Efficient Approximate Unitary Designs from Random Pauli Rotations
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Author(s)
Haah, Jeongwan
Liu, Yunchao
Tan, Xinyu
Date Issued
October 30, 2025
Journal
Communications in Mathematical Physics
Publisher
Springer Berlin Heidelberg
Citation
Haah, J., Liu, Y. & Tan, X. Efficient Approximate Unitary Designs from Random Pauli Rotations. Commun. Math. Phys. 406, 309 (2025).
Version
Author's final manuscript
Abstract
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 n is a random Pauli rotation e i θ P / 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 } n . This implies that the walk gives an ε -approximate unitary t-design in depth O ( n t 2 + t log 1 ε ) d where d = O ( log n ) is the circuit depth to implement e i θ P / 2 . Our simple proof uses quadratic Casimir operators of Lie algebras.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00220-025-05480-6
