# Determining the Ensemble N‑Representability of Reduced Density Matrices

## TL;DR

This paper introduces a new method to determine if a reduced density matrix can represent an ensemble of quantum states, using purification and variational algorithms.

## Contribution

A practical framework for ensemble N-representability using purification and variational quantum algorithms.

## Key findings

- The purification strategy successfully embeds ensemble states into pure states for N-representability analysis.
- The algorithm effectively minimizes the Hilbert-Schmidt distance to determine N-representability.
- Numerical simulations on multi-electron systems confirm the method's robustness and applicability.

## 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 that embeds
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.