Rotation gate

A one-qubit rotation gate is $R_{\alpha }(\theta )=e^{-i\theta {\sigma }_{\alpha }/2}$, where ${\sigma }_{\alpha }$ is a Pauli matrix. It rotates the Bloch vector by angle $\theta$ about the corresponding axis.

The one-qubit rotation gatesatisfies $R_y(2\theta )|0⟩=\cos \theta |0⟩+\sin \theta |1⟩$ and $R_y(\alpha )R_y(\beta )=R_y(\alpha +\beta )$. The latter identity enables a binary-angle implementation of a quantum variable rotation.

The one-qubit rotation gate is $R_z(\theta )=\mathrm{diag}(e^{-i\theta /2},e^{i\theta /2})$. It implements the two relative phases needed in simulation of a computable diagonal Hamiltonian.