Non-Hermitian expander obtained with Haar distributed unitaries

TL;DR

This paper constructs a new type of quantum expander using Haar-distributed unitaries, providing bounds on spectral gaps through eigenvalues and singular values, advancing understanding of random quantum channels.

Contribution

It introduces a method to construct quantum expanders with spectral gap estimates based on eigenvalues and singular values, extending Hastings' approach.

Findings

Constructed a quantum expander using Haar-distributed unitaries.

Derived bounds on spectral gaps in terms of eigenvalues and singular values.

Established analogs of classical bounds for quantum spectral gaps.

Abstract

We consider a random quantum channel obtained by taking a selection of $d$ independent and Haar distributed $N$ dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values \cite{hastings2007random,harrow2007quantum}. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon-Boppana bound for $d$-regular graphs. The upper bound is obtained using Schwinger-Dyson equations.