  We consider the problem of estimating the ground state energy of quantum
$p$-local spin glass random Hamiltonians, the quantum analogues of widely
studied classical spin glass models. Our main result shows that the maximum
energy achievable by product states has a well-defined limit (for even $p$) as
$n\to\infty$ and is $E_{\text{product}}^\ast=\sqrt{2 \log p}$ in the limit of
large $p$. This value is interpreted as the maximal energy of a much simpler
so-called Random Energy Model, widely studied in the setting of classical spin
glasses. The proof of the limit existing follows from an extension of Fekete's
Lemma after we demonstrate near super-additivity of the (normalized) quenched
free energy. The proof of the value follows from a second moment method on the
number of states achieving a given energy when restricting to an $\epsilon$-net
of product states.
  Furthermore, we relate the maximal energy achieved over all states to a
$p$-dependent constant $\gamma\left(p\right)$, which is defined by the degree
of violation of a certain asymptotic independence ansatz over graph matchings.
We show that the maximal energy achieved by all states $E^\ast\left(p\right)$
in the limit of large $n$ is at most
$\sqrt{\gamma\left(p\right)}E_{\text{product}}^\ast$. We also prove using
Lindeberg's interpolation method that the limiting $E^\ast\left(p\right)$ is
robust with respect to the choice of the randomness and, for instance, also
applies to the case of sparse random Hamiltonians. This robustness in the
randomness extends to a wide range of random Hamiltonian models including SYK
and random quantum max-cut.
