# Is the Matrix Completion of Reduced Density Matrices Unique?

## TL;DR

This paper shows that under certain conditions, the matrix completion of reduced density matrices in quantum systems can be uniquely determined and reconstructed.

## Contribution

The paper proves the uniqueness of matrix completion for 2-RDMs and introduces a hybrid quantum–stochastic algorithm for exact reconstruction.

## Key findings

- Matrix completion of 2-RDMs is unique under specific conditions identified in the study.
- A hybrid quantum–stochastic algorithm successfully achieves exact matrix completion.
- The method is demonstrated on the Fermi–Hubbard model with promising results.

## Abstract

Reduced density matrices are central to describing observables
in many-body quantum systems. In electronic structure theory, the
two-particle reduced density matrix (2-RDM) suffices to determine
the energy and other key properties. Recent work has used matrix completion,
leveraging the low-rank structure of RDMs and approximate theoretical
models, to reconstruct the 2-RDM from partial data and thus reduce
the computational cost. However, matrix completion is, in general,
an under-determined problem. Revisiting Rosina’s theorem (

RosinaM.,

Queen’s Papers on Pure and Applied Mathematics, 1968, No. 11, 369
), we here show that the matrix completion is unique under certain
conditions, identifying the subset of 2-RDM elements that enables
its exact reconstruction from incomplete information. Building on
this, we introduce a hybrid quantum–stochastic algorithm that
achieves exact matrix completion, demonstrated through applications
to the Fermi–Hubbard model.