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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.