N-representability Problem in the Reduced Density Matrix

Jiang Xiao

Bastiaan J. Braams, Zhengji Zhao, Mituhiro Fukuda +2 more

Gustavo E. Massaccesi, Ofelia B. Oña, P. Capuzzi +6 more

High Resolution Image Download MS PowerPoint Slide 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 for practical applications. 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 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 p -body matrix is N -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.

M. M. Mestechkin

Darwin W. Smith

The N representability problem is the problem of how to recognize whether a given pth-order reduced density matrix is derivable from an antisymmetaic N-particle wavefunction ψ by integrating out all but coordinates from ψ*ψ. The conditions under which a given second-order density matrix is N representable by a three-particle wavefunction are discussed.

Eduardo V. Ludeña, Valentin V. Karasiev, Pedro Nieto