Quantum Adiabatic Evolution Algorithm and Quantum Phase Transition in 3-Satisfiability Problem

In this paper we show that the performance of the quantum adiabatic algorithm is determined by phase transitions in underlying problem in the presence of transverse magnetic field $\Gamma$. We show that the quantum version of random Satisfiability problem with 3 bits in a clause (3-SAT) has a first-order quantum phase transition. We analyze the phase diagram $\gamma=\gamma(\Gamma)$ where $\gamma$ is an average number of clauses per binary variable in 3-SAT. The results are obtained in a closed form assuming replica symmetry and neglecting time correlations at small values of the transverse field $\Gamma$. In the limit of $\Gamma=0$ the value of $\gamma(0)\approx$ 5.18 corresponds to that given by the replica symmetric treatment of a classical random 3-SAT problem. We demonstrate the qualitative similarity between classical and quantum versions of this problem.