Efficient approximate unitary designs from random Pauli rotations Article Swipe

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Unitary state Pauli exclusion principle Mathematics Algebra over a field Pure mathematics Physics Quantum mechanics Political science Law

Jeongwan Haah , Yunchao Liu , Xinyu Tan ·

YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2402.05239 · OA: W4391709356

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 $e^{\mathrm i θP /2}$. The spectral gap of this random walk is shown to be $Ω(1/t)$, 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 $e^{\mathrm i θP /2}$. Our simple proof uses quadratic Casimir operators of Lie algebras.