Determining the Ensemble N -Representability of Reduced Density Matrices

Abstract

The N-representability problem for reduced densitymatrices remains a fundamental challenge in electronic structuretheory. Following our previous work that employs a unitaryevolution algorithm based on an adaptive derivative-assembledpseudo-Trotter variational quantum algorithm to probe pure-stateN-representability of reduced density matrices [J. Chem. TheoryComput. 2024, 20, 9968], in this work we propose a practicalframework for determining the ensemble N-representability of a pbody matrix. This is accomplished using a purification strategy thatembeds an ensemble state into a pure state defined on an extendedHilbert space, such that the reduced density matrices of the purifiedstate reproduce those of the original ensemble. By iterativelyapplying variational unitaries to an initial purified state, theproposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-bodymatrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction ofdefective ensemble reduced density matrices and quantum-state reconstruction on a quantum computer, offering a route for densitymatrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both simplemodels as well as molecular systems at finite temperature, demonstrating its robustness.