Random Quantum Circuits are Approximate 2-designs

TL;DR

This paper proves that random quantum circuits of polynomial length can efficiently approximate the first two moments of the Haar distribution, forming approximate 1- and 2-designs, which is faster than previous methods.

Contribution

It demonstrates that polynomial-length random circuits form approximate 1- and 2-designs, improving upon prior longer circuit requirements and extending to general gate sets.

Findings

Polynomial-length circuits approximate Haar distribution moments

Improved bounds on convergence rate of Clifford group random walks

Applicable to general two-qubit gate sets

Abstract

Given a universal gate set on two qubits, it is well known that applying random gates from the set to random pairs of qubits will eventually yield an approximately Haar-distributed unitary. However, this requires exponential time. We show that random circuits of only polynomial length will approximate the first and second moments of the Haar distribution, thus forming approximate 1- and 2-designs. Previous constructions required longer circuits and worked only for specific gate sets. As a corollary of our main result, we also improve previous bounds on the convergence rate of random walks on the Clifford group.