Determining the <i>N</i>‑Representability\nof a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling

The <i>N</i>-representability problem consists\nin determining\nwhether, for a given <i>p</i>-body matrix, there exists\nat least one <i>N</i>-body density matrix from which the <i>p</i>-body matrix can be obtained by contraction, that is, if\nthe given matrix is a <i>p</i>-body reduced density matrix\n(<i>p</i>-RDM). The knowledge of all necessary and sufficient\nconditions for a <i>p</i>-body matrix to be <i>N</i>-representable allows the constrained minimization of a many-body\nHamiltonian expectation value with respect to the <i>p</i>-body density matrix and, thus, the determination of its exact ground\nstate. However, the number of constraints that complete the <i>N</i>-representability conditions grows exponentially with system\nsize, and hence, the procedure quickly becomes intractable for practical\napplications. This work introduces a hybrid quantum-stochastic algorithm\nto effectively replace the <i>N</i>-representability conditions.\nThe algorithm consists of applying to an initial <i>N</i>-body density matrix a sequence of unitary evolution operators constructed\nfrom a stochastic process that successively approaches the reduced\nstate of the density matrix on a <i>p</i>-body subsystem,\nrepresented by a <i>p</i>-RDM, to a target <i>p</i>-body matrix, potentially a <i>p</i>-RDM. The generators\nof the evolution operators follow the well-known adaptive derivative-assembled\npseudo-Trotter method (ADAPT), while the stochastic component is implemented\nby using a simulated annealing process. The resulting algorithm is\nindependent of any underlying Hamiltonian, and it can be used to decide\nwhether a given <i>p</i>-body matrix is <i>N</i>-representable, establishing a criterion to determine its quality\nand correcting it. We apply the proposed hybrid ADAPT algorithm to\nalleged reduced density matrices from a quantum chemistry electronic\nHamiltonian, from the reduced Bardeen–Cooper–Schrieffer\nmodel with constant pairing, and from the Heisenberg XXZ spin model.\nIn all cases, the proposed method behaves as expected for 1-RDMs and\n2-RDMs, evolving the initial matrices toward different targets.