We study the quantum version of the random $K$-satisfiability problem in the presence of an external magnetic field $\Gamma$ applied in the transverse direction. We derive the replica-symmetric free-energy functional within the static approximation and the saddle-point equation for the order parameter: the distribution $P\left[h\left(m\right)\right]$ of functions of magnetizations. The order parameter is interpreted as the histogram of probability distributions of individual magnetizations. In the limit of zero temperature and small transverse fields, to leading order in $\Gamma$ magnetizations $m\approx 0$ become relevant in addition to purely classical values of $m\approx \pm 1$. Self-consistency equations for the order parameter are solved numerically using a quasi–Monte Carlo method for $K=3$. It is shown that for an arbitrarily small $\Gamma$ quantum fluctuations destroy the phase transition present in the classical limit $\Gamma =0$, replacing it with a smooth crossover transition. The implications of this result with respect to the expected performance of quantum optimization algorithms via adiabatic evolution are discussed. The replica-symmetric solution of the classical random $K$-satisfiability problem is briefly reexamined. It is shown that the phase transition at $T=0$ predicted by the replica-symmetric theory is of continuous type with atypical critical exponents.

• Received 31 March 2008

DOI:https://doi.org/10.1103/PhysRevE.78.061128