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

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?

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 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.