Efficient approximate unitary designs from random Pauli rotations
<|reference_start|>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 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.<|reference_end|>
@article{haah2024efficient,
    title={Efficient approximate unitary designs from random Pauli rotations},
    author={Jeongwan Haah, Yunchao Liu, Xinyu Tan},
    journal={arXiv preprint arXiv:2402.05239},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.05239},
    primaryClass={quant-ph cs.CC}
}