# IQIM Postdoctoral and Graduate Student Seminar

*,*Graduate Student

*,*Preskill Group

*,*

**Abstract**: We prove a lower bound to the spectral gap of the Davies generator for general N - qubit commuting Pauli Hamiltonians and for quantum doubles based on an Abelian group. We derive rigorous thermalization time bounds, also called mixing time bounds, for the Davies generators of these Hamiltonians. Davies generators are given in the form of a Lindblad equation and are known to converge to the Gibbs distribution of the particular Hamiltonian for which they are derived. The bound on the spectral gap essentially depends on a single number E referred to as the generalized energy barrier. When any local defect can be grown into a logical operator and in turn any product operator can be decomposed into a product of the clusters of such incomplete logical operators, then E corresponds to the largest energy barrier of the canonical logical operators. The main conclusion that can be drawn from our result is, that the presence of an energy barrier for the logical operators is in fact, although not sufficient, a necessary condition for a thermally stable quantum memory when we assume the full Davies dynamics as noise model. This rules out the possibility of entropic protection for this broad group of models.

This is a joint work with two past IQIM postdoctoral scholars, Olivier Landon-Cardinal (*McGill U.*) and Kristan Temme (*IBM*).