Determining the -Representability of a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling.

The -representability problem consists in determining whether, for a given -body matrix, there exists at least one -body density matrix from which the -body matrix can be obtained by contraction, that is, if the given matrix is a -body reduced density matrix (-RDM). The knowledge of all necessary and sufficient conditions for a -body matrix to be -representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the -body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the -representability conditions grows exponentially with system size, and hence, the procedure quickly becomes intractable for practical applications. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the -representability conditions. The algorithm consists of applying to an initial -body density matrix a sequence of unitary evolution operators constructed from a stochastic process that successively approaches the reduced state of the density matrix on a -body subsystem, represented by a -RDM, to a target -body matrix, potentially a -RDM. The generators of the evolution operators follow the well-known adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented by using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide whether a given -body matrix is -representable, establishing a criterion to determine its quality and correcting it. We apply the proposed hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, from the reduced Bardeen-Cooper-Schrieffer model with constant pairing, and from the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices toward different targets.