Self correction requires Energy Barrier for Abelian quantum doubles

TL;DR

This paper rigorously proves an Arrhenius law for the mixing time of Abelian quantum double models, showing that their energy barrier is constant and does not grow with system size, thus ruling out entropic protection.

Contribution

It provides a rigorous mathematical definition of the energy barrier in Abelian quantum doubles and evaluates it, demonstrating it remains constant regardless of system size.

Findings

Energy barrier is constant and independent of system size.

Entropic protection is ruled out for Abelian quantum doubles.

Arrhenius law is established for mixing times in these models.

Abstract

We rigorously establish an Arrhenius law for the mixing time of quantum doubles based on any Abelian group $\mathbb{Z}_d$. We have made the concept of the energy barrier therein mathematically well-defined, it is related to the minimum energy cost the environment has to provide to the system in order to produce a generalized Pauli error, maximized for any generalized Pauli errors, not only logical operators. We evaluate this generalized energy barrier in Abelian quantum double models and find it to be a constant independent of system size. Thus, we rule out the possibility of entropic protection for this broad group of models.