Construction of ${R_n(\theta)}$ using only the Hadamard and ${\pi/8}$ gates

Question

In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen & Chuang", they claim that Eqn(4.75) is a rotation about the axis along the direction ( ${cos(\pi/8)}$, ${sin (\pi/8)}$, ${cos(\pi/8)}$ ). They then defined an angle ${\theta}$ such that:

${cos(\theta/2)}$ = $cos^2$(${\pi/8}$)

and its also claimed to be an irrational multiple of 2${\pi}$.
We know that the rotational matrices about any arbitrary axis takes the form of

${cos(\theta/2)}$ ${I}$ - ${i}$ (${n_x}$${X}$ + ${n_y}$${Y}$ + ${n_z}$${Z}$) ${sin(\theta/2)}$ ,

but Eqn(4.75) gives:

${cos^2(\pi/8)}$ ${I}$ - ${i}$ [ ${cos(\pi/8)(X+Z)+sin(\pi/8)Y}$ ] ${sin(\pi/8)}$

My question is how does this ${\theta}$ be able to simultaneously satisfy ${sin(\pi/8})$? Why does ${\theta}$ be referenced from ${cos^2(\pi/8)}$ instead of ${sin(\pi/8})$?

Answer 1

In order to compare to the Pauli vector exponentiation formula, we need to write in terms a normalized unit vector:

$${\displaystyle R = \cos^2(\pi/8) I_2 -\frac{i}{\sqrt{1+\cos^2(\pi/8)} } \times \quad \times \left [\cos(\pi/8)(X+Z)+ \sin(\pi/8)Y \right ] \sqrt{1+\cos^2(\pi/8)}\sin(\pi/8)} $$

Now, the result can be seen by inspection from the comparison to the general formula:

$${\displaystyle e^{ia({\hat {n}}\cdot {\vec {\sigma }})}=I\cos {a}+i({\hat {n}}\cdot {\vec {\sigma }})\sin {a}}$$