First-order transitions and the performance of quantum algorithms in random optimization problems

Phys Rev Lett. 2010 May 21;104(20):207206. doi: 10.1103/PhysRevLett.104.207206. Epub 2010 May 20.


We present a study of the phase diagram of a random optimization problem in the presence of quantum fluctuations. Our main result is the characterization of the nature of the phase transition, which we find to be a first-order quantum phase transition. We provide evidence that the gap vanishes exponentially with the system size at the transition. This indicates that the quantum adiabatic algorithm requires a time growing exponentially with system size to find the ground state of this problem.