To make the coarse-grained ltering e cient, we rst lter out the determinants that are more than quadruply excited in the spin-up and spin-down sectors separately. Using the representation shown in Eq. (A3), this ltering does not need to run through all the internal determinants and scales as O(N ↑ det ) = O( √ N det ). It is important to notice that, at this stage, the size of J I is bounded by the number of possible quadruple excitations in both spin sectors, and does not scale any more as O(N det ). Next, we remove the determinants that are i) quadruply excited in one spin sector and excited in the other spin sector, ii) triply excited in one spin sector and more than singly excited in the other spin sector, and iii) all the determinants that are doubly excited in one spin sector and more than doubly excited in the other spin sector.