Comment on the paper "Random Quantum Circuits are Approximate 2-designs"

TL;DR

This paper critiques and corrects a key argument in a prior work demonstrating that random quantum circuits rapidly approximate 2-designs, emphasizing the importance of transpositions in the convergence process.

Contribution

It identifies and rectifies a flaw in the original proof, providing an alternative argument that clarifies the role of transpositions in convergence to quantum 2-designs.

Findings

Corrects a flaw in the original proof of convergence

Highlights the role of transpositions in the convergence process

Provides a clearer understanding of how random circuits approximate 2-designs

Abstract

In [A.W. Harrow and R.A. Low, Commun. Math. Phys. 291, 257-302 (2009)], it was shown that a quantum circuit composed of random 2-qubit gates converges to an approximate quantum 2-design in polynomial time. We point out and correct a flaw in one of the paper's main arguments. Our alternative argument highlights the role played by transpositions induced by the random gates in achieving convergence.