  We consider representing two classes of 1D quantum wave functions of spin
systems, including the AKLT and CFT correlator wave functions, in terms of
multi-layer restricted Boltzmann machines. In our prescription, the AKLT wave
function can be exactly represented by a 2-layer restricted Boltzmann machine
with five hidden spins per visible spin. The construction can be generalized to
prove that any MPS wave function on $N$ unit cells with finite bond dimension
can be approximated by a 2-layer restricted Boltzmann machine with
$\mathcal{O}(N)$ hidden spins within an error which scales linearly with $N$.
The Haldane-Shastry wave function or a chiral boson CFT correlator wave
function, as any Jastrow type of wave functions, can be exactly written as a
1-layer Boltzmann machine with $\mathcal{O}(N^2)$ hidden spins and $N$ visible
spins. Applying the cumulant expansion, we further find that the chiral boson
CFT correlator wave function (with small vertex operator conformal dimension
$\alpha$, i.e., $\alpha<0.1$) can be approximated, within 99.9\% accuracy up to
22 visible spins, by a 1-layer RBM with $\mathcal{O}(N)$ hidden spins. The
cumulant expansion also leads us to a physically inspiring result in which the
hidden spins of the restricted Boltzmann machine can be interpreted as the
conformal tower of the chiral boson CFT on the cylinder.
