Volume of $SU\left( 2^K\right)$

Abstract

Let’s make a more refined calculation of the number of operators in ${\textit{SU}}(N)$ by dividing its volume by the volume of an epsilon ball of the same dimensionality (the dimension of ${\textit{SU}}(N)$ is $N^2-1$).

Let’s make a more refined calculation of the number of operators in ${\textit{SU}}(N)$ by dividing its volume by the volume of an epsilon ball of the same dimensionality (the dimension of ${\textit{SU }}(N)$ is $N^2-1$). The volume of ${\textit{SU}}(N)$ is https://arxiv.org/abs/math-ph/0210033

$$\begin{aligned} V[{\textit{SU}}(N)] = \frac{2\pi ^{\frac{(N+2)(N-1)}{2}}}{1!2!3!\ldots (N-1)!} \end{aligned}$$

(6.1)

The volume of an epsilon-ball of dimension $N^2-1$ is

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

Using Stirling’s formula, and identifying the number of unitary operators with the number of epsilon-balls in ${\textit{SU}}(N)$

$$\begin{aligned} \#{\textit{unitaries}}\approx & {} \left( \frac{N}{\epsilon ^2} \right) ^{\frac{N^2}{2}} \\\\= & {} \left( \frac{2^K}{\epsilon ^2} \right) ^{4^K/2} \end{aligned}$$

(6.2)

Taking the logarithm,

$$\begin{aligned} \log {(\#{\textit{unitaries}})} \approx \frac{4^K}{2}K\log {2} + 4^K\log {\frac{1}{\epsilon }} \end{aligned}$$

(6.3)

which is comparable to (5.3). Again, we see the strong exponential dependence on K and the weak logarithmic dependence on $\epsilon .$ The $\log {\frac{1}{\epsilon }}$ term is multiplied by the dimension of the space.