I'm referring to the following article 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 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.


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