Efficient Approximate Unitary Designs from Random Pauli Rotations

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.