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.

翻译：我们在简单李群上构造随机游走，该游走对直至t阶的所有矩快速收敛到哈尔测度。具体而言，该游走在维度为$2^{\mathsf n}$的酉群或正交群上的每一步是一个随机泡利旋转$e^{\mathrm i \theta P /2}$。该随机游走的谱间隙被证明为$\Omega(1/t)$，这与先前在$\{0,1\}^{\mathsf n}$置换群上随机游走的已知最优界一致。这意味着该游走在深度为$O(\mathsf n t^2 + t \log 1/\varepsilon)d$（其中$d=O(\log \mathsf n)$是实现$e^{\mathrm i \theta P /2}$的电路深度）时给出一个$\varepsilon$-近似酉$t$-设计。我们简洁的证明利用了李代数的二次卡西米尔算子。