 Solution

With $B_y=0$, a pulse of the $x$ field realizes $R_x(\theta )=e^{-i\theta {\sigma }_x/2}$ when $\int B_x(t)dt=\theta /2$. Similarly the $y$ field realizes $R_y(\theta )$ with area $\theta /2$. These are instances of the quantum pulse area rule. A rotation about the z-axis can be synthesized from the two available axes:because $R_x(\pi /2){\sigma }_y{R}_x(-\pi /2)={\sigma }_z$.

For a target $V=\left(\begin{array}{cc}u& v\\ -v^{\ast }& u^{\ast }\end{array}\right)$, one convenient construction isTake $\beta =2atan2(|v|,|u|)$ and choose $\alpha +\gamma =-2\arg u$, $\alpha -\gamma =-2\arg (-v)$ when both entries are nonzero; if an entry vanishes its phase constraint can be omitted. Substitute the three-pulse construction for each $z$ rotation. Apply the factors from right to left in time. Pulses along the available $x,y$ axes therefore implement every single-qubit gate up to global phase. Negative angles use negative field area, or an equivalent full-period rotation.