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

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. $$(4.77)$$?

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 $$\pi$$.

• It can be proved that there are distinct $j$ and $k$ in the range $1, . . . , N$ such that $|θ_k−θ_j|≤2π/N<δ$. Assuming $k>j$ we will get $|θ_k−θ_j|=|θ_{k-j}|<\delta$. How do I make sense of "sequence $θ_{l(k−j)}$ fills up the interval $[0, 2π)$ as l is varied, so that adjacent members of the sequence are no more than $δ$ apart" ? Oct 20, 2021 at 15:30
• How do you ay that "angles of rotations around $\hat{n}$ attainable by $R_\hat{n}(θ)^k$ for $k∈Z$, fills up the interval $[0,2π)$ in the sense that for any rotation angle $α$ and any desired accuracy $δ>0$ there exists $\tilde{θ}∈Θ$ such that $|α−\tilde{θ}|<δ$ " ? Oct 20, 2021 at 15:33
• So the text before your quote shows that the sequence $\theta_i$ contains a non-zero element (namely, $\theta_{k-j}$) whose absolute value is less than $\delta$. Then multiples of this element (i.e. elements of the form $\theta_{l(k-j)}=l(k-j)\theta \mod 2\pi = l\theta_{k-j}\mod 2\pi$) are less than $\delta$ apart and so end up filling the interval $[0, 2\pi)$ with density at least one element every $\delta$. Finally, since $\delta$ is arbitrarily small, we end up with $[0, 2\pi)$ being filled in the sense I described. Oct 20, 2021 at 17:42
• I think it makes sense now, except that where does precisely $\epsilon/3$ comes in the picture ? Oct 21, 2021 at 10:29
• Definitely. They make it a third to get a nice whole $\epsilon$ later in inequality $(4.81)$. The $3$ comes from the fact that any single-qubit unitary can be decomposed into three rotations, see $(4.80)$. Oct 21, 2021 at 17:28