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

Understand

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.