Hello everybody,

I'm referring to the following article https://arxiv.org/abs/1810.11563 [1], in particular to section 6.

The goal is to estimate the number of unitary transformations in $SU(N)$, identifying unitaries within balls of radius $\epsilon$. The strategy is to take the total volume of $SU(N)$ (see also https://arxiv.org/abs/math-ph/0210033 [2]), and then dividing for the volume of an $\epsilon$-ball. Here and in the following $N=2^K$, where $K$ is an integer.

The total volume of $SU(N)$ is:

\begin{equation}

\frac{2 \pi^{\frac{(N+2)(N-1)}{2}}}{1! 2! 3!\cdots(N-1)!}

\end{equation}

Here the author misses a total factor of $\sqrt{N 2^{N-3}}$, see the original article [2], equation 5.13. Despite this, the author then states that the volume of an $\epsilon$-ball of dimension $N^2-1$ is:

\begin{equation}

\frac{\pi^{\frac{N^2-1}{2}}}{\left(\frac{N^2-1}{2}\right)!}

\end{equation}

I don't get this last formula. First of all, I believe that there would be a factor $\epsilon$ to some power of $N$. Secondly, it seems to me the volume of an $\epsilon$-ball in an Euclidean space off even dimension, while considering that $N=2^K$, $N^2-1$ is odd. Thirdly, in my opinion it would be better to consider the volume of an $\epsilon$-ball in $SU(N)$ (if I didn't misunderstood the formula above).

My questions are:

- Has someone, reading the article [1], reached a better comprehension than mine on the above formulas?

- If no, may you have other references about estimating the number of unitary transformations in $SU(N)$ within a precision $\epsilon$?

- And, finally, has someone a reference for the volume of an $\epsilon$-ball in $SU(N)$? Looking on the net I didn't find anything until now.

Thank you very much!

[1]: https://arxiv.org/abs/1810.11563

[2]: https://arxiv.org/abs/math-ph/0210033