Determining the N-representability of a reduced density matrix via unitary evolution and stochastic sampling

• URL: http://arxiv.org/abs/2503.17303v1
• Date: Fri, 21 Mar 2025 16:52:22 GMT
• Title: Determining the N-representability of a reduced density matrix via unitary evolution and stochastic sampling
• Authors: Gustavo. E. Massaccesi, Ofelia. B. Oña, Pablo Capuzzi, Juan I. Melo, Luis Lain, Alicia Torre, Juan E. Peralta, Diego R. Alcoba, Gustavo E. Scuseria,
• Abstract summary: This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions.<n>The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide if a given p-body matrix is N-representable.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The N-representability problem consists in determining whether, for a given p-body matrix, there exists at least one N-body density matrix from which the p-body matrix can be obtained by contraction, that is, if the given matrix is a p-body reduced density matrix (p-RDM). The knowledge of all necessary and sufficient conditions for a p-body matrix to be N-representable allows the constrained minimization of a many-body Hamiltonian expectation value with respect to the p-body density matrix and, thus, the determination of its exact ground state. However, the number of constraints that complete the N-representability conditions grows exponentially with system size, and hence the procedure quickly becomes intractable in practice. This work introduces a hybrid quantum-stochastic algorithm to effectively replace the N-representability conditions. The algorithm consists of applying to an initial N-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 p-body subsystem, represented by a p-RDM, to a target p-body matrix, potentially a p-RDM. The generators of the evolution operators follow the adaptive derivative-assembled pseudo-Trotter method (ADAPT), while the stochastic component is implemented using a simulated annealing process. The resulting algorithm is independent of any underlying Hamiltonian, and it can be used to decide if a given p-body matrix is N-representable, establishing a criterion to determine its quality and correcting it. We apply this hybrid ADAPT algorithm to alleged reduced density matrices from a quantum chemistry electronic Hamiltonian, the reduced BCS model with constant pairing, and the Heisenberg XXZ spin model. In all cases, the proposed method behaves as expected for 1-RDMs and 2-RDMs, evolving the initial matrices towards different targets.

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