  Quantum annealers aim at solving non-convex optimization problems by
exploiting cooperative tunneling effects to escape local minima. The underlying
idea consists in designing a classical energy function whose ground states are
the sought optimal solutions of the original optimization problem and add a
controllable quantum transverse field to generate tunneling processes. A key
challenge is to identify classes of non-convex optimization problems for which
quantum annealing remains efficient while thermal annealing fails. We show that
this happens for a wide class of problems which are central to machine
learning. Their energy landscapes is dominated by local minima that cause
exponential slow down of classical thermal annealers while simulated quantum
annealing converges efficiently to rare dense regions of optimal solutions.
