2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/108338In 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 $Γ$. 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 $γ=γ(Γ)$ where $γ$ 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 $Γ$. In the limit of $Γ=0$ the value of $γ(0)\approx$ 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.30 pages, 7 figuresDisordered Systems and Neural NetworksStatistical MechanicsQuantum Physicstext