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In this article, we define a new mathematical object, called a pair$D=\left(A,C\right)$ of anti-commutation matrices (ACMP) based on theanti-commutation relation $a^\dag_{i}a_{j} + a_{j}a^\dag_{i} = \delta_{ij}$applied to the scalar product between the many-body wavefunctions. This ACMPexplicitly separates the different levels of correlation. The one-bodycorrelations are defined by a ACMP $D^0=\left(A^0,C^0\right)$ and the two-bodyones by a set of $n$ ACMPs $D^i=\left(A^i,C^i\right)$ where $n$ is the numberof states. We show that we can have a compact and exact parametrization with$n^4$ parameters of the two-body reduced density matrix (\TRDM) of any pure ormixed $N$-body state to determine the ground state energy with a$\mathcal{O}\left(n^6\right)$ complexity.

Exact solution of the many-body problem with a $\mathcal{O}\left(n^6\right)$ complexity