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.

For chronological equatorial rotations $R_{\varphi }(\pi )R_{2\varphi }(\pi )R_{\varphi }(\pi )$, the ideal propagator is $iX$. If all pulse angles have the same fractional error $ϵ$, choose $\cos \varphi =-1/2$ to cancel the quadratic term of the trace-overlap gate fidelity. With $x=\pi ϵ/2$, its exact fidelity is $|\cos x(1+{\sin }^{2}x/2)|$, whose leading infidelity is $3x^4/8$. This correction compensates a systematic amplitude error, not arbitrary axis or timing errors.

The rotation gate about the Pauli X gate axis commutes with $X$. The J gate convention gives $J(\alpha )J(0)=H\mathrm{diag}(1,e^{i\alpha })H=e^{i\alpha /2}{R}_x(\alpha )$, differing from the standard axis rotation only by global phase.

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.