  We construct random walks on simple Lie groups that quickly converge to the
Haar measure for all moments up to order $t$. Specifically, a step of the walk
on the unitary or orthognoal group of dimension $2^{\mathsf n}$ is a random
Pauli rotation $e^{\mathrm i \theta P /2}$. The spectral gap of this random
walk is shown to be $\Omega(1/t)$, which coincides with the best previously
known bound for a random walk on the permutation group on $\{0,1\}^{\mathsf
n}$. This implies that the walk gives an $\varepsilon$-approximate unitary
$t$-design in depth $O(\mathsf n t^2 + t \log 1/\varepsilon)d$ where $d=O(\log
\mathsf n)$ is the circuit depth to implement $e^{\mathrm i \theta P /2}$. Our
simple proof uses quadratic Casimir operators of Lie algebras.
