Matrix concentration inequalities and efficiency of random universal sets of quantum gates

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