N-representability Problem in the Reduced Density Matrix

Xiaoqin Jiang, Jiang Mao-Sen, LI Bo-Fu +1 more

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.

Gustavo E. Massaccesi, P. Capuzzi, Ofelia B. Oña +4 more

The N-representability problem of reduced density matrices represents a fundamental challenge in electronic structure theory. In this work, we focus on the N-representability of transition reduced density matrices and propose a practical approach to embed a p-body transition reduced density matrix (RDM) of an N-particle system into a (p + 1)-body RDM of an (N + 1)-particle system. This embedding allows us to apply a unitary evolution algorithm based on a recently developed adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to determine the N-representability of reduced density matrices. The algorithm works by effectively applying a sequence of unitary transformations to a given (N + 1)-particle initial state in such a way that the distance of a projection of its corresponding (p + 1)-body RDM (embedded p-body transition RDM) to a target is minimized. Our methodology enables the purification of transition reduced density matrices and the reconstruction of approximate wave functions involved in the studied transitions, providing an effective strategy to correct and refine transition density matrices. We validate our approach with numerical simulations on N-particle systems, including a three-electron model system and the H3 molecule, demonstrating its robustness and accuracy.