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

TL;DR

This paper introduces a new mathematical framework using anti-commutation matrices to exactly parametrize the two-body reduced density matrix of many-body quantum states, enabling precise ground state energy calculations with polynomial complexity.

Contribution

The paper presents a novel exact parametrization of the two-body reduced density matrix using anti-commutation matrices, achieving a computational complexity of 5(n^6) for ground state energy determination.

Findings

Exact parametrization of the two-body RDM with n^4 parameters.

Polynomial 5(n^6) complexity for ground state energy calculation.

Explicit separation of correlation levels via anti-commutation matrices.

Abstract

In this article, we define a new mathematical object, called a pair $D=\left(A,C\right)$ of anti-commutation matrices (ACMP) based on the anti-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 ACMP explicitly separates the different levels of correlation. The one-body correlations are defined by a ACMP $D^0=\left(A^0,C^0\right)$ and the two-body ones by a set of $n$ ACMPs $D^i=\left(A^i,C^i\right)$ where $n$ is the number of 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 or mixed $N$-body state to determine the ground state energy with a $\mathcal{O}\left(n^6\right)$ complexity.