  We study a system of $N$ qubits with a random Hamiltonian obtained by drawing
coupling constants from Gaussian distributions in various ways. This results in
a rich class of systems which include the GUE and the fixed $q$ SYK theories.
Our motivation is to understand the system at large $N$. In practice most of
our calculations are carried out using exact diagonalisation techniques (up to
$N=24$). Starting with the GUE, we study the resulting behaviour as the
randomness is decreased. While in general the system goes from being chaotic to
being more ordered as the randomness is decreased, the changes in various
properties, including the density of states, the spectral form factor, the
level statistics and out-of-time-ordered correlators, reveal interesting
patterns. Subject to the limitations of our analysis which is mainly numerical,
we find some evidence that the behaviour changes in an abrupt manner when the
number of non-zero independent terms in the Hamiltonian is exponentially large
in $N$. We also study the opposite limit of much reduced randomness obtained in
a local version of the SYK model where the number of couplings scales linearly
in $N$, and characterise its behaviour. Our investigation suggests that a more
complete theoretical analysis of this class of systems will prove quite
worthwhile.
