$\mathcal{PQG}$ results, however, do not follow the same trend and deviate significantly from the other results. This discrepancy becomes even worse when we move away from half filling. There the $\mathcal{PQG}$ energies are almost twice the $\mathcal{PQGT}'$ results at large values of $U$. The gap between the upper bound provided by the DMRG results and the lower bound $\mathcal{PQGT}'$ results also becomes larger when we dope the lattice. This means that the $\mathcal{PQGT}'$ conditions perform worse when the system is more delocalized. The DMRG results also become less accurate when the system is delocalized, but are closer to the exact result. In Appendix \[error\_est\] an estimate of the error on the DMRG result is made by looking at the relation between the discarded weight and the energy. In the remainder of this paper we look for ways to bridge the gap between the $\mathcal{PQG}$ and $\mathcal{PQGT}'$ results while avoiding the considerable computational burden that is associated with the three-index constraints.

\[pure\_cluster\]Cluster constraints
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As the 2D Hubbard Hamiltonian only connects nearest neighbouring sites, it is reasonable to assume that the local correlations, occuring between adjacent sites, are particularly important. Various many-body methods take advantage of this observation and treat the 