Random Unitaries Give Quantum Expanders

TL;DR

This paper demonstrates that randomly selecting matrices from the unitary group produces quantum expanders, providing tight bounds on eigenvalues using advanced mathematical techniques, thus advancing understanding of quantum information processing.

Contribution

It introduces a novel method of generating quantum expanders via random unitaries and applies Schwinger-Dyson equations to analyze their spectral properties.

Findings

Random unitaries produce quantum expanders.

Asymptotically tight bounds on second eigenvalue in Hermitian case.

Applicable to both Hermitian and non-Hermitian maps.

Abstract

We show that randomly choosing the matrices in a completely positive map from the unitary group gives a quantum expander. We consider Hermitian and non-Hermitian cases, and we provide asymptotically tight bounds in the Hermitian case on the typical value of the second largest eigenvalue. The key idea is the use of Schwinger-Dyson equations from lattice gauge theory to efficiently compute averages over the unitary group.