{2})

Action on states

R_{y} (θ) ∣ ψ ⟩ = e^{- i θ Y /2} ∣ ψ ⟩

Bloch sphere

|0⟩

→

RY(π/2)|0⟩

Circuit

State comparison

Before RY Gate

|0⟩

|0\u27E9

100%

|1\u27E9

Bloch: (0.00, 0.00, 1.00)

→

After RY Gate

RY(π/2)|0⟩

|0\u27E9

85%

|1\u27E9

15%

Bloch: (0.71, 0.00, 0.71)

Precise explanation

RY(θ) = exp(−iθY/2) = cos(θ/2)I − i·sin(θ/2)Y. Since Y is purely imaginary in the computational basis, the resulting matrix has only real entries: [[cos(θ/2), −sin(θ/2)], [sin(θ/2), cos(θ/2)]]. This means RY maps real states to real states. At θ = π, it becomes the Y gate up to global phase.

Observable effect

Creates real-valued superpositions with tunable probabilities.

Hidden effect

At θ = π this becomes the Y gate (up to global phase).

Bloch sphere

Rotates by θ around the Y axis. Stays in the XZ plane. At θ = π/2 from |0⟩, reaches the equator with a real-valued superposition.

Worked example

Preparing a state with known probabilities

1Goal: create a state with 75% probability of |0⟩ and 25% probability of |1⟩.
2We need cos²(θ/2) = 0.75, so θ/2 = arccos(√0.75) ≈ π/6.
3Apply RY(π/3) to |0⟩: state becomes cos(π/6)|0⟩ + sin(π/6)|1⟩ ≈ 0.866|0⟩ + 0.5|1⟩.
4Measurement gives |0⟩ with probability 75% and |1⟩ with probability 25%.

Common use cases

Preparing states with specific amplitude distributions
Variational circuits (VQE, QAOA) for parameter optimization
Amplitude encoding of classical data into quantum states
ZYZ decomposition of arbitrary single-qubit gates

Relation to other gates

RY(π) = −iY (Y gate up to global phase)
Any single-qubit gate = RZ(α)RY(β)RZ(γ) (Euler decomposition)
RY keeps amplitudes real when starting from a computational basis state
H ≈ RY(π/2) followed by a Z correction

Common misconception

RY and RX might seem interchangeable, but RY uniquely produces real amplitudes from basis states. This simplifies analysis and is why RY appears in many decomposition theorems.

Why this gate matters

RY is the workhorse of state preparation. When you need to encode specific probability distributions into a qubit, RY gives you direct control over the amplitudes without introducing unwanted phases.

Test your understanding

Why does RY uniquely produce real amplitudes from |0⟩?

Quick reference →↗ Nielsen and Chuang, Quantum Computation and Quantum Information

← Back to all lessons

HomeQuantum PhysicsLessonsRY Gate (Y-Rotation)

RY Gate (Y-Rotation)

RotationSingle-qubitY-rot

Rotates around the Y axis by angle θ.

Intuition

Matrix representation

(cos \frac{θ}{2} sin \frac{θ}{2} - sin \frac{θ}{2} cos \frac{θ}{2})

Action on states

R_{y} (θ) ∣ ψ ⟩ = e^{- i θ Y /2} ∣ ψ ⟩

Bloch sphere

|0⟩

→

RY(π/2)|0⟩

Circuit

State comparison

Before RY Gate

|0⟩

|0\u27E9

100%

|1\u27E9

Bloch: (0.00, 0.00, 1.00)

→

After RY Gate

RY(π/2)|0⟩

|0\u27E9

85%

|1\u27E9

15%

Bloch: (0.71, 0.00, 0.71)

Precise explanation

Observable effect

Creates real-valued superpositions with tunable probabilities.

Hidden effect

At θ = π this becomes the Y gate (up to global phase).

Bloch sphere

Rotates by θ around the Y axis. Stays in the XZ plane. At θ = π/2 from |0⟩, reaches the equator with a real-valued superposition.

Worked example

Preparing a state with known probabilities

1Goal: create a state with 75% probability of |0⟩ and 25% probability of |1⟩.
2We need cos²(θ/2) = 0.75, so θ/2 = arccos(√0.75) ≈ π/6.
3Apply RY(π/3) to |0⟩: state becomes cos(π/6)|0⟩ + sin(π/6)|1⟩ ≈ 0.866|0⟩ + 0.5|1⟩.
4Measurement gives |0⟩ with probability 75% and |1⟩ with probability 25%.

Common use cases

Preparing states with specific amplitude distributions
Variational circuits (VQE, QAOA) for parameter optimization
Amplitude encoding of classical data into quantum states
ZYZ decomposition of arbitrary single-qubit gates

Relation to other gates

RY(π) = −iY (Y gate up to global phase)
Any single-qubit gate = RZ(α)RY(β)RZ(γ) (Euler decomposition)
RY keeps amplitudes real when starting from a computational basis state
H ≈ RY(π/2) followed by a Z correction

Common misconception

RY and RX might seem interchangeable, but RY uniquely produces real amplitudes from basis states. This simplifies analysis and is why RY appears in many decomposition theorems.

Why this gate matters

RY is the workhorse of state preparation. When you need to encode specific probability distributions into a qubit, RY gives you direct control over the amplitudes without introducing unwanted phases.

Test your understanding

Why does RY uniquely produce real amplitudes from |0⟩?

Quick reference →↗ Nielsen and Chuang, Quantum Computation and Quantum Information

← Back to all lessons