Using the quantum cavity method, we present an analytical study of the phase diagram of a random optimization problem in a transverse field, in the thermodynamic limit and for a uniformly random distribution of instances. We include the effect of replica symmetry breaking, and unveil the occurrence of a first-order quantum phase transition. We back up our results with Quantum Monte Carlo and Exact Diagonalization methods, and provide evidence that the gap vanishes exponentially with the system size at the transition. This suggests that the Quantum Adiabatic Algorithm requires an exponential time to find the ground state of such problems.