space. On the other hand there are first-order algorithms such as the boundary point method [@maz_bp; @rendl], which have a better scaling of $\mathcal{O}(M^9)$ when the three-index conditions are included, but these do not have the nice convergence properties of the interior point methods. In this study a boundary point method has been used.

\[symm\]Symmetry adaptation of the constraints
----------------------------------------------

   $U$                                                                                                                                                                                                      
  ----- --------- ------------------- ----------------------------------- ----------------- --------------------------------- --------- ------------------- ----------------- --------- ------------------- -----------------
          exact    $\mathcal{PQGT}'$   $\mathcal{PQGT}'$ [@nakata_2dhub]   $\mathcal{PQG}$   $\mathcal{PQG}$ [@nakata_2dhub]    exact    $\mathcal{PQGT}'$   $\mathcal{PQG}$    exact    $\mathcal{PQGT}'$   $\mathcal{PQG}$
    4    -0.8514        -0.8607                     -0.8607                    -0.9343                   -0.9341               -1.1246        -1.1486            -1.2664       -1.9581        -1.9595            -2.0106
    8    -0.5329        -0.5476                     -0.5476                    -0.6604                   -0.6603               -0.8478        -0.9124            -1.1162       -1.7510        -1.7597            -1.8806
   12    -0.3745        -0.3909                       \*                       -0.5054                     \*                  -0.7180        -0.8024            -1.1050       -1.6455        -1.6626            -1.8284
   16    -0.2882       -0.30101                       \*                       -0.4073                     \*                  -0.6475        -0.7409            -1.0136       -1.5843        -1.6085            -1.8014

  : \[4x4\]Ground-state energy per particle for the $4\times4$ lattice with different fillings. $\mathcal{PQG}$ and $\mathcal{PQGT}'$ results are compared with exact diagonalization result from [@fano; @wardje], and the previous v2DM reference results from [@nakata_2dhub].

   $U$                                                $N=24$         $N=18$         $N=12$              $ N=6$                                                
  ----- --------- ------------------- --------- ------------------- --------- ------------------- ------------------- ------------------- ------------------- -------------------
          DMRG     $\mathcal{PQGT}'$    DMRG     $\mathcal{PQGT}'$    DMRG     $\mathcal{PQGT}'$   $\mathcal{PQGT}'$   $\mathcal{PQGT}'$   $\mathcal{PQGT}'$   $\mathcal{PQGT}'$
    1    -1.3256        -1.3399        -1.4272        -1.4398        -1.6710        