A group with at least subexponential hyperlinear profile

TL;DR

This paper constructs a finitely-presented group with a hyperlinear profile growing at least subexponentially, demonstrating a two-player non-local game requiring subexponential Hilbert space dimension for near-perfect play.

Contribution

It introduces a new finitely-presented group with subexponential hyperlinear profile, linking group theory to quantum information theory.

Findings

Constructed a group with at least subexponential hyperlinear profile

Provided an example of a non-local game requiring subexponential Hilbert space dimension

Established a connection between group properties and quantum game complexity

Abstract

The hyperlinear profile of a group measures the growth rate of the dimension of unitary approximations to the group. We construct a finitely-presented group whose hyperlinear profile is at least subexponential, i.e. at least $\exp(1/\epsilon^{k})$ for some $0 < k < 1$. We use this group to give an example of a two-player non-local game requiring subexponential Hilbert space dimension to play near-perfectly.