Exact representations of many-body interactions with restricted-Boltzmann-machine neural networks

Restricted Boltzmann machines (RBMs) are simple statistical models defined on a bipartite graph which have been successfully used in studying more complicated many-body systems, both classical and quantum. In this work, we exploit the representation power of RBMs to provide an exact decomposition of many-body contact interactions into one-body operators coupled to discrete auxiliary fields. This construction generalizes the well known Hirsch's transform used for the Hubbard model to more complicated theories such as pionless effective field theory in nuclear physics, which we analyze in detail. We also discuss possible applications of our mapping for quantum annealing applications and conclude with some implications for RBM parameter optimization through machine learning.

Figure 1

Restricted Boltzmann machine; visible layer below (in blue) and hidden layer above (in red).

Figure 2

Two-body interaction from the RBM mapping.

Figure 3

Three-body interaction from the RBM mapping.

Figure 4

Two-body coupling as a function of $\mathcal{K}$. The dashed line corresponds to the lower bound in Eq. (37).

Figure 5

Three-body coupling as a function of $\mathcal{K}$. The dashed line corresponds to the lower bound in Eq. (39).

Figure 6

Cell of Chimera graph in D-Wave systems.

Figure 7

Three-body interactions implemented on the Chimera topology.

Figure 8

Training by CD of fully connected (solid lines) and sparse (dashed lines) RBM; the average energy difference in black, and relative errors in couplings ${\mathrm{\Lambda }}_{X,T}$. For the fully connected RBM we also show the average values for spurious couplings denoted by $\tilde{\mathrm{\Lambda }}$.

Figure 9

Training by minimizing the difference in energy of fully connected (solid lines) and sparse (dashed lines) RBM; the average energy difference in black, and relative errors in couplings ${\mathrm{\Lambda }}_{X,T}$. For the fully connected RBM we also show the average values for spurious couplings denoted by $\tilde{\mathrm{\Lambda }}$.

Figure 10

Training by minimizing the relative importance of fully connected (solid lines) and sparse (dashed lines) RBM; the average energy difference in black, and relative errors in couplings ${\mathrm{\Lambda }}_{X,T}$. For the fully connected RBM we also show the average values for spurious couplings denoted by $\tilde{\mathrm{\Lambda }}$.

Figure 11

Relative errors in couplings ${\mathrm{\Lambda }}_{X,T}$ for the sparse RBM for each of the 3 optimization methods discussed in this section.

Figure 12

Training by minimizing the relative importance of the sparse RBM; the average energy difference in black, and relative errors in couplings ${\mathrm{\Lambda }}_{X,T}$.