  Neural-network quantum states (NQS) have become a powerful tool in many-body
physics. Of the numerous possible architectures in which neural-networks can
encode amplitudes of quantum states the simplicity of the Restricted Boltzmann
Machine (RBM) has proven especially useful for both numerical and analytical
studies. In particular devising exact NQS representations for important classes
of states, like Jastrow and stabilizer states, has provided useful clues into
the strengths and limitations of the RBM based NQS. However, current
constructions for a system of $N$ spins generate NQS with $M \sim O(N^2)$
hidden units that are very sparsely connected. This makes them rather atypical
NQS compared to those commonly generated by numerical optimisation. Here we
focus on compact NQS, denoting NQS with a hidden unit density $\alpha = M/N
\leq 1$ but with system-extensive hidden-visible unit connectivity. By unifying
Jastrow and stabilizer states we introduce a new exact representation that
requires at most $M=N-1$ hidden units, illustrating how highly expressive
$\alpha \leq 1$ can be. Owing to their structural similarity to numerical NQS
solutions our result provides useful insights and could pave the way for more
families of quantum states to be represented exactly by compact NQS.
