Determining the ensemble N-representability of Reduced Density Matrices

TL;DR

This work tackles the challenging ensemble $N$-representability problem for $p$-body RDMs by embedding an ensemble state into a pure state on an extended Hilbert space (purification) and applying an ensemble extension of the ADAPT-VQA variational protocol. The method evolves a purified state using a fermionic antihermitian excitation pool to minimize the Hilbert-Schmidt distance $\mathcal{D}$ between the evolved $^p\rho_k$ and a target $^p\rho_{\text{target}}$, enabling both classification of pure versus ensemble representability and correction of non-$N$-representable targets, with potential quantum-state reconstruction. Validation on model (4e) systems and H$_2$/H$_3$ at finite temperature shows consistency with Klyachko inequalities for pure cases and demonstrates robustness to representability defects through $\mathcal{D}_{\min}$ as a defect metric. The framework provides a practical density-matrix refinement route and supports quantum-device implementations for thermal ensembles and open-system simulations in quantum chemistry.

Abstract

The N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy consisting of embedding an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices, and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both, simple models as well as molecular systems at finite temperature, demonstrating its robustness.

Figures (1)

• Figure : Distance ($\mathcal{D}$) minimization algorithm