Parametric Rotation Gates (Rx, Ry, Rz) | Continuous Angle Unitaries
Quick Technical Answer:
Parametric rotation gates Rx(θ), Ry(θ), and Rz(θ) rotate a single-qubit state vector by an arbitrary continuous angle θ around the X, Y, and Z Cartesian axes of the Bloch sphere, forming the primary parameterization in VQE and QML circuits.
Formula / Unitary:
R_x(\theta) = e^{-i\frac{\theta}{2}X} = \begin{pmatrix} \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\ -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{pmatrix}, \quad R_z(\theta) = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}
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Frequently Asked Questions
How are rotation gates differentiated in VQA optimization?
They are differentiated analytically using the Parameter-Shift Rule: d⟨H⟩/dθ = (⟨H⟩(θ + π/2) - ⟨H⟩(θ - π/2)) / 2 without finite-difference approximations.