Fig. 1: RNN wave functions architecture details.

a An illustration of a positive RNN wave function. Each RNN cell receives an input σ_n−1 and a hidden state h_n−1 and outputs a new hidden state h_n. This vector is taken as an input to the Softmax layer (denoted S) that computes the conditional probability P_i. b RNN autoregressive sampling scheme: after obtaining the probability vector y_i from the Softmax layer (S) in step i, we sample it to produce σ_i. The latter is taken again as an input to the RNN along with the hidden state h_i to sample the following degree of freedom σ_i+1. c Mapping of a Kagome lattice to a square lattice by embedding three atoms in a larger local Hilbert space. d A two-dimensional (2D) RNN with periodic boundary conditions for a 3 × 3 lattice for illustration purposes. A bulk RNN cell receives two hidden states h_i,j−1 and \({{{{\boldsymbol{h}}}}}_{i-{(-1)}^{j},j}\), as well as two input vectors σ_i,j−1 and \({{{{\boldsymbol{\sigma }}}}}_{i-{(-1)}\,^{j},j}\) (not shown) as illustrated by the black solid arrows. RNN cells at the boundary receive additional hidden states h_i,j+1 and \({{{{\boldsymbol{h}}}}}_{i+{(-1)}\,^{j},j}\), as well as two input vectors σ_i,j+1 and \({{{{\boldsymbol{\sigma }}}}}_{i+{(-1)}^{j},j}\) (not shown), as demonstrated by the blue curved and solid arrows. The sampling path is taken as a zigzag path, as demonstrated by the dashed red arrows. The initial memory states of the 2D RNN and the initial inputs are null vectors, as indicated by the dashed black arrows.