Quantum expanders and property (T) discrete quantum groups

TL;DR

This paper constructs quantum expanders using discrete quantum groups with property (T), extending classical expander graph constructions to the quantum setting through two innovative methods.

Contribution

It introduces two new methods for constructing quantum expanders from discrete quantum groups with property (T), generalizing classical group-based approaches.

Findings

Constructed quantum expanders from finite-dimensional irreducible representations.

Extended Margulis' classical construction to the quantum setting.

Provided examples using discrete quantum groups from compact bicrossed products.

Abstract

Families of expander graphs were first constructed by Margulis from discrete groups with property (T). Within the framework of quantum information theory, several authors have generalised the notion of an expander graph to the setting of quantum channels. In this work, we use discrete quantum groups with property (T) to construct quantum expanders in two ways. The first approach obtains a quantum expander family by constructing the requisite quantum channels directly from finite-dimensional irreducible unitary representations, extending earlier work of Harrow using groups. The second approach directly generalises Margulis' original construction and is based on a quantum analogue of a Schreier graph using the theory of coideals. To obtain examples of quantum expanders, we apply our machinery to discrete quantum groups with property (T) coming from compact bicrossed products.