Rotation Gates: Rx, Ry, Rz

Rotation Gates: Rx, Ry, Rz

The rotation gates provide continuous control over qubit states. They're essential for variational quantum algorithms (VQE, QAOA) and parameterized circuits.

The Bloch Sphere Geometry

Every single-qubit state corresponds to a point on the Bloch sphere. The rotation gates spin a qubit around one of the three principal axes:

The Three Rotation Gates

Rx(θ)|0⟩ = cos(θ/2)|0⟩ − i·sin(θ/2)|1⟩ Ry(θ)|0⟩ = cos(θ/2)|0⟩ + sin(θ/2)|1⟩ $R_z(\theta )|0⟩=e^{-i\theta /2}|0⟩$ (adds phase only)

import math
qc.rx(math.pi/2, 0)   # 90° rotation around X
qc.ry(math.pi, 0)     # 180° rotation around Y (same as Y gate)
qc.rz(math.pi/4, 0)   # 45° rotation around Z (same as T gate)

Special Cases: Pauli Gates

The Pauli gates are rotation gates with θ = π (180°):

"Up to global phase" means the measurement statistics are identical: only the relative phase between |0⟩ and |1⟩ components matters.

Why Rotation Gates Matter for VQE/QAOA

In variational algorithms, the circuit contains parameters (angles) that a classical optimizer adjusts to minimize a cost function. Rotation gates are the natural parameterized gate:

theta = 0.7  # parameter, adjusted by optimizer
qc.ry(theta, 0)
qc.rz(theta * 2, 0)

The circuit runs many times with different θ values until the optimizer finds the minimum energy configuration.