Computational phase transition signature in Gibbs sampling

Abstract

Gibbs sampling is fundamental to a wide range of computer algorithms. Such algorithms are set to be replaced by physics based processors—be it quantum or stochastic annealing devices—which embed problem instances and evolve a physical system into a low-energy ensemble to recover a probability distribution. At a critical constraint to variable ratio, satisfiability (SAT) problem instances exhibit a SAT-UNSAT transition (frustrated to frustration free). Algorithms require increasing computational resources from this critical point. This is a so called, algorithmic or computational phase transition and has extensively been studied. In this paper we consider the complexity in sampling and recovering ground states from resultant distributions of a physics based processor. In particular, we first consider the ideal Gibbs distributions at some fixed inverse temperature and observe that the success probability in sampling and recovering ground states decrease for instances starting at the critical density. Furthermore, simulating the Gibbs distribution, we employ Ising spin dynamics, which play a crucial role in understanding of non-equilibrium statistical physics, to find their steady states of 2-SAT Hamiltonians. We observe that beyond the critical density, the probability of sampling ground states decreases. Our results apply to several contemporary devices and provide a means to experimentally probe a signature of the computational phase transition.