Quantum expanders and property (T) discrete quantum groups
Michael Brannan, Eric Culf, Matthijs Vernooij
TL;DR
This work extends classical expander constructions to the quantum realm by exploiting property (T) for discrete quantum groups. It provides two explicit, representation-theoretic routes to quantum expanders: one via finite-dimensional irreducible representations yielding bounded-degree quantum channels, and another via a Margulis-like Schreier-graph framework built from coideals, including a spectral-gap analysis. The results cover both tracial and non-tracial settings, with a detailed exposition of bicrossed-product quantum groups as natural sources of examples. The article also develops the theory of quantum Cayley and Schreier graphs, establishing quantum expansion bounds and outlining significant open problems for finding and understanding non-classical, finite-dimensional coideals.
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.