Why can the Hamiltonian $H=P_x(t)X+P_y(t)Y$ make an arbitrary unitary $U=R_x(b)R_y(c)R_x(d)$?

Question

p.281 of Nielsen and Chuang's book says that

A single spin might evolve under the Hamiltonian $H = P_x(t)X + P_y(t)Y$, where $P_{\{xy\}}$ are classically controllable parameters. From Exercise 4.10, we know that by manipulating $P_x$ and $P_y$ are appropriately, one can perform arbitrary single spin rotations.

And Exercise 4.10 is that an arbitrary 2X2 unitary operator can be decomposed into $e^{ia}R_x(b)R_y(c)R_x(d)$.

I can't understand why the Hamiltonian $H = P_x(t)X + P_y(t)Y$ can make an arbitrary unitary $U = R_x(b)R_y(c)R_x(d)$.

$H = P_x(t)X + P_y(t)Y$ makes an unitary $U^{-i(P_x(t)X + P_y(t)Y)/\hbar}$.

And it can be $U^{-i(P_{x1}(t)X + P_y(t)Y+P_{x2}(t)X)/\hbar}$.

But I think it cannot be decomposed into $U = R_x(P_{x1})R_y(P_y)R_x(P_{x2})$, since X and Y don't commute.

Can you tell me how the proposition above in Nielsen and Chuang's book is correct?

Thanks in advance.

Answer 1

If you set $P_y(t)=0$ for a particular length of time, then you just get an $X$ rotation. Similarly, if you set $P_x(t)=0$ for a certain length of time, you just get a $Y$ rotation. So, you do a sequence that looks something like \begin{align*} P_X(t)=\left\{\begin{array}{cc} J & 0\leq t\leq \frac{d}{2J} \\ 0 & 0<t-\frac{d}{2J}\leq \frac{c}{2J} \\ J & 0<t-\frac{c+d}{2J}\leq\frac{b}{2J} \end{array}\right. \\ P_Y(t)=\left\{\begin{array}{cc} 0 & 0\leq t\leq \frac{d}{2J} \\ J & 0<t-\frac{d}{2J}\leq \frac{c}{2J} \\ 0 & 0<t-\frac{c+d}{2J}\leq\frac{b}{2J} \end{array}\right. \end{align*} where $J$ is just a number, perhaps the maximum strength that you can achieve. Alternatively, they can time vary so long as certain integrals come out the same.