First-Order Phase Transition in the Quantum Adiabatic Algorithm

We simulate the quantum adiabatic algorithm (QAA) for the exact cover problem for sizes up to $N=256$ using quantum Monte Carlo simulations incorporating parallel tempering. At large $N$, we find that some instances have a discontinuous (first-order) quantum phase transition during the evolution of the QAA. This fraction increases with increasing $N$ and may tend to 1 for $N\to \infty$.