Computational Bottlenecks of Quantum Annealing

A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of $N$ qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by $O(\log N)$ bottlenecks inside the spin glass phase, where the gap scales as a stretched exponential. For smaller $N$, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield Model. Qualitative comparison with the Sherrington-Kirkpatrick Model leads to similar conclusions.