= 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:
$$
R_z(\theta)=R_x(\pi/2)R_y(\theta)R_x(-\pi/2),
$$
because $R_x(\pi/2)\sigma_yR_x(-\pi/2)=\sigma_z$.
For a target $V=\begin{pmatrix}u&v\\-v^*&u^*\end{pmatrix}$, one convenient construction is
$$
V=R_z(\alpha)R_y(\beta)R_z(\gamma),\qquad
u=e^{-i(\alpha+\gamma)/2}\cos(\beta/2),\quad v=-e^{-i(\alpha-\gamma)/2}\sin(\beta/2).
$$
Take $\beta=2\operatorname{atan2}(|v|,|u|)$ 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. \b[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.
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