  Quantum annealing is a general purpose optimization algorithm that is based
on the quantum adiabatic theorem. Quantum annealing involves an evolving
Hamiltonian that is local. Thus, we expect that short-time quantum annealing
algorithms to be inherently local and limited as well. In this paper, we
validate this intuition by proving some limitations of short-time quantum
annealing algorithms. We show that the distribution of the measurement output
of short-time (at most logarithmic) quantum annealing computations are
\emph{concentrated} and satisfy an \emph{isoperimetric inequality}. To showcase
explicit applications, we also study the \textsc{MaxCut} problem and conclude
that quantum annealing needs at least a run-time that scales logarithmically in
the problem size to beat classical algorithms. To establish our results, we
also prove a Lieb-Robinson bound that works for time-dependent Hamiltonians
which might be of independent interest.
