Efficiency of quantum vs. classical annealing in nonconvex learning problems

Significance Quantum annealers are physical quantum devices designed to solve optimization problems by finding low-energy configurations of an appropriate energy function by exploiting cooperative tunneling effects to escape local minima. Classical annealers use thermal fluctuations for the same computational purpose, and Markov chains based on this principle are among the most widespread optimization techniques. The fundamental mechanism underlying quantum annealing consists of exploiting a controllable quantum perturbation to generate tunneling processes. The computational potentialities of quantum annealers are still under debate, since few ad hoc positive results are known. Here, we identify a wide class of large-scale nonconvex optimization problems for which quantum annealing is efficient while classical annealing gets stuck. These problems are of central interest to machine learning. Quantum annealers aim at solving nonconvex optimization problems by exploiting cooperative tunneling effects to escape local minima. The underlying idea consists of 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 nonconvex 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 are dominated by local minima that cause exponential slowdown of classical thermal annealers while simulated quantum annealing converges efficiently to rare dense regions of optimal solutions.