Multi-Layer Restricted Boltzmann Machine Representation of 1D Quantum Many-Body Wave Functions

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 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 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 O(N) hidden spins. The cumulant expansion also suggests that the hidden spins of the restricted Boltzmann machine can be interpreted as the conformal tower of the chiral boson CFT on the cylinder.