The Quantum Random Energy Model is the Limit of Quantum $p$-Spin Glasses

We consider the free energy of a class of spin glass models with $p$-spin interactions in a transverse magnetic field. As $p\to \mathrm{\infty }$, the infinite system-size free energy is proven to converge to that of the quantum random energy model. This is accomplished by combining existing analytical techniques addressing the non-commutative properties of such quantum glasses, with the description of the typical geometry of extreme negative deviations of the classical $p$-spin glass. We also review properties of the corresponding classical free energy and conjectures addressing $1\mathrm{/}p$-corrections in the quantum case.