For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider
the uniform measure $\nu_\mathcal{S}$ whose support is given by $\mathcal{S}$.
The measure $\nu_\mathcal{S}$ can be regarded as a
$\delta(\nu_\mathcal{S},t)$-approximate $t$-design, $t\in\mathbb{Z}_+$. We
propose a random matrix model that aims to describe the probability
distribution of $\delta(\nu_\mathcal{S},t)$ for any $t$. Our model is given by
a block diagonal matrix whose blocks are independent, given by Gaussian or
Ginibre ensembles, and their number, size and type is determined by $t$. We
prove that, the operator norm of this matrix, $\delta({t})$, is the random
variable to which $\sqrt{|\mathcal{S}|}\delta(\nu_\mathcal{S},t)$ converges in
distribution when the number of elements in $\mathcal{S}$ grows to infinity.
Moreover, we characterize our model giving explicit bounds on the tail
probabilities $\mathbb{P}(\delta(t)>2+\epsilon)$, for any $\epsilon>0$. We also
show that our model satisfies the so-called spectral gap conjecture, i.e. we
prove that with the probability $1$ there is $t\in\mathbb{Z}_+$ such that
$\sup_{k\in\mathbb{Z}_{+}}\delta(k)=\delta(t)$. Numerical simulations give
convincing evidence that the proposed model is actually almost exact for any
cardinality of $\mathcal{S}$. The heuristic explanation of this phenomenon,
that we provide, leads us to conjecture that the tail probabilities
$\mathbb{P}(\sqrt{\mathcal{S}}\delta(\nu_\mathcal{S},t)>2+\epsilon)$ are
bounded from above by the tail probabilities $\mathbb{P}(\delta(t)>2+\epsilon)$
of our random matrix model. In particular our conjecture implies that a Haar
random set $\mathcal{S}\subset U(d)$ satisfies the spectral gap conjecture with
the probability $1$.
