by the symmetry operations. For consistency the symmetry then has to be enforced in these blocks by imposing linear constraints on the 2DM during the optimization.

Reference results
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![\[N36\] Ground-state energy per particle as a function of on-site repulsion $U$ for 36 particles on a $6\times6$ lattice (cfr. Table \[6x6\]). DMRG results are compared with v2DM results using $\mathcal{PQGT}'$ and $\mathcal{PQG}$ conditions.](energy36.pdf)

![\[N34\] Ground-state energy per particle as a function of on-site repulsion $U$ for 34 particles on a $6\times6$ lattice (cfr. Table \[6x6\]). DMRG results are compared with v2DM results using $\mathcal{PQGT}'$ and $\mathcal{PQG}$ conditions.](energy34.pdf)

![\[N30\] Ground-state energy per particle as a function of on-site repulsion $U$ for 30 particles on a $6\times6$ lattice (cfr. Table \[6x6\]). DMRG results are compared with v2DM results using $\mathcal{PQGT}'$ and $\mathcal{PQG}$ conditions.](energy30.pdf)

Using the method discussed in Sections \[intro\_2DM\] and \[symm\] we have performed ground-state $\mathcal{PQGT}'$ calculations for the $6\times6$ Hubbard model, for different values of on-site repulsion $U$ and at various fillings. For reference, we compare our calculations of the energy per particle number on a $4\times4$ lattice with exact diagonalization results by Fano [*et al.*]{} [@fano] and an exact diagonalization program written by one of us [@wardje]. As can 