Error bound on approximating arbitrary rotation gates

Question

I'm working on the following problem form Nielsen and Chuang (Ex. 4.40) regarding the universality of Hadamard, phase and $\pi/8$ gates for single qubit gates:

For arbitrary $\alpha$, $\beta$,

$E(R_{\vec{n}}(\alpha), R_{\vec{n}}(\alpha + \beta)) = |1 - exp(i\beta/2)|$, where $E(U, V) = max_{|\psi\rangle} || (U-V)|\psi\rangle||$

Show (using the previous equality) $\forall \epsilon >0$ , $\exists n$, such that $E(R(\alpha)_{\vec{n}}, {R(\theta)_{\vec{n}}}^n) < \frac{\epsilon}{3}$, where $\theta$ is an irrational multiple of $2\pi$.

My attempt:

Prior to the exercise, it discusses how for any arbitrary error $\epsilon > 0$, we can find some multiple $(mod$ $2\pi)$ of irrational $\theta$ (found using the Pigeonhole principle, which I think I understand, and I'm skipping and taking as given), call it $\theta_k \in (0, 2\pi]$, |$\theta_k| < \epsilon$ such that all multiples $(mod$ $2\pi)$ of $\theta_k$ differ by at most $\epsilon$. Thus $\forall \alpha \in [0, 2\pi)$, $\exists m$, such that $|\alpha - (m\theta_k) | =|\alpha - (mk\theta)| < \epsilon$. Let $n = mk$, then

$E(R(\alpha), {R_{\vec{n}}}(\theta)^n) = E(R_{\vec{n}}(\alpha), {R_{\vec{n}}}(n\theta)) = E(R_{\vec{n}}(\alpha), {R_{\vec{n}}}(\alpha + (n\theta -\alpha))) < E(R_{\vec{n}}(\alpha), {R_{\vec{n}}}(\alpha + \epsilon)) = |1 - exp(i\epsilon/2)| = \sqrt{((1 - cos(\epsilon/2))^2 + sin(\epsilon/2)^2} = \sqrt{2(1 - cos(\epsilon/2))} < 2$

(which is not correct).

I feel like I'm missing something very obvious in this whole exercise. A hint would be appreciated.

Answer 1

Firstly, I think there's a reason why the bit in the textbook before the question is using $\delta$ instead of $\epsilon$ for the results. So, replacing what you've written, it should really be $$ E<\sqrt{2(1-\cos(\delta/2))} $$

Now, start with a double-angle formula: $1-\cos(\delta/2)=2\sin^2(\delta/4)$. Thus, $$ E<2\sin(\delta/4) $$ Now if you apply the small $\epsilon$ approximation (which should be more obvious following that manipulation), you get $$ E<\delta/2. $$ Finally, if you want $E<\delta/2$, you simply have to select $\delta=2\epsilon/3$.

You might ask why you want the factor of $1/3$ in there at all? Why not just leave it as $1/2$? This point is that there's going to be a sequence of three of these rotations, and you want to bound the final error by $\epsilon$.