  For a random set $\mathcal{S} \subset U(d)$ of quantum gates we provide
bounds on the probability that $\mathcal{S}$ forms a $\delta$-approximate
$t$-design. In particular we have found that for $\mathcal{S}$ drawn from an
exact $t$-design the probability that it forms a $\delta$-approximate
$t$-design satisfies the inequality $\mathbb{P}\left(\delta \geq x \right)\leq
2D_t \, \frac{e^{-|\mathcal{S}| x \,
\mathrm{arctanh}(x)}}{(1-x^2)^{|\mathcal{S}|/2}} = O\left( 2D_t \left(
\frac{e^{-x^2}}{\sqrt{1-x^2}} \right)^{|\mathcal{S}|} \right)$, where $D_t$ is
a sum over dimensions of unique irreducible representations appearing in the
decomposition of $U \mapsto U^{\otimes t}\otimes \bar{U}^{\otimes t}$. We use
our results to show that to obtain a $\delta$-approximate $t$-design with
probability $P$ one needs $O( \delta^{-2}(t\log(d)-\log(1-P)))$ many random
gates. We also analyze how $\delta$ concentrates around its expected value
$\mathbb{E}\delta$ for random $\mathcal{S}$. Our results are valid for both
symmetric and non-symmetric sets of gates.
