Efficient approximate unitary designs from random Pauli rotations

TL;DR

This paper introduces a method to efficiently generate approximate unitary t-designs using random Pauli rotations, with proven spectral gap bounds and implications for quantum circuit depth.

Contribution

It provides a new construction of approximate unitary t-designs via random walks on Lie groups using Pauli rotations, with a simple proof based on quadratic Casimir operators.

Findings

Spectral gap of the random walk is Ω(1/t).

Achieves ε-approximate unitary t-design in depth O(n t^2 + t log(1/ε)).

Uses quadratic Casimir operators for a simple proof.

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 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.