local degrees of freedom on a higher level than the rest of the system [@dmft; @review_cluster; @dmet]. In a related approach we propose to impose the three-index conditions on smaller local clusters of $2\times 2$ or $3\times 3$ sites, while the full system is treated on a $\mathcal{PQG}$ level. This means we impose non-negativity only on those blocks of the $\mathcal{T}_1$ and $\mathcal{T}_2'$ matrices for which all single-particle indices are on the local cluster. A problem with imposing these local constraints is that we have to Fourier transform the 2DM back from the quasi-momentum basis to the site basis, which leads to some overhead. However, the matrix computations are the bottleneck of the program, and these become cheaper, since only subcluster matrices are considered. As a result the algorithm runs much faster than the full $6\times6$ $\mathcal{PQGT}'$ calculations. Because translational invariance is still imposed, only one cluster constraint has to be taken into account. In Figures \[36c\], \[34c\] and \[30c\] it is shown what the improvement upon $\mathcal{PQG}$ is when implementing the three-index constraints on clusters of $2\times2$ and $3\times3$ sites. The $2\times2$ clusters seem to be too small, and the constraints only become active for larger values of on-site 