  We prove Goldschmidt's formula [Phys. Rev. B 47 (1990) 4858] for the free
energy of the quantum random energy model. In particular, we verify the
location of the first order and the freezing transition in the phase diagram.
The proof is based on a combination of variational methods on the one hand, and
percolation bounds on large-deviation configurations in combination with simple
spectral bounds on the hypercube's adjacency matrix on the other hand.
