Efficient Approximate Unitary Designs from Random Pauli Rotations

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 orthogonal group of dimension $2^{\mathrm{n}}$ 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 $\{0,1\}^{\mathrm{n}}$. This implies that the walk gives an $\varepsilon$ -approximate unitary t-design in depth $\mathcal{O}(\mathrm{n}t^{2}+t\log\frac{1}{\varepsilon})d$ where $d=\mathrm{O}(\log \mathrm{n})$ is the circuit depth to implement $e^{\mathrm{i}\theta P/2}$. Our simple proof uses quadratic Casimir operators of Lie algebras.