Mathematical Physics Seminar

We prove a lower bound to the spectral gap of the Davies generator for
general N - qubit commuting Pauli Hamiltonians. We expect this bound
to provide the correct asymptotic scaling of the gap with the systems
size up to a factor of 1/N in the low temperature regime. 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 ε referred to as the generalized energy barrier. When
any local defect can be grown into a logical operator of a stabilizer
code S by applying single qubit Pauli operators and in turn any Pauli
operator can be decomposed into a product of the clusters of such
excitations, ε corresponds to the largest energy barrier of the
canonical logical operators.
The main conclusion that can be drawn from the 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.