Fig. 10: Representation of symmetry-aware backflow and quantum number projection.
Panel a) showcases how the backflow determinants are constructed. The neural network CNNμ for a given orbital μ takes as input the configuration n, and produces a backflow vector output Fμ(n). The reduced matrix \({\bar{F}}_{\mu }\) is obtained by selecting the active indices i from Fμ. These active indices are linked to the occupied sites, represented by the dark green blocks in the input n, using the canonical ordering. Panel b) demonstrates the equivariance property of the backflow function. Applying a symmetry transformation to the input as \({\hat{g}}^{-1}n\) and then extracting the active indices, we get \({\bar{F}}_{\mu }({\hat{g}}^{-1}n)\). This is equivalent to applying the symmetry transformation directly to the active indices represented by \(\hat{g}i\). This results equivalently in the reduced matrix \({\bar{F}}_{\mu \hat{g}}(n)\). Panel c) represents the quantum number projection. Following a single CNN evaluation, we obtain the reduced matrices \({\bar{F}}_{\mu g}\), \(\forall \hat{g}\in G\) by properly constructing the reduced matrix, without reevaluating the backflow CNN. Subsequently, we compute the symmetry-averaged wavefunction \(\psi ({\hat{g}}^{-1}n)\).
