Prove $E(R_n(\alpha),R_n(\theta)^n)<\epsilon/3$ from $E\big(R_n(\alpha),R_n(\alpha+\beta)\big)=|1-\exp(i\beta/2)|$

Question

where $E(U,V)=\max_{|\psi\rangle}||(U-V)|\psi\rangle ||=||U-V||$ is the error when $V$ is implemented instead of $U$. See page 196, Quantum Computation and Quantum Information by Nielsen and Chuang.

I have performed calculations for the exercise

$$ E\big(R_n(\alpha),R_n(\alpha+\beta)\big)=|(R_n(\alpha)-R_n(\alpha+\beta))|\phi\rangle|=|(R_n(\alpha)-(R_n(\alpha)R_n(\beta))|\phi\rangle|=\Big|R_n(\alpha)\big[1-R_n(\beta)\big]|\phi\rangle\Big|=\sqrt{\langle\phi|\big[1-R_n(\beta)\big]^\dagger R_n^\dagger(\alpha)R_n(\alpha)\big[1-R_n(\beta)\big]|\phi\rangle}\\ =\sqrt{\langle\phi|\big[1-R_n^\dagger(\beta)\big] \big[1-R_n(\beta)\big]|\phi\rangle}=\sqrt{\langle\phi|\big[1-R_n(-\beta)\big] \big[1-R_n(\beta)\big]|\phi\rangle}=\sqrt{\langle\phi|\big[1-R_n(-\beta)-R_n(\beta)+1\big]|\phi\rangle}=\sqrt{2-2\cos(\beta/2)}=|1-\exp(i\beta/2)|. $$

Now, how do we prove that for any $\epsilon>0$ there exists an $n$ such that $E(R_n(\alpha),R_n(\theta)^n)<\epsilon/3$ from Eq. $(4.77)$?

Answer 1

The proof consists in connecting together two arguments. The first, covered by the exercise, reduces the problem of approximating the rotation gate $R_\hat{n}(\alpha)$ to the problem of approximating the rotation angle $\alpha$. The second, described in the quoted text from Nielsen & Chuang, shows that one can achieve arbitrarily fine approximations of all angles using a rotation by an irrational multiple of $\pi$.

Reducing gate approximation to angle approximation

From $(4.77)$ we have

$$ \lim_{\beta\to 0}E(R_\hat{n}(\alpha), R_\hat{n}(\alpha+\beta))=\lim_{\beta\to 0}|1 -\exp(i\beta/2)| = 0. $$

In other words, for any sequence of angles $\gamma_k$ such that $\lim_{k\to\infty}\gamma_k=\alpha$, we have

$$ \lim_{k\to\infty}E(R_\hat{n}(\alpha), R_\hat{n}(\gamma_k)) = 0. $$

This means that if we can apply a rotation around $\hat{n}$ by a angle that approximates the rotation angle $\alpha$ to arbitrary accuracy then we can approximate $R_\hat{n}(\alpha)$ to arbitrarily small error $E$.

Achieving arbitrarily fine approximations of all angles

Now, as shown in the quoted paragraph from Nielsen & Chuang, the set

$$ \Theta=\{\theta_k\,|\,\theta_k=(k\theta)\mod{2\pi}\}, $$

of angles of rotations around $\hat{n}$ attainable by $R_\hat{n}(\theta)^k$ for $k\in\mathbb{Z}$, fills up the interval $[0, 2\pi)$ in the sense that for any rotation angle $\alpha$ and any desired accuracy $\delta>0$ there exists $\tilde\theta\in\Theta$ such that $|\alpha - \tilde\theta|<\delta$. In other words, the set of attainable angles $\Theta$ contains arbitrarily fine approximations of all angles. In particular, $\Theta$ contains $\theta^*$ that approximates $\alpha$ to whatever accuracy is needed for $E(R_\hat{n}(\alpha), R_\hat{n}(\theta)^n)<\frac{\epsilon}{3}$.

Connecting the arguments

The connection between the two parts is as follows. The first argument proves the implication that if we can approximate the rotation angle $\alpha$ arbitrarily well then we can approximate $R_\hat{n}(\alpha)$ to arbitrarily small error $E$. The second argument establishes the premise for that implication, namely that we can indeed approximate the rotation angle $\alpha$ arbitrarily well using repeated applications of $R_\hat{n}(\theta)$, as long as $\theta$ is an irrational multiple of $\pi$.